\\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\Wave Motion\\Eng\\Exercise.p65 JEE-Physics 8 . Statement–1 : The superposition of the waves y1 = A sin (kx –t) and y2 = 3A sin(kx + t) is a pure standing wave plus a travelling wave moving in the negative direction along X–axis. and S t a t e m e n t – 2 : The resultant of y1 & y2 is y=y1+y2 = 2A sin kx cos t+2A sin (kx+t) (A) Statement–1 is true, Statement–2 is true ; Statement–2 is correct explanation for Statement–1. (B) Statement–1 is true, Statement–2 is true ; Statement–2 is NOT a correct explanation for Statement–1 (C) Statement–1 is true, Statement–2 is false (D) Statement–1 is false, Statement–2 is true 9 . Statem ent–1 : Mechanical transverse waves cannot be generated in gaseous medium. and Statement–2 : Mechanical transverse waves can be produced only in such medium which have shearing property. (A) Statement–1 is true, Statement–2 is true ; Statement–2 is correct explanation for Statement–1. (B) Statement–1 is true, Statement–2 is true ; Statement–2 is NOT a correct explanation for Statement–1 (C) Statement–1 is true, Statement–2 is false (D) Statement–1 is false, Statement–2 is true 1 0 . Statem ent–1 : Description of sound as pressure wave is preferred over displacement wave. and Statem ent–2 : Sound sensors (ear or mike) detect pressure changes. (A) Statement–1 is true, Statement–2 is true ; Statement–2 is correct explanation for Statement–1. (B) Statement–1 is true, Statement–2 is true ; Statement–2 is NOT a correct explanation for Statement–1 (C) Statement–1 is true, Statement–2 is false (D) Statement–1 is false, Statement–2 is true 1 1 . Statem ent–1 : During thunderstorm, light is seen much earlier than the sound is heard and Statement–2 : Light travels faster than sound. (A) Statement–1 is true, Statement–2 is true ; Statement–2 is correct explanation for Statement–1. (B) Statement–1 is true, Statement–2 is true ; Statement–2 is NOT a correct explanation for Statement–1 (C) Statement–1 is true, Statement–2 is false (D) Statement–1 is false, Statement–2 is true 1 2 . Statem ent–1 : The flash of lightening is seen before the sound of thunder is heard. and Statem ent–2 : The sound of thunder is produced after the flash of lightening. (A) Statement–1 is true, Statement–2 is true ; Statement–2 is correct explanation for Statement–1. (B) Statement–1 is true, Statement–2 is true ; Statement–2 is NOT a correct explanation for Statement–1 (C) Statement–1 is true, Statement–2 is false (D) Statement–1 is false, Statement–2 is true 1 3 . Statem ent–1 : Earthquakes cause vast devastation. Sometimes short and tall structures remain unaffected while the medium height structures fall. and S t a t e m e n t – 2 : The natural frequency of the medium structures coincides with the frequency of the seismic wave. (A) Statement–1 is true, Statement–2 is true ; Statement–2 is correct explanation for Statement–1. (B) Statement–1 is true, Statement–2 is true ; Statement–2 is NOT a correct explanation for Statement–1 (C) Statement–1 is true, Statement–2 is false (D) Statement–1 is false, Statement–2 is true E 51
JEE-Physics \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\Wave Motion\\Eng\\Exercise.p65 1 4 . Statem ent–1 : Shock waves produced by supersonic aircraft may be visible. and Statem ent–2 : The sudden decrease in air pressure in the shock waves caused water molecules in the air to condense, forming a fog. (A) Statement–1 is true, Statement–2 is true ; Statement–2 is correct explanation for Statement–1. (B) Statement–1 is true, Statement–2 is true ; Statement–2 is NOT a correct explanation for Statement–1 (C) Statement–1 is true, Statement–2 is false (D) Statement–1 is false, Statement–2 is true 1 5 . S t atem ent –1 : In the case of a stationary wave, a person hear a loud sound at the nodes as compared to the antinodes. and Statem ent–2 : In a stationary wave all the particles of the medium vibrate in phase. (A) Statement–1 is true, Statement–2 is true ; Statement–2 is correct explanation for Statement–1. (B) Statement–1 is true, Statement–2 is true ; Statement–2 is NOT a correct explanation for Statement–1 (C) Statement–1 is true, Statement–2 is false (D) Statement–1 is false, Statement–2 is true 1 6 . Statem ent–1 : Waves generated in a metal piece can be transverse or longitudinal. and Statement–2 : Waves generated depend upon the method of creating waves in the metal. (A) Statement–1 is true, Statement–2 is true ; Statement–2 is correct explanation for Statement–1. (B) Statement–1 is true, Statement–2 is true ; Statement–2 is NOT a correct explanation for Statement–1 (C) Statement–1 is true, Statement–2 is false (D) Statement–1 is false, Statement–2 is true 1 7 . Statem ent–1 : A balloon filled with CO gas acts as a converging lens for a sound wave. 2 and St atem ent –2 : Sound waves travel faster in air than in CO2. (A) Statement–1 is true, Statement–2 is true ; Statement–2 is correct explanation for Statement–1. (B) Statement–1 is true, Statement–2 is true ; Statement–2 is NOT a correct explanation for Statement–1 (C) Statement–1 is true, Statement–2 is false (D) Statement–1 is false, Statement–2 is true 1 8 . Statem ent–1 : Node of pressure wave is formed at the open end of an organ pipe. and S t a t e m e n t – 2 : Due to huge volume of the atmosphere outside the tube, deformation in its volume is negligible. (A) Statement–1 is true, Statement–2 is true ; Statement–2 is correct explanation for Statement–1. (B) Statement–1 is true, Statement–2 is true ; Statement–2 is NOT a correct explanation for Statement–1 (C) Statement–1 is true, Statement–2 is false (D) Statement–1 is false, Statement–2 is true 1 9 . S t atem ent –1 : If transverse waves are produced in a very long string fixed at one end. Near the free end only progressive wave is observed, in practice. and Statement–2 : Energy of reflected wave does not reach the free end. (A) Statement–1 is true, Statement–2 is true ; Statement–2 is correct explanation for Statement–1. (B) Statement–1 is true, Statement–2 is true ; Statement–2 is NOT a correct explanation for Statement–1 (C) Statement–1 is true, Statement–2 is false (D) Statement–1 is false, Statement–2 is true 52 E
JEE-Physics 2 0 . S t a t e m e n t – 1 : When two vibrating tuning forks have f = 300 Hz and f = 350 Hz and held close to each other; 12 beats cannot be heard. and Statem ent–2 : The principle of superposition is valid only when f – f < 10 Hz 12 (A) Statement–1 is true, Statement–2 is true ; Statement–2 is correct explanation for Statement–1. (B) Statement–1 is true, Statement–2 is true ; Statement–2 is NOT a correct explanation for Statement–1 (C) Statement–1 is true, Statement–2 is false (D) Statement–1 is false, Statement–2 is true 2 1 . Statement–1 : The fundamental frequency of an organ pipe increases as the temperature increases. and S t atem ent –2 : As the temperature increases, the velocity of sound increases more rapidly than length of the pipe. (A) Statement–1 is true, Statement–2 is true ; Statement–2 is correct explanation for Statement–1. (B) Statement–1 is true, Statement–2 is true ; Statement–2 is NOT a correct explanation for Statement–1 (C) Statement–1 is true, Statement–2 is false (D) Statement–1 is false, Statement–2 is true COMPREHENSION TYPE QUESTIONS Comprehension#1 The figure represents the instantaneous picture of a transverse harmonic wave travelling along the negative x-axis. Choose the correct alternative(s) related to the movement of the 9 points shown in the figure. (Instanteous velocity) y b a c hx o g d e f 1 . The points moving upward is/are :– (A) a (B) c (C) f (D) g 2 . The points moving downwards is/are :– (A) o (B) b (C) d (D) h \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\Wave Motion\\Eng\\Exercise.p65 3 . The stationary points is/are:– (A) o (B) b (C) f (D) h 4 . The points moving with maximum velocity is/are:– (A) b (B) c (C) d (D) h 5 . A perfectly elastic uniform string is suspended vertically with its upper end fixed to the ceiling and the lower end loaded with the weight. If a transverse wave is imparted to the lower end of the string, the pulse will (A) not travel along the length of the string (B) travel upwards with increasing speed (C) travel upwards with decreasing speed (D) travelled upwards with constant acceleration E 53
JEE-Physics Comprehension#2 A narrow tube is bent in the form of a circle of radius R, as shown in the figure. Two small holes S and D are made in the tube at the positions right angle to each other. A source placed at S generated a wave of intensity I which is equally 0 divided into two parts : One part travels along the longer path, while the other travels along the shorter path. Both the parts waves meet at the point D where a detector is placed. R S D 1 . If a maxima is formed at the detector then, the magnitude of wavelength of the wave produced is given by :– (A) R R R R (B) 2 (C) 4 (D) 3 2 . If the minima is formed at the detector then, the magnitude of wavelength of the wave produced is given by:– (A) R 3 R 2 R 2 R (B) (C) (D) 2 3 5 (D) 3I 3 . The maximum intensity produced at D is given by :– 0 (A) 4I (B) 2 I (C) I 0 0 0 4 . The maximum value of to produce a maxima at D is given by :– (A) R (B) 2R R 3 R (C) 2 (D) 2 5 . The maximum value of to produce a minima at D is given by :– 3 R (D) (A) R (B) 2R R (C) 2 2 MISCELLANEOUS TYPE QUESTION ANSWER KEY EXERCISE –3 \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\Wave Motion\\Eng\\Exercise.p65 Match the Column 1. (A) q, (B) p, (C) p 2. (A) p, q (B) s (C) p,r (D) s 3. (A) q, (B) p (C) r 4. (A) r, (B) p (C) q (D) t 5. (A) q, (B) r (C) s (D) p Assertion – Reason 1 2 3 4 5 6 7 A AABAAA 14 A 8 9 10 11 12 13 21 A AAAACA 15 16 17 18 19 20 CAAAAC Comprehension Based Comprehension #1: 1.A,D 2. C 3. B,C 4. C,D 5. B,D Comprehension #2: 1. A,B,C,D 2. A,C,D 3. B 4. A 5. B 54 E
EXERCISE–04 [A] JEE-Physics CONCEPTUAL SUBJECTIVE EXERCISE 1 . A stone dropped from the top of a tower of height 300 m high splashes into the water of a pond near the base of the tower. When is the splash heard at the top, given that the speed of sound in air is 340 m s–1 ? (g=9.8 ms–2) 2 . Determine resultant amplitude after super position of given four waves with help of phasor diagram. y1 = 15 sin t mm, y2 = 9 sin (t–/2) mm, y3 =7 sin (t+/2) mm & y4 = –13 sin t mm 3 . Calculate the ratio of intensity of wave train A to wave train B. Wavetrain A 2 Wavetrain B 1 1 01 2 3 t t 0 1 1 3 25 3 7 2222 4 . Two audio speakers are kept some distance apart and are driven by the same amplifier system. A person is sitting at a place 6.0 m from one of the speakers and 6.4 m from the other. If the sound signal is continuously varied from 500 Hz to 5000 Hz, what are the frequencies for which there is a destructive interference at the place of the listener? Speed of sound in air = 320 m/s. 5 . One end of a long string of linear mass density 8.0 × 10–3 kg m–1 is connected to an electrically driven tuning fork of frequency 256 Hz. The other end passes over a pulley and is tied to a pan containing a mass of 90 kg. The pulley end absorbs all the incoming energy so that reflected waves at this end have negligible amplitude. At t = 0, the left end (fork end) of the string x = 0 has zero transverse displacement (y = 0) and is moving along positive y–direction. The amplitude of the wave is 5.0 cm. Write down the transverse displacement y as function of x and t that describes the wave on the string. 6 . A uniform rope of length 12 m and mass 6 kg hangs vertically from a rigid support. A block of mass 2 kg is attached to the free end of the rope. A transverse pulse of wavelength 0.06 m is produced at the lower end of the rope. What is the wavelength of the pulse when it reaches the top of the rope ? 7 . A steel wire of length 1m, mass 0.1 kg and uniform cross–sectional area 10–6 m2 is rigidly fixed at both ends. The temperature of the wire is lowered by 20°C. If transverse waves are set up by plucking the string in the middle calculate the frequency of the fundamental mode of vibration. [Given: Y = 2 × 1011 N/m2, stell = 1.21 × 10–5/°C] stell 8 . In the given figure the string has mass 4.5 g. Find the time taken by a 25cm \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\Wave Motion\\Eng\\Exercise.p65transverse pulse produced at the floor to reach the pulley. (g = 10 ms–2). 2m 2kg \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 9. Figure shows a string of linear mass density 1.0 g/cm on which floor \\\\\\\\ a wave pulse is travelling. Find the time taken by the pulse in \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ travelling through a distance of 50 cm on the string. (g=10 m/s2). 1kg x 1 0 . A string vibrate according to the equation y = 5 sin 3 cos (40t) where x and y are in cm's and t is in second. (i) What is the equation of incident and reflected wave ? (ii) What is the distance between the adjacent nodes? 9 (iii) What is the velocity of the particle of the string at the position x = 1.5 cm when t = 8 sec ? E 55
JEE-Physics 1 1 . A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a water surface, what is the wavelength of (i) the reflected sound, (ii) the transmitted sound ? Speed of sound in air is 340 ms–1 and in water 1486 ms–1. 1 2 . The vibrations of a string of length 60 cm fixed at both ends are represented by the equation : x y = 4 sin 15 cos(96t) where x and y are in cm and t in seconds. (i) What is the maximum displacement of a point at x = 5 cm ? (ii) Where are the nodes located along the string ? (iii) What is the velocity of the particle at x = 7.5 cm at t = 0.25 s ? (iv) Write down the equations of the component waves whose superposition gives the above wave 1 3 . Given below are some functions of x and t to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (a) a travelling wave, (b) a stationary wave or (c) none at all : (i) y = cos (3x) sin (10t) (ii) y = 2 x vt (iii) y = 3 sin(5x – 0.5t) + 4 cos (5x – 0.5t) (iv) y = cosx sint + cos2xsin2t 1 4 . If the bulk modulus of water is 4000 MPa, what is the speed of sound in water? 1 5 . A steel rod 100 cm long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod are given to be 2.53 kHz. What is the speed of sound in steel ? 1 6 . Two successive resonant frequencies in an open organ pipe are 1944 and 2592 Hz. If the speed of sound in air 324 ms–1, then find the length of tube. 1 7 . A flute which we treat as a pipe open at both ends is 60 cm long. How far from the mouth piece should a hole be uncovered for the fundamental frequency to be 330 Hz ? Take the speed of sound in air as 340 m/sec. And also calculate fundamental frequency when all the holes are covered ? 1 8 . A string 25 cm long and having a mass of 2.5 g is under tension. A pipe closed at one end is 40 cm long. When the string is set vibrating in its first overtone and the air in the pipe in its fundamental frequency. 8 beats/s are heard. It is observed that decreasing the tension in the string decreases the beat frequency. If the speed of sound in air is 320 m/s find the tension in the string. 1 9 . Two tunning fork having frequency. 300 Hz & 305 Hz produce beat phenomena. Then (i) How many beats produce in 5 sec. (ii) Determine minimum time interval in which maximum intensity become min. 2 0 . Two sitar strings A and B playing the note 'Dha' are slightly out of tune and produce beats of frequency 5 Hz. The tension of the string B is slightly increased and the beat frequency is found to decrease to 3 Hz. What is the original frequency of B if the frequency of A is 427 Hz ? 2 1 . A source of sound of frequency 256 Hz is moving rapidly towards a wall with a velocity of 5 m/s. How many \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\Wave Motion\\Eng\\Exercise.p65 beats per second will be heard by the observer on source it self if sound travels at a speed of 330 m/s ? 2 2 . A person going away from a factory on his scooter at a speed of 36 km/hr listens to the siren of the factory. If the actual frequency of the siren is 700Hz and a wind is blowing along the direction of the scooter at 36 km/hr, find the observed frequency heard by the person. (Given speed of sound = 340 m/s) 2 3 . A car has two horns having a difference in frequency of 180 Hz. The car is approaching a stationary observer with a speed of 60 ms–1. Calculate the difference in frequencies of the notes as heard by the observer, if velocity of sound in air is 330 ms–1. 2 4 . A whistle emitting a sound of frequency 440 Hz is tied to a string of 1.5 m length and rotated with an angular velocity of 20 rad/s in the horizontal plane. Calculate the range of frequencies heard by an observer stationed at a large distance from the whistle. [Speed of sound = 330m/s] 2 5 . Two tuning forks with natural frequencies of 340 Hz each move relative to a stationary observer. One fork moves away from the observer, While the other moves towards him at the same speed. The observer hears beats of frequency 3 Hz. Find the speed of the tuning fork. [Speed of sound = 330m/s] E 56
JEE-Physics 2 6 . A SONAR system fixed in a submarine operates at a frequency 40.0 kHz. An enemy submarine moves towards the SONAR with a speed of 360 km h–1. What is the frequency of sound reflected by the submarine ? Take the speed of sound in water to be 1450 ms–1. 2 7 . A sonometer wire under tension of 64 N vibrating in its fundamental mode is in resonance with a vibrating tuning fork. The vibrating portion of the sonometer wire has a length of 10 cm and mass of 1g. The vibrating tuning fork is now moved away from the vibrating wire with a constant speed and an observer standing near the sonometer hears one beat per second. Calculate the speed with which the tuning fork is moved, if the speed of sound in air is 300 m/s. 2 8 . A train approaching a hill at a speed of 40 km/h sounds a whistle of frequency 580 Hz when it is at a distance of 1 km from a hill. A wind with a speed of 40 km/h is blowing in the direction of motion of the train. Find (i) The frequency of the whistle as heard by an observer on the hill. (ii) The distance from the hill at which the echo from the hill is heard by the driver and its frequency. (Velocity of sound in air = 1200 km/h) 2 9 . A train of length is moving with a constant sped v along a circular track of radius R, the engine of the train emits a whistle of frequency f. Find the frequency heard by a guard at the rear end of the train. CONCEPTUAL SUBJECTIVE EXERCISE ANSWER KEY EXERCISE–4(A) 1 . 8.707 s 13. (i) Stationary wave 2 . 2.83 mm (ii) Unacceptable function for any travelling wave 3. 1 (iii) Travelling harmonic wave 4. 1200 Hz, 2000 Hz, 2800 Hz, 3600Hz, 4400 Hz (iv) Superposition of two stationary waves. 5 . y = 0.05 sin (1609 t – 4.84 x) 1 4 . 2000 ms–1 (where x and y in m) 1 5 . 5.06 × 103 ms–1 1 6 . 0.25 m 6 . 0.12 m 1 7 . 283.33 Hz 7 . 11 Hz 1 8 . 27.04 N 8 . 0.02 s 9 . 0.05 s 19. (i) 25 beats(ii) 1 s \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\Wave Motion\\Eng\\Exercise.p65 10 1 0 . (i) y = 2.5 sin 40 t x and 2 0 . 422 Hz incident 3 2 1 . 7.87 Hz y= – 2.5 sin 40t x (ii) 3 cm (iii) 0 2 2 . 680 Hz reflected 3 2 3 . 220 Hz 1 1 . (i) 3.4 × 10–4 m (ii) 1.49 × 10–3 m 2 4 . 403.3 Hz to 484 Hz 1 2 . (i) 2 3 cm (ii) x = 0, 15 cm, 30 cm .. etc. 2 5 . 1.5 m/s x 9 6 t 2 6 . 45.93 kHz 15 (iii) 0 (iv)y = 2 sin and 2 7 . 0.073 m/s 1 x t 2 8 . (i) 599.33 Hz (ii) 0.935 km, 621.43 Hz 15 y = 2 sin 96 29. f 2 E 57
JEE-Physics EXERCISE–04 [B] BRAIN STORMING SUBJECTIVE EXERCISE 1 . A source of sound is moving along a circular orbit of radius 3m 3m 6m 6m with an angular velocity of 10 rad/s. A sound detector located A B CD far away from the source is executing linear simple harmonic motion along the line BD (see figure) with an amplitude BC = CD 5 = 6 m. The frequency of oscillation of the detector is per second. The source is at the point A when the detector is at the point B. If the source emits a continuous sound wave of frequency 340 Hz, find the maximum and the minimum frequencies recorded by the detector. (Speed of sound = 340 m/s) 2 . Two radio stations broadcast their programmes at the same amplitude A and at slightly different frequencies 1 and 2 respectively, where 1–2 = 103 Hz. A detector receives the signals from the two stations simultaneously. It can only detect signals of intensity 2A2. (i) Find the time interval between successive maximum of the intensity of the signal received by the detector. (ii) Find the time for which the detector remains idle in each cycle of the intensity of the signal 3 . A band playing music at a frequency ƒ is moving towards a wall at a speed v . A motorist is following the band b with a speed v . If v is the speed of sound. Obtain an expression for the beat frequency heard by the motorist. m 4 . The figure shows a snap photograph of a vibrating string at t=0. y The particle P is observed moving up with velocity 20 3 cm/s. The 4 P 60° x tangent at P makes an angle 600 with x–axis (i) Find the direction in which the wave is moving (ii) the equation of the wave (iii) the total 22 3.5 5.5 7.5 energy carried by the wave per cycle of the string. [Assuming that (in 102m) , the mass per unit length of the string = 50 gm/m] 0 1.5 5. The harmonic wave y = (2.0 × 10–3) cos (2.0 x –50t) travels along a string toward a boundary at x=0 with i a second string. The wave speed on the second string is 50 m/s. Write expressions for reflected and transmitted waves. Assume SI units. 6 . An open organ pipe filled with air has a fundamental frequency 500 Hz. The first harmonic of another organ pipe closed at one end and filled with carbon dioxide has the same frequency as that of the first harmonic of the open organ pipe. Calculate the length of each pipe. Assume that the velocity of sound in air and in carbondioxide to be 330 and 264 m/s respectively. 7 . Two speakers are driven by the same oscillator with frequency of 200 Hz. \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\Wave Motion\\Eng\\Exercise.p65 They are located 4 m apart on a vertical pole. A man walks straight towards L d the lower speaker in a direction perpendicular to the pole, as shown in figure. (i) How many times will he hear a minimum in sound intensity, and (ii) how far is he from the pole at these moments? Take the speed of sound to be 330 m/s, and ignore any sound reflections coming off the ground. 8 . A cylinder ABC consists of two chambers 1 and 2 which contains two different A B gases. The wall C is rigid but the walls A and B are thin diaphragms. A 1 2 vibrating tuning fork approaches the wall A with velocity u= 30 m/s and air columns in chamber 1 and 2 vibrates with minimum frequency such that there is node (displacement) at B and antinode (displacement) at A. Find : v1= 100 m/s v2= 300 m/s (i) the fundamental frequency of air column 0.5m 1.0m (ii) find the frequency of tuning fork. Assume velocity of sound in the first and second chamber be 1100 m/s and 300 m/s respectively. Velocity of sound in air 330 m/s. 58 E
JEE-Physics 9 . A string of length 1 m fixed at one end and on the other end a block of mass M = 4 kg is suspended. The string is set into vibration and represented by equation y 6 sin x cos(100t) where x and y are in cm and t is in seconds. 10 ///////////// (i) Find the number of loops formed in the string . ///////// / ////////////////////////////////////////////// (ii) Find the maximum displacement of a point at x=5/3 cm (iii) Calculate the maximum kinetic energy of the string M (iv) Write down the equations of the component waves whose superposition gives the wave. 1 0 . The following equation represent transverse wave; z = Acos(kx – t), z = Acos(kx + t), z = Acos(ky–t) 1 2 3 Identify the combination (s) of the waves which will produce. (i) standing wave (s) (ii) a wave travelling in the direction making an angle of 45° with the positive x and positive y–axis. In each case, find the position at which the resultant intensity is always zero. 1 1 . The displacement of the medium in a sound wave is given by the equation y = Acos(ax + bt) where A, a and 1 b are positive constants. The wave is reflected by an obstacle situated a x = 0. The intensity of the reflected wave is 0.64 times that of the incident wave. (i) What are the wavelength and frequency of incident wave ? (ii) Write the equation for the reflected wave. (iii) In the resultant wave formed after reflection, find the maximum and minimum values of the particle speeds in the medium. (iv) Express the resultant wave as a superposition of a standing wave and a travelling wave. What are the positions of the antinodes of the standing wave ? What is the direction of propagation of travelling wave? 1 2 . A metallic rod of length 1m is rigidly clamped at its mid point. Longitudinal stationary waves are setup in the rod in such a way that there are two nodes on either side of the midpoint. The amplitude of an antinode is 2 × 10–6m. Write the equation of motion at a point 2 cm from the midpoint and those of the constituent waves in the rod. (Young's modulus of the material of the rod = 2 × 1011 Nm–2; density = 8000 kg–m–3). Both ends are free. 1 3 . A parabolic pulse given by equation y (in cm) = 0.3 – 0.1 (x–5t)2 (y 0) travelling in a uniform string. The pulse passes through a boundary beyond which its velocity becomes 2.5 m/s. What will be the amplitude of pulse in this medium after transmission? BRAIN STORMING SUBJECTIVE EXERCISE ANSWER KEY EXERCISE–4(B) 1 . 438.7Hz, 257.3Hz 10. (i) z and z : x = (2n + 1) (2n+1) /4 12 2 . (i) 6.28 × 10–3 S, (ii)1.57 × 10–3 S 2k 2f vb v vm where n = 0, ± 1, ± 2 ..... etc. 3 . v2 v 2 (ii) z and z : x – y = (2n + 1) b \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\Wave Motion\\Eng\\Exercise.p65 13 k 4. (i) Negative x, (ii) y = 0.4 sin 1 0 t 2 x 4 where n = 0, ± 1, ± 2 ......etc. (x, y are in cm) (iii) 1.6 × 10–5 J 2, b a 2 5 . (i) 6.67 × 10–4 2.0x 50t 11. (i) (ii) 2.67 × 10–3 cos 1.0x 50t (ii) y = – 0.8 A cos (ax – bt) r 6 . 33 cm and 13.2 cm (iii) 1.8 Ab, 0 (iv) y = – 1.6 A sin ax sinbt + 0.2 A cos (ax + bt) 7 . (i) 2 (ii) 9.28 m and 1.99 m 1n 8 . (i) 1650 Hz (ii) 1500 Hz Antinodes are at x n . Travelling 2 a 9 . (i) 10 (ii) 3cm (iii) 36 J wave is propagating in negative x–direction x 100t x + 100t 12. y = 10–6 sin (0.1) sin (25000 t), y =10–6 sin 10 10 1 (iv) y1= 3sin , y2 = 3sin (25000 t–5x), y = 10–6 sin (25000 t + 5x) 2 13. 0.2 cm E 59
JEE-Physics PREVIOUS YEAR QUESTIONS EXERCISE–05(A) 1 . Tube A has both ends open while tube B has one end closed, otherwise they are identical. The ratio of fundamental frequency of tubes A and B is- [AIEEE - 2002] (1) 1 : 2 (2) 1 : 4 (3) 2 : 1 (4) 4 : 1 2 . A tuning fork arrangement (pair) produces 4beats/s with one fork of frequency 288 cps. A little wax is placed on the unknown fork and it then produces 2 beats/s. The frequency of the unknown fork is-[AIEEE - 2002] (1) 286 cps (2) 292 cps (3) 294 cps (4) 288 cps 3 . A wave y = asin(t – kx) on a string meets with another wave producing a node at x = 0. Then the equation of the unknown wave is- [AIEEE-2002] (1) y = asin(t + kx) (2) y = – asin(t + kx) (3) y = asin(t – kx) (4) y = – asin(t – kx) 4 . Length of a string tied to two rigid supports is 40 cm. Maximum length (wavelength in cm) of a stationary wave produced on it, is- [AIEEE-2002] (1) 20 (2) 80 (3) 40 (4) 120 5. The displacement y of a wave travelling in the x-direction is given by y = 10 –4 si n 6 00 t 2x metre, 3 where, x is expressed in metres and t in seconds. The speed of the wave-motion, in ms–1 is- [AIEEE-2003] (1) 300 (2) 600 (3) 1200 (4) 200 6 . A tuning fork of known frequency 256 Hz makes 5 beats per second with the vibrating string of a piano. The beat frequency decreases to 2 beats per second when the tension in the piano string is slightly increased. The frequency of the piano string before increasing the tension was- [AIEEE-2003] (1) (256 + 2) Hz (2) (256 – 2) Hz (3) (256 – 5) Hz (4) (256 + 5) Hz 7 . When two tuning forks (fork 1 and fork 2) are sounded simultaneously, 4 beats per second are heard. Now, some tape is attached on the prong of the fork 2. When the tuning forks are sounded again, 6 beats per second are heard. If the frequency of fork 1 is 200 Hz, then what was the original frequency of fork 2? (1) 200 Hz (2) 202 Hz (3) 196 Hz [AIEEE - 2005] (4) 204 Hz 8 . An observer moves towards a stationary source of sound, with a velocity one-fifth of the velocity of sound. what is the percentage increase in the apparent frequency ? [AIEEE - 2005] (1) zero (2) 0.5% (3) 5% (4) 20% 9 . A whistle producing sound waves of frequencies 9500 Hz and above is approaching a stationary person with speed v ms–1. The velocity of sound in air is 300 ms–1. If the person can hear frequencies upto a maximum of 10,000 Hz, the maximum value of v upto which he can hear the whistle is- [AIEEE - 2006] (1) 15 2 ms–1 (2) 15/ 2 ms–1 (3) 15 ms–1 (4) 30 ms–1 1 0 . A sound absorber attenuates the sound level by 20 dB. The intensity decreases by a factor of-[AIEEE - 2007] \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\Wave Motion\\Eng\\Exercise.p65 (1) 1000 (2) 10000 (3) 10 (4) 100 1 1 . While measuring the speed of sound by performing a resonance column experiment, a student gets the first resonance condition at column length of 18 cm during winter. Repeation the same experiment during sumer, student measures the column length to be x cm for the second resonance. Then [AIEEE - 2008] (1) 18 > x (2) x > 54 (3) 54 > x > 36 (4) 36 > x > 18 1 2 . A wave travelling along the x-axis is described by the equation y (x, t ) = 0.005 cos ( x t ). If the wavelength and the time period of the wave in 0.08m and 2.0 s respectively then and in appropriate units are [AIEEE - 2008] (1) 25.00, (2) 0.08 , 2.0 (3) 0.04 , 1.0 (4) 12.50 , 2.0 60 E
JEE-Physics 1 3 . Three sound waves of equal amplitudes have frequencies (–1), , (+1). They superpose to give beats. The number of beats produced per second will be :- [AIEEE - 2009] (1) 2 (2) 1 (3) 4 (4) 3 1 4 . A motor cycle starts from rest and accelerates along a straight path at 2 m/s2. At the starting point of the motor cycle there is a stationary electric siren. How far has the motor cycle gone when the driver hears the frequency of the siren at 94% of its value when the motor cycle was at rest ? (Speed of sound = 330 ms–1) :- [AIEEE - 2009] (1) 147 m (2) 196 m (3) 49 m (4) 98 m 1 5 . The equation of a wave on a string of linear mass density 0.04 kg m–1 is given by y = 0.02(m) sin t – x . The tension in the string is : 2 0.04 (s ) 0.50(m ) [AIEEE - 2010] (1) 6.25 N (2) 4.0 N (3) 12.5 N (4) 0.5 N 16. The transverse displacement y(x, t) of a wave on a string is given by ax2 bt2 2 abxt . This represents y x, t e a :- [AIEEE - 2011] (1) standing wave of frequency b 1 (2) standing wave of frequency b a b (3) wave moving in +x directionwith speed (4) wave moving in –x direction with speed b a 1 7 . Statement-1: Two longitudinal waves given by equations : y1(x, t) = 2a sin (t – kx) and y2(x, t) = a sin (2t – 2kx) will have equal intensity. Statement-1: Intensity of waves of given frequency in same medium is proportional to square of amplitude only. [AIEEE - 2011] (1) Statement-1 is false, statement-2 is true (2) Statement-1 is ture, statement-2 is false (3) Statement-1 is ture, statement-2 true; statement-2 is the correct explanation of statement-1 (4) Statement-1 is true, statement-2 is true; statement -2 is not correct explanation of statement-1. 1 8 . A travelling wave represented by y = A sin(t–kx) is superimposed on another wave represented by y = A sin (t + kx). The resultant is :- [AIEEE-2011] (1) A standing wave having nodes at x n 1 ,n 0,1, 2 2 2 (2) A wave travelling along + x direction (3) A wave travelling along –x direction \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\Wave Motion\\Eng\\Exercise.p65 (4) A standing wave having nodes at x n ; n = 0, 1, 2 2 1 9 . A cylindrical tube, open at both ends, has a fundamental frequency, f, in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air-column is now :- [AIEEE-2012] (1) 2f (2) f (3) f/2 (4) 3f/4 2 0 . A sonometer wire of length 1.5m is made of steel. The tension in it produces an elastic strain of 1%. What is the fundamental frequency of steel if density and elasticity of steel are 7.7 × 103 kg/m3 and 2.2 × 1011 N/ m2 respectively? [AIEEE-2013] (1) 188.5 Hz (2) 178.2 Hz (3) 200.5 Hz (4) 770 Hz ANSWER-KEY Que. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Ans. 3 2 2 2 1 3 3 4 3 4 2 1 1 4 1 4 2 1 2 2 E 61
JEE-Physics EXERCISE–05(B) PREVIOUS YEAR QUESTIONS MCQ'S WITH ONE CORRECT ANSWER 1 . A string of length 0.4 m and mass 10–2 kg is tightly clamped at its ends. The tension in the string is 1.6 N. Identicals wave pulses are produced at one end at equal intervals of time t. The minimum value of t, which allows constructive interference between successive pulses, is :– [IIT-JEE 1998] (A) 0.05 s (B) 0.10 s (C) 0.20 s (D) 0.40 s 2 . A train moves towards a stationary observer with speed 34 m/s. The train sounds a whistle and its frequency registered by the observer is ƒ . If the train's speed is reduced to 17 m/s, the frequency registered is ƒ . 12 If the speed of sound is 340 m/s then the ratio ƒ1 is :– [IIT-JEE 2000] 18 1 ƒ2 19 (A) (B) (C) 2 (D) 19 2 18 3 . Two vibrating strings of the same material but of lengths L and 2L have radii 2r and r respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental modes, the one of length L with frequency f and the other with frequency f . The ratio ƒ1 is given by :– [IIT-JEE 2000] 1 2 ƒ2 (A) 2 (B) 4 (C) 8 (D) 1 4 . The ends of a stretched wire of length L are fixed at x=0 and x = L. In one experiment the displacement of the wire is y= x is E and in experiment displacement is = Asin 2x sin2t and 1 Asin sint and energy 1 other its y L 2 L energy is E . Then :– [IIT-JEE 2001] 2 (A) E = E (B) E = 2E (C) E = 4E (D) E = 16E 21 21 21 21 5 . Two pulses in a stretched string, whose centres are initially 8 cm apart, are moving towards each other as shown in the figure. The speed of each pulse is 2 cm/s. After 2 s the total energy of the pulses will be :– [IIT-JEE 2001] (A) Zero (B) Purely kinetic (C) Purely potential (D) Partialy kinetic and partialy potential 8cm 6 . A siren placed at a railway platform is emitting sound of frequency 5 kHz. A passenger sitting in a moving train A records a frequency of 5.5 kHz, while the train approaches the siren. During his return journey in a different train B he records a frequency of 6.0 kHz while approaching the same siren. The ratio of the velocity of train B to that train A is :– [IIT-JEE 2002] 242 (B) 2 5 11 (A) (C) (D) 252 6 6 7 . A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a mass of 9 kg is suspended from the wire. When this mass is replaced by mass M. The wire resonates with the same tuning fork forming three antinodes for the same positions of the bridges. \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\Wave Motion\\Eng\\Exercise.p65 The value of M is :– [IIT-JEE 2002] (A) 25 kg (B) 5 kg (C) 12.5 kg 1 (D) kg 25 8 . A police car moving at 22 m/s chases a motocyclist. The police man sounds his horn at 176 Hz, while both of them move towards a stationary siren of frequency 165 Hz. Calculate the speed of the motorcycle. If it is given that the motorcyclist does not observe any beats :– [IIT-JEE 2003] Police car Motorcycle 22 m/s, 176 Hz v Stationary siren (165 Hz) (A) 33 m/s (B) 22 m/s (C) zero (D) 11 m/s 62 E
JEE-Physics 9 . In the experiment for the determination of the speed of sound in air using the resonance column method, the length of the air column that resonates in the fundamental mode, with a tuning fork is 0.1 m. When this length is changed to 0.35 m, the same tuning fork resonates with the first overtone. Calculate the end correction :– [IIT-JEE 2003] (A) 0.012 m (B) 0.025 m (C) 0.05 m (D) 0.024 m 1 0 . A source of sound of frequency 600 Hz is placed inside water. The speed of sound in water is 1500 m/ s and in air it is 300 m/s. The frequency of sound recorded by an observer who is standing in air is :– (A) 200 Hz (B) 3000 Hz (C) 120 Hz (D) 600 Hz [IIT-JEE 2004] 1 1 . A closed organ pipe of length L and an open organ pipe contain gases of densities 1 and 2 respectively. The compressibility of gases are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency. The length of the open organ pipe is :– [IIT-JEE 2004] L 4L 4L 1 4L 2 (A) (B) (C) 3 2 (D) 3 1 3 3 1 2 . A source emits sound of frequency 600 Hz inside water. The frequency heard in air will be equal to (velocity of sound in water = 1500 m/s, velocity of sound in air = 300 m/s) :– [IIT-JEE 2004] (A) 3000 Hz (B) 120 Hz (C) 600 Hz (D) 6000 Hz 1 3 . An open pipe is in resonance in 2nd harmonic with frequency ƒ . Now one end of the tube is closed and frequency 1 is increased to ƒ such that the resonance again occurs in nth harmonic. Choose the correct option :–[IIT-JEE 2005] 2 3 5 5 3 (A) n = 3, ƒ = ƒ (B) n = 3, ƒ = ƒ (C) n = 5, ƒ = ƒ (D) n = 5, ƒ = ƒ 2 41 2 41 2 41 2 41 1 4 . A tuning fork of 512 Hz is used to produce resonance in a resonance tube experiment. The level of water at first resonance is 30.7 cm and at second resonance is 63.2 cm. The error in calculating velocity of sound is :– (A) 204.1 cm/s (B) 110 cm/s (C) 58 cm/s (D) 280 cm/s[IIT-JEE 2005] 1 5 . A massless rod BD is suspended by two identical massless strings AB and CD of equal lengths. A block of mass m is suspended at point P such that BP is equal to x, if the fundamental frequency of the left wire is twice the fundamental frequency of right wire, then the value of x is :– [IIT-JEE 2006] AC xP D B m \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\Wave Motion\\Eng\\Exercise.p65 4 3 (A) (B) (C) (D) 5 4 5 4 1 6 . A transverse sinusoidal wave moves along a string in the positive x–direction at a speed of 10 cm/s. The wavelength of the wave is 0.5 m and its amplitude is 10 cm. At a particular time t, the snap–shot of the wave is shown in figure. The velocity of point P when its displacement is 5 cm is :– [IIT-JEE 2008] y P x (A) 3 ˆj m/s (B) 3 ˆj m/s (C) 3 ˆi m/s (D) 3 ˆi m/s 50 50 50 50 E 63
JEE-Physics 1 7 . A vibrating string of certain length under a tension T resonates with a mode corresponding to the first overtone (third harmonic) of an air column of length 75 cm inside a tube closed at one end. The string also generates 4 beats per second when excited along with a tuning fork of frequency n. Now when the tension of the string is slightly increased the number of beats reduces to 2 per second. Assuming the velocity of sound in air to be 340 m/s, the frequency n of the tuning fork in Hz is. [IIT-JEE 2008] (A) 344 (B) 336 (C) 117.3 (D) 109.3 MCQS (one or more than one answer may be correct) 1 . The (x, y) coordinates of the corners of a square plate are (0, 0), (L, 0), (L, L) and (0, L). The edges of the plate are clamped and transverse standing waves are set–up in it. If u(x, y) denotes the displacement of the plate at the point (x, y) at some instant of time, the possible expression (s) for u is (are) (a = positive constant) :– [IIT-JEE 1998] (A) acos x y (B) a sin x sin y (C) asin x sin 2 y (D) 2x sin y 2L cos 2L L L L L acos L L 2 . A transverse sinusoidal wave of amplitude a, wavelength and frequency ƒ is travelling on a stretched string. The v maximum speed of any point on the string is 10 , where v is the speed of propagation of the wave. If a = 10–3 m and v = 10 m/s, then and ƒ are given by :– [IIT-JEE 1998] (A) = 2 × 10–2 m (B) = 10–3 m (C) ƒ = 103 Hz (D) ƒ = 104 Hz 2 [IIT-JEE 1999] 3 . As a wave propagates :– (A) The wave intensity remains constant for a plane wave (B) The wave intensity decreases as the inverse of the distance from the source for a spherical wave (C) The wave intensity decreases as the inverse square of the distance from the source for a spherical wave (D) Total intensity of the spherical wave over the spherical surface centred at the source remains constant at all time 0.8 4 . y(x,t)= [(4x 5t)2 5] represents a moving pulse where x and y are in metres and t in second. Then: (A) pulse is moving in positive x–direction (B) in 2 s it will travel a distance of 2.5 m (C) its maximum displacement is 0.16 m (D) it is a symmetric pulse [IIT-JEE 1999] 5 . In a wave motion y=asin(kx–t), y can represent :– [IIT-JEE 1999] (A) electric field (B) magnetic field (C) displacement (D) pressure 6 . Standing wave can be produced :– [IIT-JEE 1999] \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\Wave Motion\\Eng\\Exercise.p65 (A) on a string clamped at both ends (B) on a string clamped at one end and free at the other (C) when incident wave gets reflected from a wall (D) when two identical waves with a phase difference of are moving in the same direction 7 . A horizontal stretched string, fixed at two ends, is vibrating in its fifth harmonic according to the equation, y(x, t) = (0.01m) sin [(62.8 m–1)x]cos [(628 s–1)t]. Assuming p = 3.14, the correct statement(s) is (are) (A) The number of nodes is 5. [IIT-JEE 2013] (B) The length of the string is 0.25 m. (C) The maximum displacement of the midpoint of the string, from its equilibrium position is 0.01m. (D) The fundamental frequency is 100 Hz. 64 E
JEE-Physics Comprehension Based Question Comprehension#1 [IIT-JEE 2006] Two plane harmonic sound waves are expressed by the equations. (All parameters are in MKS) : y (x, t) = Acos(0.5x – 100t), y (x, t) = Acos(0.46x – 92t) 1 2 1 . How many times does an observer hear maximum intensity in one second :– (A) 4 (B) 10 (C) 6 (D) 8 2 . What is the speed of the sound :– (A) 200 m/s (B) 180 m/s (C) 192 m/s (D) 96 m/s 3 . At x = 0 how many times the amplitude of y + y is zero in one second :– 12 (A) 192 (B) 48 (C) 100 (D) 96 Comprehension#2 [IIT-JEE 2007] Two trains A and B are moving with speed 20 m/s and 30 m/s respectively in the same direction Intensity on the same straight track, with B ahead of A. The engines are at the front ends. The engine of ƒ1 ƒ2 Frequency train A blows a long whistle. Assume that the sound of the whistle is composed of components varying in frequency from ƒ = 800 Hz to ƒ = 1120 Hz, as shown in the figure. The spread in the frequency 12 (highest frequency–lowest frequency) is thus 320 Hz. The speed of sound in still air is 340 m/s. 1 . The speed of sound of the whistle is :– (A) 340 m/s for passengers in A and 310 m/s for passengers in B (B) 360 m/s for passengers in A and 310 m/s for passengers in B (C) 310 m/s for passengers in A and 360 m/s for passengers in B (D) 340 m/s for passengers in both the trains 2 . The distribution of the sound intensity of the whistle as observed by the passengers in train A is best represented by Intensity Intensity Intensity Intensity (A) (B) (C) (D) ƒ1 ƒ2 ƒ1 ƒ2 ƒ1 ƒ2 ƒ1 ƒ2 Frequency Frequency Frequency Frequency 3 . The spread of frequency as observed by the passengers in train B is :– (A) 310 Hz (B) 330 Hz (C) 350 Hz (D) 290 Hz Subjective Questions \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\Wave Motion\\Eng\\Exercise.p65 1 . The air column in a pipe closed at one end is made to vibrate in its second overtone by tuning fork of frequency 440 Hz. The speed of sound in air is 330 m/s. End corrections may be neglected. Let P0 denote the mean pressure at any point in the pipe and P0 the maximum amplitude of pressure variation. [IIT-JEE 1998] (i) Find the length L of the air column. (ii) What is the amplitude of pressure variation at the middle of the column ? (iii) What are the maximum and minimum pressures at the open end of the pipe ? (iv) What are the maximum and minimum pressures at the closed end of the pipe ? 2 . A long wire PQR is made by joining two wires PQ and QR of equal radii. PQ has length 4.8 m and mass 0.06 kg. QR has length 2.56 m and mass 0.2 kg. The wire PQR is under a tension of 80 N. A sinusoidal wave pulse of amplitude 3.5 cm is sent along the wire PQ from the end P. No power is dissipated during the propagation of the wave pulse. Calculate : (i) The time taken by the wave pulse to reach the other end R and (ii) The amplitude of the reflected and transmitted wave pulse after the incident wave pulse crosses the joint Q. [IIT-JEE 1999] 3. A 3.6 m long pipe resonates with a source of frequency 212.5 Hz when water level is at certain heights in the pipe. E Find the heights of water level (from the bottom of the pipe) at which resonance occur. Neglect end correction. Now the pipe is filled to a height H ( 3.6 m). A small hole is drilled very close to its bottom and water is allowed to leak. Obtain an expression for the rate of fall of water level in the pipe as a function of H. If the radii of the pipe and the hole are 2 × 10–2 m and 1 × 10–3 m respectively. Calculate the time interval between the occurrence of first two resonance. Speed of sound in air is 340 m/s and g = 10 m/s2. [IIT-JEE 2010] 65
JEE-Physics 4 . A boat is travelling in a river with a speed 10 m/s along the stream flowing with a speed 2 m/s. From this boat a sound transmitter is lowered into the river through a rigid support. The wavelength of the sound emitted from the transmitter inside the water is 14.45 mm. Assume that attenuation of sound in water and air is negligible. (i) What will be the frequency detected by a receiver kept inside the river downstream ? [IIT-JEE 2001] (ii) The transmitter and the receiver are now pulled up into air. The air is blowing with a speed 5 m/s in the direction opposite the river stream. Determine the frequency of the sound detected by the receiver. (Temperature of the air and water = 20°C; Density of river water = 103 kg/m3 ; Bulk modulus of the water = 2.088 × 109 Pa; Gas constant R = 8.31 J/mol–K; Mean molecular mass of air = 28.8 × 10–3 kg/mol; C /C for air = 1.4) PV 5 . Two narrow cylindrical pipes A and B have the same length. Pipe A is open at both ends and is filled with a monatomic gas of molar mass MA. Pipe B is open at one end and closed at the other end, and is filled with a diatomic gas of molar mass MB. Both gases are at the same temperature. [IIT-JEE 2002] (i) If the frequency to the second harmonic of the fundamental mode in pipe A is equal of the frequency of the third harmonic of the fundamental mode in pipe B, determine the value of M /M . AB (ii) Now the open end of the pipe B is closed (so that the pipe is closed at both ends). Find the ratio of the fundamental frequency in pipe A to that in pipe B. 6 . In a resonance tube experiment to determine the speed of sound in air, a pipe of diameter 5 cm is used. The air column in pipe resonates with a tuning fork of frequency 480 Hz when the minimum length of the air column is 16 cm. Find the speed of sound in air at room temperature. [IIT-JEE 2003] 7 . A string of mass per unit length µ is clamped at both ends such that one end of the string is at x = 0 and the other is at x = . When string vibrates in fundamental mode, amplitude of the midpoint O of the string is a, tension in the string is T and amplitude of vibration is A. Find the total oscillation energy stored in the string. [IIT-JEE 2003] 8 . An observer standing on a railway crossing receives frequency of 2.2 kHz and 1.8 kHz when the train approaches and recedes from the observer. Find the velocity of the train. (The speed of the sound in air is 300 m/s) [IIT-JEE 2005] 9 . A harmonically moving transverse wave on a string has a maximum particle velocity and acceleration of 3 m/ s and 90 m/s2 respectively. Velocity of the wave is 20 m/s. Find the waveform. [IIT-JEE 2005] 1 0 . A 20 cm long string having a mass of 1.0 g, is fixed at both the ends. The tension in the string is 0.5 N. The string is set into vibrations using an external vibrator of frequency 100 Hz. Find the separation (in cm) between the successive nodes on the string. [IIT-JEE 2009] PREVIOUS YEARS QUESTIONS ANSWER KEY EXERCISE –5 MCQ's One correct answers 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\Wave Motion\\Eng\\Exercise.p65 B DDCB B AB B DCCCDA AA MCQ's with one or more than one correct answer 1. B,C 2.A,C 3.A,C,D 4.B,C,D 5.A,B,C 6.A,B,C 7. B,C Comprehension based question Comprehension #1 1. A 2. A 3. D Comprehension #2 1. B 2. A 3. A Subjective Questions 15 P0 (iii) Pmax Pmin P0 , (iv) Pmax P0 P0 , Pmin P0 P0 1. (i) m (ii) 2 16 2. (i) 140 ms (ii) Ar v2 v1 Ai 1.5cm , At 2v2 Ai 2 cm v 2 v1 v 2 v1 3. h = 3.2, 2.4, 1.6, 0.8, 0, v= 5 103 5H , t 80 4 2 3 4. (i) 100696Hz (ii) 103038Hz 5. (i) 2.116 (ii) 3 6. 336 ms–1 7. A 22 T 8. 30 ms–1 9. y = (10 cm) sin 30 t 3 x 10. 5 4 4 2 66 E
JEE-Physics EXERCISE–01 CHECK YOUR GRASP SELECT THE CORRECT ALTERNATIVE (ONLY ONE CORRECT ANSWER) 1 . A person A of 50 kg rests on a swing of length 1m making an angle 37O with the vertical. Another person B pushes him to swing on other side at 53O with vertical. The work done by person B is : [ g = 10 m/s2 ] (A) 50 J (B) 9.8 J (C) 100 J (D) 10 J 2 . The work done by the frictional force on a pencil in drawing a complete circle of radius r = 1/ metre on the surface by a pencil of negligible mass with a normal pressing force N = 5 N (µ = 0.5) is : (A) + 4J (B) –3 J (C) – 2 J (D) – 5J 3 . A rope is used to lower vertically a block of mass M by a distance x with a constant downward acceleration g/2. The work done by the rope on the block is : (A) Mgx 1 1 (D) Mgx² (B) Mgx² (C) – Mgx 2 2 4 . The work done in moving a particle under the effect of a conservative A force, from position A to B is 3 joule and from B to C is 4 joule. The B C work done in moving the particle from A to C is : (A) 5 joule (B) 7 joule (C) 1 joule (D) –1 joule 5 . Work done in time t on a body of mass m which is accelerated from rest to a speed v in time t as a function of 1 time t is given by : 1 m v t2 m v t2 1 mv 2 1 v2 2 t1 t1 2 t1 t 2 t12 (A) (B) (C) t2 (D) m t2 6 . Velocity–time graph of a particle of mass 2 kg moving in a straight line v (m/s) is as shown in figure. Work done by all the forces on the particle is : 20 (A) 400 J (B) –400 J (C) –200 J (D) 200 J t (s) 2 3 7 . A particle moves on a rough horizontal ground with some initial velocity say v0. If 4 of its kinetic energy is lost due to friction in time t0 then coefficient of friction between the particle and the ground is : node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 v0 v0 3v0 v0 (A) 2gt0 (B) 4gt0 (C) 4gt0 (D) gt0 8 . A block of mass m moving with speed v compresses a spring through distance x before its speed is halved. What is the value of spring constant ? 3mv2 mv2 mv2 2mv2 (A) 4 x2 (B) 4 x2 (C) 2 x2 (D) x2 9. An engine can pull 4 coaches at a maximum speed of 20 m/s. Mass of the engine is twice the mass of E every coach. Assuming resistive forces proportional to the weight, approximate maximum speeds of the engine when it pulls 12 and 6 coaches are : (A) 8.5 m/s and 15 m/s respectively (B) 6.5 m/s and 8 m/s respectively (C) 8.5 m/s and 13 m/s respectively (D) 10.5 m/s and 15 m/s respectively 31
JEE-Physics 1 0 A small sphere starts falling from a very large height and after falling a distance of 100 m it attains the terminal velocity and continues to fall with this velocity. The work done by the atmosphere during the first fall of 100m is : (A) Greater than the work done for next fall of 100 m (B) Less than the work done for next fall of 100 m (C) Equal to 100 mg (D) Greater than 100 mg 1 1 A force acts on a 3 gm particle in such a way that the position of the particle as a function of time is given by x= 3t – 4t2 + t3, where x is in meters and t is in seconds. The work done during the first 4 second is : (A) 384 mJ (B) 168 mJ (C) 528 mJ (D) 541 mJ 1 2 . A body is moved along a straight line by a machine delivering constant power. The distance moved by the body in time t is proportional to : (A) t1/2 (B) t3/4 (C) t3/2 (D) t2 1 3 . A particle of mass m is moving in a circular path of constant radius r such that its centripetal acceleration a is varying C with time t as a = k2rt2, where k is a constant. The power delivered to the particle by the force acting on it is : C (A) 2mk2r2 (B) mk2r2t (mk4r2t5 ) (D) zero (C) 3 1 4 . In the figure shown the potential energy (U) of a particle is plotted against U its position 'x' from origin. Then which of the following statement is correct. A particle at : (A) x is in stable equilibrium (B) x is in stable equilibrium 1 2 O X1 X2 X3 X (C) x3 is in stable equilibrium (D) None of these 1 5 . The given plot shows the variation, the potential energy (U) of interaction between two particles with the separating distance (r) between them. Which of the above statements are correct ? (1) B and D are equilibrium points UA (2) C is a point of stable equilibrium points (3) The force of interaction between the two particles is attractive between E points C and D and repulsive between points D and E on the curve. BD F C r (4) The force of interaction between the particles is repulsive between points E and F on the curve. (A) 1 and 3 (B) 1 and 4 (C) 2 and 4 (D) 2 and 3 16. A weight is hung freely from the end of a spring. A boy then slowly pushes the weight upwards until the node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 spring becomes slack. The gain in gravitational poetential energy of the weight during this process is equal to : (A) The work done by the boy against the gravitational force acting on the weight. (B) The loss of the stored energy by the spring minus the work done by the tension in the spring. (C) The work done on the weight by the boy plus the stored energy lost by the spring. (D) The work done on the weight by the boy minus the workdone by the tension in the spring plus the stored energy lost by the spring. 17. A rope ladder with a length carrying a man of mass m at its end is attached to the basket of balloon with a mass M. The entire system is in equilibrium in the air. As the man climbs up the ladder into the balloon, the balloon descends by a height h. Then the potential energy of the man : (A) Increases by mg (–h) (B) Increases by mg (C) Increases by mgh (D) Increases by mg (2–h) 32 E
JEE-Physics 1 8 . A block attached to a spring, pulled by a constant horizontal force, is k F kept on a smooth surface as shown in the figure. Initially, the spring is F2 in the natural state. Then the maximum positive work that the applied (D) force F can do is : [Given that spring does not break] 2k F2 2F2 (C) (A) (B) k k 1 9 . A simple pendulum has a string of length and bob of mass m. When the bob is at its lowest position, it is given the minimum horizontal speed necessary for it to move in a circular path about the point of suspension. The tension in the string at the lowest position of the bob is : (A) 3mg (B) 4mg (C) 5mg (D) 6mg 2 0. In the previous question, when the string is horizontal, the net force on the bob is : (A) mg (B) 3mg (C) 10mg (D) 4mg 2 1 . A particle of mass m is fixed to one end of a light rigid rod of length and rotated in a vertical circular path about its other end. The minimum speed of the particle at its highest point must be : (A) zero (B) g (C) 1.5 g (D) 2g 2 2 . A stone tied to a string of length L is whirled in a vertical circle, with the other end of the string at the centre. At a certain instant of time, the stone is at its lowest position and has a speed u. The magnitude of the change in its velocity as it reaches a position where the string is horizontal is : (A) u2 2gL (B) 2gL (C) u2 gL (D) 2(u2 gL ) 2 3 . A marble of mass m and radius b is placed in a hemispherical bowl of radius r. The minimum velocity to be given to the marble so that it reaches the highest point is : (A) 2g(r b) (B) 2 g r (C) 2 g (r b ) (D) g (r b ) 2 4 . A particle is placed at the top of a sphere of radius r. It is given a little jerk so that it just starts slipping down. Find the point where it leaves the sphere. (A) r/2 (B) r/3 (C) r/4 (D) r 2 5 . A particle is moving in a circular path with a constant speed v. If is the angular displacement, then starting from = 00, the maximum and minimum change in the linear momentum will occur when value of is respectively : (A) 45º & 90º (B) 90º & 180º (C) 180º & 360º (D) 90º & 270º node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 2 6 . In a simple pendulum, the breaking strength of the string is double the weight of the bob. The bob is released from rest when the string is horizontal. The string breaks when it makes an angle with the vertical– (A) cos1 1 (B) = 60° (C) cos1 2 (D) = 0 3 3 CHECK YOUR GRASP ANSWER KEY EXERCISE –1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 C D CBDBA A ABCCBDC 16 17 18 19 20 21 22 23 24 25 26 C ABDCADA BCC E 33
JEE-Physics EXERCISE–02 BRAIN TEASERS SELECT THE CORRECT ALTERNATIVES (ONE OR MORE THEN ONE CORRECT ANSWERS) 1 . In the figure shown, the system is released from rest. Find the velocity of block A when block B has fallen a distance 'l'. Assume all pulleys to be massless and frictionless. g (B) g Am P (A) mB 5 (C) 5g (D) None of these 2 . A block of mass m is attached to two spring of spring constant k and k as shown in figure. The block is 12 displaced by x towards right and released. The velocity of the block when it is at x/2 will be : k1 k2 m (A) (k1 k2 )x2 (B) 3 (k1 k2 )x2 (C) (k1 k2 )x2 (D) (k1 k2 )x2 2m 4m m 4m 3 . An object of mass m slides down a hill of height h of arbitrary shape and after travelling a certain horizontal path stops because of friction. The friction coefficient is different for different segments for the entire path but is independent of the velocity and direction of motion. The work that a force must perform to return the object to its initial position along the same path is : (A) mgh (B) 2mgh (C) 4mgh (D) –mgh 4 . A bob hangs from a rigid support by an inextensible string of length l. If it is displaced through a distance l (from the lowest position) keeping the string straight & released, the speed of the bob at the lowest position is: (A) g (B) 3g (C) 2g (D) 5g 5 . A cube of mass M starts at rest from point 1 at a height 4R, where R is the 1 4R 2 radius of the circular track. The cube slides down the frictionless track and node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 around the loop. The force which the track exerts on the cube at point 2 is R : (A) 3 mg (B) mg (C) 2 mg (D) cube will not reach the point 2. 6 . Two bodies of mass m1 and m2 (m2 > m1) are connected by a light inextensible string which passes through a smooth fixed pulley. The instantaneous power delivered by an external agent to pull m1 with constant velocity v is : (A) (m2 – m1) g/v (B) (m2 – m1) v/g m1 (C) (m2 – m1) gv (D) (m1 – m2) gv Fext m2 34 E
JEE-Physics 7 . A small block slides with velocity 0.5 gr on the horizontal frictionless surface as shown in the figure. The block leaves the surface at point C. The angle in the figure is : v0 AB C r vC r mg OD (A) cos1 4 (B) cos1 3 (C) cos1 1 (D) cos1 4 9 4 2 5 8 . A man places a chain of mass 'm' and length '' on a table slowly. Initially the lower end of the chain just touches the table. The man drops the chain when half of the chain is in vertical position. Then work done by the man in this process is : mg 3mg mg (A) mg (B) (C) (D) 2 4 8 8 9 . The potential energy of a particle of mass m free to move along x–axis is given by U =½ kx2 for x < 0 and U = 0 for x 0 (x denotes the x–coordinate of the particle and k is a positive constant). If 2E the total mechanical energy of the particle is E, then its speed at x = – is : k (A) zero 2E E E (B) (C) (D) m m 2m 10. The blocks A and B shown in the figure have masses M = 5 kg and M = 4 kg. \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 11. AB 12. The system is released from rest. The speed of B after A has travelled a distance E 1 m along the incline is : (A) 3 g (B) 3 g A 2 4 5m g g B 37° (C) (D) 23 2 A collar 'B' of mass 2 kg is constrained to move along a horizontal smooth and fixed circular track of radius 5m. The spring lying in the plane of the circular track and having spring constant 200 N/m is undeformed when the collar is at 'A'. If the collar starts from rest at B' the normal reaction exerted by the track on the collar when it passes through 'A' is : node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 B 5m AC 7m D (A) 360 N (B) 720 N (C) 1440 N (D) 2880 N A A particle is projected along a horizontal field whose coefficient of friction varies as µ = r2 where r is the distance from the origin in meters and A is a positive constant. The initial distance of the particle is 1 m from the origin and its velocity is radially outwards. The minimum initial velocity at this point so that particle never stops is : (A) (B) 2 gA (C) 2gA (D) 4 gA 35
JEE-Physics 1 3 . Two identical blocks A and B are placed on two inclined planes as shown in diagram. Neglect air resistance and other friction. Choose the correct statement : A L NB hh Fixed Fixed J KM O Statement I: Kinetic energy of 'A' on sliding to J will be greater than the kinetic energy of B on falling to M. Statement II: Acceleration of 'A' will be greater than acceleration of 'B' when both are released to slide on inclined plane. Statement III : Work done by external agent to move block slowly from position B to O is negative (A) statement I is true (B) statement II is true (C) statement I and III are true (D) statement II and III are true 1 4 . Figure shows the roller coaster track. Each car will start from rest at A point A and will roll with negligible friction. It is important that there should be at least some small positive normal force exerted by the track on the car at all points, otherwise the car would leave the track. With 25m B 15m the above fact, the minimum safe value for the radius of curvature at point B is (g = 10 m/s2) : ////// / //////// (A) 20 m (B) 10 m ///////// ////// (C) 40 m (D) 25 m 10 1 5 . A particle 'A' of mass 7 kg is moving in the positive x–direction. Its initial position is x = 0 & initial velocity is 1 m/s. The velocity at x = 10m is : (use the graph given) Power (in watts) 4 2 10 x (in m) (A) 4 m/s (B) 2 m/s (C) 3 2m / s (D) 100/3 m/s 1 6 . A fire hose has a diameter of 2.5 cm and is required to direct a jet of water to a height of at least 40m. The minimum power of the pump needed for this hose is : (A) 21.5 kW (B) 40 kW (C) 36.5 kW (D) 48 kW 1 7 . A particle is projected vertically upwards with a speed of 16 m/s, after some time, when it again passes through node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 the point of projection, its speed is found to be 8 m/s. It is known that the work done by air resistance is same during upward and downward motion. Then the maximum height attained by the particle is : (Take g = 10 m/ s2) (A) 8 m (B) 4.8 m (C) 17.6 m (D) 12.8 m 1 8 . A force F 3ˆi 4ˆj N acts on a 2 kg movable object that moves from an initial position di 3ˆi 2ˆj m to final position df 5ˆi 4ˆj in 6s. The average power delivered by the force during the interval is equal to : (A) 8 watt 50 (C) 15 watt 50 (B) watt (D) watt 36 6 3 E
JEE-Physics 1 9 . A wedge of mass M fitted with a spring of stiffness 'k' is kept on a smooth horizontal surface. A rod of mass m is kept on the m k wedge as shown in the figure. System is in equilibrium. Assuming M that all surfaces are smooth, the potential energy stored in the ( spring is: m2g2 tan2 mg2 tan2 m2g tan2 m2g2 tan2 (D) (A) (B) (C) k 2k 2k 2k 2 0 . In the figure, a block slides along a track from one level to a higher level, by moving through an intermediate valley. The track is frictionless untill the block reaches the higher level. There a frictional force stops the block in a distance d. The block's initial speed v0 is 6 m/s, the height difference h is 1.1 m and the coefficient of kinetic friction µ is 0.6. The value of d is µ=0.6 v0 h µ=0 (A) 1.17 m (B) 1.71 m (C) 7.11 m (D) 11.7 m 2 1 . A ball rolls down an inclined plane figure. The ball is first released from rest from P and then later from Q. Which of the following statement is/are correct ? Q (A) The ball takes twice as much time to roll from Q to O as it does to roll from P to O. (B) The acceleration of the ball at Q is twice as large as the acceleration at P. P 2h (C) The ball has twice as much K.E. at O when rolling from Q as it does when h rolling from P O (D) None of the above 2 2 . A car of mass m starts moving so that its velocity varies according to the law v a s , where a is a constant, and s is the distance covered. The total work performed by all the forces which are acting on the car during the first t seconds after the beginning of motion is : (A) ma4t2/8 (B) ma2t4/8 (C) ma4t2/4 (D) ma2t4/4 2 3 . A block of mass m is attached with a massless spring of force constant k. m M The block is placed over a rough inclined surface for which the coefficient of friction is = 3/4. The minimum value of M required to move the 37° block up the plane is : (Neglect mass of string and pulley and friction in pulley) node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 3 4 (C) 2 m 3 (A) m (B) m (D) m 5 5 2 2 4 . A bob is suspended from a crane by a cable of length = 5 m. The crane and load are moving at a constant speed v0. The crane is stopped by a bumper and the bob on the cable swings out an angle of 60°. The initial speed v0 is– (g = 9.8 m/s2) v0 (A) 10 m/s (B) 7 m/s (C) 4 m/s (D) 2 m/s E 37
JEE-Physics 2 5 . If one of the forces acting on a particle is conservative then : (A) Its work is zero when the particle moves exactly once around any closed path. (B) Its work equals the change in the kinetic energy of the particle. (C) It obeys Newton's second law. (D) Its work depends on the end points of the motion, not on the path between. 2 6 . A particle of mass m = 1 kg lying on x–axis experiences a force given by law F=x(3x–2) Newton, where x is the x–coordinate of the particle in meters. The points on x–axis where the particle is in equilibrium are : (A) x = 0 (B) x = 1/3 (C) x = 2/3 (D) x = 1 2 7 . With what minimum velocity v should block be projected from left end A towards end B such that it reaches 0 the other end B of conveyer belt moving with constant velocity v ? Friction coefficient between block and belt is . A m v0 B v (A) gL (B) 2gL (C) 3gL (D) 2 gL 2 8 . A light spring of length 20 cm and force constant 2 N/cm is placed vertically on a table. A small block of mass 1 kg falls on it. The length h from the surface of the table at which the block will have the maximum velocity is : (A) 20 cm (B) 15 cm (C) 10 cm (D) 5cm 2 9 . In the figure shown all the surfaces are frictionless, and mass of the block, 10m/s2 m = 1 kg. The block and wedge are held initially at rest. Now wedge is given m a horizontal acceleration of 10 m/s2 by applying a force on the wedge, so that the block does not slip on the wedge. Then work done by the normal M force in ground frame on the block in 3 seconds is : (A) 30J (B) 60 J (C) 150 J (D) 100 3 J 3 0 . When a conservative force does positive work on a body : (A) The potential energy increases (B) The potential energy decreases (C) Total energy increases (D) Total energy decreases 3 1 . A 1.0 kg block collides with a horizontal weightless spring of force constant 2.75 Nm–1. The block compresses the spring 4.0 m from the rest position. If the coefficient of kinetic friction between the block and horizontal surface is 0.25, the speed of the block at the instant of collision is : (A) 0.4 ms–1 (B) 4 ms–1 (C) 0.8 ms–1 (D) 8 ms–1 node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 3 2 . Acceleration versus time graph of a particle moving in a straight line is as a shown in adjoining figure. If initially particle was at rest then corresponding kinetic energy versus time graph will be: 2 (m/s ) KE KE KE KE t(s) (A) (B) (C) (D) t t t t E 38
JEE-Physics y 3 3 . A particle is moved from (0, 0) to (a, a) under a force F (3ˆi 4ˆj) P(a, a) from two paths. Path 1 is OP and path 2 is OQP. Let W1 and W2 be the work done by this force in these two paths. Then : (A) W1 = W2 (B) W1 = 2W2 45° x (C) W2 = 2W1 (D) W2 = 4W1 OQ 3 4 . A particle of mass m begins to slide down a fixed smooth sphere from the top. What is the tangential acceleration when it breaks off the sphere? 2g 5g (C) g g (A) 3 (B) (D) 3 3 3 5 . A machine, in an amusement park, consists of a cage of the end of one E arm, hinged at O. The cage revolves along a vertical circle of radius r FD (ABCDEFGH) about its hinge O, at constant linear speed v= gr . The G OC H cage is so attached that the man of weight 'W' standing on a weighing B machine, inside the cage, is always vertical. Then which of the following is A correct (A) The weight reading at A is greater than the weight reading at E by 2W (B) The weight reading at G = W (C) The ratio of the weight reading at E to that at A = 0 (D) The ratio of the weight reading at A to that at C = 2 3 6 . A hollow vertical cylinder of radius r and height h has a smooth internal surface. A small Au particle is placed in contact with the inner side of the upper rim, at point A, and given a horizontal speed u, tangential to the rim. It leaves the lower rim at point h B, vertically below A. If n is an integer then– B u 2h h n (B) n (A) 2r g 2r (C) 2r n u h n (D) 2gh 3 7 . The kinetic energy K of a particle moving along a circle of radius R depends upon the distance s, as K = as². The force acting on the particle is– 1 (A) 2a s² (B) 2as 1 s² 2 (C) 2as (D) 2a R R2 3 8 . A simple pendulum of length L and mass (bob) M is oscillating in a plane about a vertical line between angular limits – and . For an angular displacement , [||<] the tension in the string and velocity of the bob are T and v respectively. The following relations hold good under the above conditions node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 (A) T cos Mg Mv2 (B) T Mg cos L (C) Tangential acceleration = g sin (D) T=Mg cos BRAIN TEASERS ANSWER KEY EXERCISE –2 Que. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Ans. A B B A A C B C A C C C D A A Que. 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 A n s . A A A C A C A A B A,C,D A,C B B C B Que. 31 32 33 34 35 36 37 38 A n s . D A A B A,B,C,D A B B,C E 39
JEE-Physics EXERCISE–03 MISCELLANEOUS TYPE QUESTIONS TRUE FALSE 1 . Mechanical energy of a body cannot be negative. 2 . Net work done by external forces on a moving body is same as observed from different inertial frames. 3 . Work done by a non-conservative force is always negative. 4 . Work done by all internal conservative forces is required to change in its potential energy. FILL IN THE BLANKS 1 . The work done in holding a 15 kg suitcase while waiting for a bus for 15 minute is..... 2 . A body of mass M tied to a string is lowered at a constant acceleration of (g/4) through a vertical distance h. The work done by the string will be.............. 3 . An open knife edge of mass M is dropped from a height h on a wooden floor. If the blade penetrates 's' into the wood, the average resistance offered by the wood to the blade is............ 4 . The power of a pump which raises 100 kg of a water in 10 s to a height 100 m is.......... kW. 5 . A ball suspended by a thread swings in a vertical plane so that its acceleration values in the extreme and lowest position are equal. The thread deflection angle in the extreme position is _ _ _ _. MATCH THE COLUMN 1. A block of mass m is stationary with respect to a rough wedge as shown in figure. Starting from rest, in time t work done on the block : (m = 1kg, = 30°, a = 2m/s2, t = 4s) a Column I Column II (A) By gravity (p) 144 J (B) By normal reaction (q) 32 J (C) By friction (r) 56 J (D) By all the forces (s) 48 J (t) None 2. Acceleration 'a' versus x and potential energy 'U' versus x graph of a particle moving along x–axis is as shown node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 in figure. Mass of the particle is 1kg and velocity at x = 0 is 4 m/s. At x = 8 m :– a(m/s2) U(J) 20 120 4 8 x(m) - 120 4 8 x(m) Column I Column II (A) Kinetic energy (p) 120J (B) Work done by conservative forces (q) 240 J (C) Total work done (r) 128 J (D) Work done by external forces (s) 112 J (t) –120 J 40 E
JEE-Physics 3 . A block of mass m lies on wedge of mass M. The wedge in turn lies on smooth horizontal surface. Friction is absent everywhere. The wedge block system is released from rest. All situation given in column–I are to be estimated in duration the block undergoes a vertical displacement 'h' starting from rest (assume the block to be still on the wedge, g is acceleration due to gravity). m M Column I Column II (p) Positive (A) Work done by normal reaction acting on the block is (q) Negative (B) Work done by normal reaction (r) Zero (exerted by block) acting on wedge is (s) Less than mgh in magnitude (C) The sum of work done by normal reaction on block and work done by normal reaction (exerted by block) on wedge is (D) Net work done by all forces on block is 4 . In vertical circular motion of a bob, match the entries of column–I with entries of column–II. Here v0 is the velocity of bob at lowest point & T is tension in string. Column – I (Speed at lowest point) Column–II (Possible situation) (A) v0 = 5g (p) Tlowest – Thighest = 6mg (B) v0 = g (q) String will slack for a finite time (C) v0 = 2 g (r) bob will oscillate (D) v0 = 3 g (s) bob will complete the circle ASSERTION – REASON 1 . Statement 1 : The work done in pushing a block is more than the work done in pulling the block in a rough surface. and Statement 2 : In the pushing condition more normal reaction increases the frictional force. (A) Statement–I is true, Statement–II is true ; Statement–II is correct explanation for Statement–I. (B) Statement–I is true, Statement–II is true ; Statement–II is NOT a correct explanation for statement–I (C) Statement–I is true, Statement–II is false (D) Statement–I is false, Statement–II is true 2. Statement 1 : One end of ideal massless spring is connected to fixed vertical wall and other end to a block of mass m initially at rest on smooth horizontal surface. The spring is initially in natural node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 length. Now a horizontal force F acts on block as shown. Then the maximum extension in spring is equal to maximum compression in spring. mF and Statement 2 : To compress or to expand an ideal unstretched spring by equal amount, same work is to done on spring. (A) Statement–I is true, Statement–II is true ; Statement–II is correct explanation for Statement–I. (B) Statement–I is true, Statement–II is true ; Statement–II is NOT a correct explanation for statement–I (C) Statement–I is true, Statement–II is false (D) Statement–I is false, Statement–II is true E 41
JEE-Physics 3 . Statement 1 : The work done by friction is always negative. and Statement 2 : If frictional force acts on a body its kinetic energy may decrease. (A) Statement–I is true, Statement–II is true ; Statement–II is correct explanation for Statement–I. (B) Statement–I is true, Statement–II is true ; Statement–II is NOT a correct explanation for statement–I (C) Statement–I is true, Statement–II is false (D) Statement–I is false, Statement–II is true 4 . Statement 1 : A particle is rotated in a vertical circle with the help of a string. Power produced by tension in the string on the particle is zero. and S t a t e m e n t 2 : Tension is always perpendicular to instantaneous velocity. (A) Statement–I is true, Statement–II is true ; Statement–II is correct explanation for Statement–I. (B) Statement–I is true, Statement–II is true ; Statement–II is NOT a correct explanation for statement–I (C) Statement–I is true, Statement–II is false (D) Statement–I is false, Statement–II is true 5 . S t a t e m e n t – 1 : A body can have energy without having momentum. and S t a t e m e n t – 2 : A body can have momentum without having mechanical energy. (A) Statement–I is true, Statement–II is true ; Statement–II is correct explanation for Statement–I. (B) Statement–I is true, Statement–II is true ; Statement–II is NOT a correct explanation for statement–I (C) Statement–I is true, Statement–II is false (D) Statement–I is false, Statement–II is true 6 . S t atem ent –1 : If the internal forces are conservative, the work done by the external force is equal to the change in mechanical energy. and S t a t e m e n t – 2 : Work done on a system by all the (external and internal) force is equal to the change in its kinetic energy and the change in the potential energy of a system corresponding to conservative internal forces is equal to negative of the work done by these forces. (A) Statement–I is true, Statement–II is true ; Statement–II is correct explanation for Statement–I. (B) Statement–I is true, Statement–II is true ; Statement–II is NOT a correct explanation for statement–I (C) Statement–I is true, Statement–II is false (D) Statement–I is false, Statement–II is true 7 . S t a t e m e n t – 1 : When a gas is allowed to expand, work done by gas is positive. node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 and S t a t e m e n t – 2 : In expansion of a gas the force due to gaseous pressure and displacement (of piston) are in the same direction. (A) Statement–I is true, Statement–II is true ; Statement–II is correct explanation for Statement–I. (B) Statement–I is true, Statement–II is true ; Statement–II is NOT a correct explanation for statement–I (C) Statement–I is true, Statement–II is false (D) Statement–I is false, Statement–II is true 8 . S t a t e m e n t – 1 : A body at rest may be in equilibrium. E and S t a t e m e n t – 2 : A body in equilibrium is at rest. (A) Statement–I is true, Statement–II is true ; Statement–II is correct explanation for Statement–I. (B) Statement–I is true, Statement–II is true ; Statement–II is NOT a correct explanation for statement–I (C) Statement–I is true, Statement–II is false (D) Statement–I is false, Statement–II is true 42
JEE-Physics COMPREHENSION QUESTIONS Comprehension # 1 A block of mass m is kept in an elevator which starts moving downward with an acceleration a as shown in figure. 0 The block is observed by two observers A and B for a time interval t . 0 a0 A B 1 . The observer B finds that the work done by gravity is 1 1 1 1 (A) mg2t 2 (B) – mg2t 2 (C) mgat 2 (D) – mgat 2 20 20 20 20 2 . The observer B finds that work done by normal reaction N is (A) zero (B) –Nat 2 Nat2 (D) None of these 0 (C) + (D) – mgat 2 0 3 . The observer B finds that work done by pseudo force is (A) zero (B) –ma2t (C) + ma2t 0 0 4 . According to observer B, the net work done on the block is 1 1 1 1 (A) – ma2t 2 (B) ma2t 2 (C) mgat 2 (D)– mgat 2 20 20 20 20 5 According to the observer A (B) the work done by normal reaction is zero (A) the work done by gravity is zero (D) all the above (C) the work done by pseudo force is zero Comprehension # 2 Two identical beads are attached to free ends of two identical springs of spring constant k (2 3 )mg . Initially both 3R node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 springs make an angle of 60° at the fixed point N. Normal length of each spring is 2R. Where R is the radius of smooth ring over which bead is sliding. Ring is placed on vertical plane and beads are at symmetry with respect to vertical line as diameter. N mm 1. Normal reaction on one of the bead at initial moment due to ring is E (A) mg/2 (B) 3 mg/2 (C) mg (D) Insufficient data 43
JEE-Physics 2 . Relative acceleration between two beads at the initial moment (A) g/2 vertically away from each other (B) g/2 horizontally towards each other (C) 2g/ 3 vertically away from each other (D) 2g/ 3 horizontally towards each other 3 . The speed of bead when spring is at normal length 2 3 gR 2 3 gR 2gR (D) 3gR (C) (A) (B) 3 3 3 4 . Choose the correct statement (A) Maximum angle made by spring after collision is same as that of initial moment (B) If the collision is perfectly inelastic, the total energy is conserved (C) If the collision is perfectly elastic, each bead undergoes SHM (D) Both Linear momentum and angular momentum with respect to centre of smooth ring are conserved only at the instant of collision. Comprehension # 3 The bob of simple pendulum of length is released from a point in the same horizontal line as the point of suspension (O) and at a distance from it. o N 1 . The velocity of the bob at the lowest point of the string will be (A) v 2g (B) v g (C) v 3g (D) v 2 g 2 . If the string is catched by a nail located vertically below the point of suspension and the bob just swings around a complete circle around the nail, then the distance of the nail from point of suspension. (A) (2/5) (B) (2/3) (C) (3/5) (D) (1/3) 3 . If the string of the pendulum is made of rubber then how much will it be stretched on reaching the bob at the lowest point. (A) 2mg/k (B) 3mg/k (C) 5 mg/k (D) mg/k Comprehension # 4 node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 A ball is released from point A as shown in figure. The ball leaves the track at B. All surfaces are smooth. A B 6m 2m 1 . Let h be the maximum height from ground reached by ball after leaving track at B. Then :– (A) h = 6m (B) h < 6 m (C) h > 6m (D) speed of ball at B will change if shape of track is changed keeping h and h constant AB 2 . If track makes an angle 30° with horizontal at B then maximum height attained by ball will be : (A) 3m (B) 4m (C) 4.5 m (D) 5m 44 E
JEE-Physics Comprehension # 5 In the figure shown upper block is given a velocity of 6 m/s and lower block 3 m/s. When relative motion between them is stopped. Rough 6m/s 1kg 2kg 3m/s Smooth 1 . (A) Work done by friction on upper block is negative (B) Work done by friction on both the blocks is positive (C) Work done by friction on lower block is negative (D) Work done by friction on both the blocks is negative 2 . (A) Work done by friction on upper block is – 10 J (B) Work done by friction on lower block is + 10 J (C) Net work done by friction is zero (D) All of the above Comprehension # 6 In a conservative force field we can find the radial component of force from the potential energy function by using dU F = – . Here, a positive force means repulsion and a negative force means attraction. From the given potential dr energy function U(r) we can find the equilibrium position where force is zero. We can also find the ionisation energy which is the work done to move the particle from a certain position to infinity. Let us consider a case in which a particle is bound to a certain point at a distance r from the centre of the force. The AB potential energy of the particle is : U(r) = r2 – r where r is the distance from the centre of the force and A and B are positive constants. 1 . The equilibrium distance is given by : A 2A 3A B (A) B (B) B (C) B (D) 2A 2 . The equilibrium is : (B) Unstable (C) Neutral (D) Cannot be predicted (A) Stable 3 . The work required to move the particle from equilibrium distance to infinity is : B 4B B2 4B2 (A) (B) (C) (D) 4A A 4A A 3B2 4 . If the total energy of the particle is E = – , and it is known that the motion is radial only then the velocity 16A is zero at : (here, r = equilibrium distance) 0 node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 (A) r0 (B) 2r0 (C) r (D) 2r0 3 3 0 5 Comprehension # 7 Ram and Shyam are two fast friends since childhood. Shyam neglected studies and now has no means to earn money other than a camel whereas Ram becomes an engineer. Now both are working in the same factory. Shyam uses camel to transport the load within the factory. E 45
JEE-Physics Due to low salary & degradation in health of camel, Shyam becomes worried and meet his friend Ram and discusses his problem. Ram collected some data & with some assumptions concluded the following: • The load used in each trip is 1000 kg and has friction coefficient µk = 0.1 and µs = 0.2. • Mass of camel is 500 kg. • Load is accelerated for first 50 m with constant acceleration, then it is pulled at a constant speed of 5m/s for 2 km and at last stopped with constant retardation in 50 m. • From biological data, the rate of consumption of energy of camel can be expressed as P = 18 × 103 v+ 104 J/s where P is the power and v is the velocity of the camel. After calculations on different issues Ram suggested proper food, speed of camel etc. to his friend. For the welfare of Shyam, Ram wrote a letter to the management to increase his salary. (Assuming that the camel exerts a horizontal force on the load) : 1 . Sign of work done by the camel on the load during parts of motion : accelerated motion, uniform motion and retarted motion respectively are : (A) +ve, +ve, +ve (B) +ve, +ve, –ve (C) +ve, zero, –ve (D) +ve, zero, +ve 2 . The ratio of magnitude of work done by camel on the load during accelerated motion to retarded motion is– (A) 3 : 5 (B) 2.2 : 1 (C) 1 : 1 (D) 5 : 3 3 . Maximum power transmitted by the camel to load is– (A) 6250 J/s (B) 5000 J/s (C) 105 J/s (D) 1250 J/s 4 . The ratio of the energy consumed of the camel during uniform motion for the two cases when it moves with speed 5 m/s to the case when it moves with 10 m/s. 19 19 10 20 (A) 20 (B) 10 (C) 19 (D) 19 (D) 4.22 × 106 J 5 . The total energy consumed of the camel during the trip of 2100 m is– (A) 2.1 × 106 J (B) 4.22 × 107 J (C) 2.22 × 104 J Comprehension # 8 node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 A block of mass m moving with a velocity v0 on a smooth horizontal surface strikes and compresses a spring of stiffness k till mass comes to rest as shown in the figure. This phenomenon is observed by two observers : v0 k m A : Standing on the horizontal surface; B : Standing on the block (B) – 1 m v 2 1 . To an observer A, the work done by spring force is 2 0 (A) negative but nothing can be said about its magnitude (D) + 1 m v 2 2 0 (C) positive but nothing can be said about its magnitude E 46
JEE-Physics 2 . To an observer A, the work done by the normal reaction N between the block and the spring on the block is (A) zero (B) – 1 m v 2 (C) + 1 m v 2 (D) None of these 2 0 2 0 3 . To an observing A, the net work done on the block is (A) – m v 2 (B) + m v 2 (C) 1 m v 2 (D) zero 0 0 2 0 4 . According to the observer A (A) the kinetic energy of the block is converted into the potential energy of the spring (B) the mechanical energy of the spring–mass system is conserved (C) the block loses its kinetic energy because of the negative work done by the conservative force of spring (D) all the above 5 . To an observer B, when the block is compressing the spring (A) velocity of the block is decreasing (B) retardation of the block is increasing (C) kinetic energy of the block is zero (D) all the above 6 . According to observer B, the potential energy of the spring increases (A) due to the positive work done by pseudo force (B) due to the negative work done by pseudo force (C) due to the decrease in the kinetic energy of the block (D) all the above Comprehension # 9 A small sphere of mass m suspended by a thread is first taken aside so that the thread forms the right angle with the vertical and then released, then 1 . The total acceleration of the sphere and the thread tension as function of , the angle of deflection of the thread from the vertical will be (A) g 1 3 cos2 , T = 3 mg cos (B) gcos, T = 3mg cos (C) g 1 3 sin2 , T = 5mg cos (D) gsin, T = 5 mg cos node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 2 . The thread tension at the moment when the vertical component of the sphere's velocity is maximum will be (A) mg (B) mg 2 (C) mg 3 mg (D) 3 3. The angle between the thread and the vertical at the moment when the total acceleration vector of the sphere is directed horizontally will be E 1 1 (C) sin 1 (D) sin 1 (A) cos = 3 (B) cos 3 2 3 47
JEE-Physics Comprehension # 10 A vertical frictionless semicircular track of radius R is fixed on the edge of movable trolley. Initially the system is at rest and a mass m is kept at the top of the track. The trolley starts moving to the right with a uniform horizontal acceleration 2g a = . The mass slides down the track, eventually losing contact with it and dropping to the floor h below the trolley. 9 m R 4R 2g h= 3 a= 9 1 . The angle with vertical, at which it losses contact with the trolley is (A) 37° (B) 53° 2 (D) co s 1 2 (C) cos–1 3 2 3 2 . The height at which mass m losing contact is 4 17 (C) R 32 (A) R (B) R (D) R 5 15 15 3 . The time taken by the mass to drop on the floor, after losing contact is 2R 2R (C) 2 2R (D) Can't be determined (A) 3g (B) g 3g MISCELLANEOUS TYPE QUESTION ANSWER KEY EXERCISE –3 True / False 1. F 2. F 3. F 4. T Fill in the Blanks 3 mg h s 4 1 Match the Column 1. zero 2. Mgh 3. 4. 9.8 5. sin–1 5 or 2 tan–1 2 4 s 1. (A)–t, (B) –p, (C)–s, (D)–q 2. (A)–r, (B)–q, (C)–p, (D)–t 3. (A)–q, s (B)–p,s (C)–r,s (D)–p,s 4. (A)–p,s (B) –r, (C)–q, (D)–p,s Assertion – Reason 1.A 2.D 3.D 4.A 5. B node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 6.A 7.A 8.C Comprehension Based Questions Comprehension #1 : 1. C 2. D 3. A 4.B 5. D 3. C 4. D Comprehension #2 : 1. C 2. D 3. B Comprehension #3: 1. A 2. C 3. C 3. A Comprehension #4 : 1. B 2. A 3. C 3. A Comprehension #5 : 1. A 2. A 3. B Comprehension #6 : 1. B 2. A 4. B 4. D Comprehension #7 : 1. A 2. D 4. D 5. B 5. C Comprehension #8 : 1. B 2. B 6. B Comprehension #9 : 1. A 2. C E Comprehension #10 : 1. A 2. D 48
JEE-Physics EXERCISE–04 (A) CONCEPTUAL SUBJECTIVE EXERCISE 1 . In the figure (A) and (B) AC, DG and GF are fixed inclined planes, BC = EF = x and AB = DE = y. A small block of mass M is released from the point A. It slides down AC and reaches C with a speed v . The same block is released from rest from the point D. It slides down DGF and reaches the point C F with speed v . The coefficients of kinetic friction between the block and both the surfaces AC and DGF F are µ. Calculate v and v . CF AD G B CE F (a) (b) 2 . A body of mass 2 kg is being dragged with a uniform velocity of 2 m/s on a rough horizontal plane. The coefficient of friction between the body and the surface is 0.20, J = 4.2 J/cal and g = 9.8 m/s2. Calculate the amount of heat generated in 5 s. 3 . Two blocks A and B are connected to each other by a string and a spring; the string pass overs a frictionless pulley as shown in the figure. Block B slides over the horizontal top surface of a stationary block C and the block A slides along the vertical side of C, both with the same uniform speed. The coefficient of friction between the surface and the block is 0.2. Force constant of the spring is 1960 N/m. If mass of block A is 2 kg, calculate the mass of block B and the energy stored in the spring. B C A node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 4 . A particle is moving in x direction, under the influence of force F = sinx. Find the work done by another external agent in slowly moving a particle from x=0 to x = 0.5m. 5 . A particle of mass 5 kg is free to slide on a smooth ring of radius r=20 cm fixed in a vertical plane. The particle is attached to one end of a spring whose other end is fixed to the top point O of the ring. Initially the particle is at rest at a point A of the ring such that OCA = 600, C being the centre of the ring. The natural length of the spring is also equal to r = 20 cm. After the particle is released and slides down the ring the contact force between the particle & the ring becomes zero when it reaches the lowest position B. Determine the force constant of the spring. O A 600 C 6. B E A particle of mass m moves along a circle of radius R with a normal acceleration varying with time as an bt2 , where b is a constant. Find the time dependence of the power developed by all the forces acting on the par- ticle, and the mean value of this power averaged over the first t seconds after the beginning of motion. 49
JEE-Physics 7 . A light string ABCDE whose mid point is C passes through smooth rings B and D, which are fixed in a horizontal plane distance 2a apart. To each of the points A, C and E is attached a mass m. Initially C is held at rest at O (mid point BD) and is then set free. What is the distance OC when C comes to instantaneous rest? B 2a \\\\\\\\\\\\\\\\\\ D A E C \\\\\\\\\\\\\\\\\\ F Ix2 GH KJ8 . The potential energy of a particle of mass 1kg free to move along x-axis is given by V(x) = 2 x joule. If total mechanical energy of the particle is 2J, then find the maximum speed of the particle. 9 . For what minimum value of m1 the block of mass m will just leave the contact with surface ? K m1 m 1 0 . A particle of mass 3 kg is rotating in a circle of radius 1m such that the angle rotated by its radius is given by =3 (t + sint). Find the net force acting on the particle when t = / 2 . 1 1 . The bob of a simple pendulum of length is released from point P.What is the angle made by the direction of net acceleration of the bob with the string at point Q. P node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 h Q 1 2 . A ball of mass 1 kg is released from position, A inside a fixed wedge with a hemispherical cut of radius 0.5 m as shown in the figure. Find the force exerted by the vertical wall OM on wedge, when the ball is in position B. (neglect friction ever y where). (Take g = 10 m/s2) M C A 600 B ON 50 E
JEE-Physics 1 3 . For a particle rotating in a vertical circle with uniform speed, the maximum and minimum tension in the string are in the ratio 5:3. If the radius of vertical circle is 2m, then find the speed of revolving body. 1 4 . In the shown arrangement a bob of mass 'm' is suspended by means of a string connected to peg P. If the bob is given a horizontal velocity u having magnitude 3g , find the minimum speed of the bob in subsequent motion. P node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 u mg R 1 5 . A bead of mass m is tied at one end of a spring of spring constant R and unstretched length 2 and other end to fixed point O. The smooth semicircular wire frame is fixed in vertical plane. Find the normal reaction between bead and wire just before it reaches the lowest point. Am Rk R/2 O B 1 6 . A small block of mass m is projected horizontally from the top of the smooth hemisphere of radius r with speed u as shown. For values of u u0 , it does not slide on the hemisphere (i.e. leaves the surface at the top itself). u r O (i) For u = 2u , it lands at point P on the ground Find OP. 0 (ii) For u=u /3, find the height from the ground at which it leaves the hemisphere. 0 (iii) Find its net acceleration at the instant it leaves the hemisphere. CONCEPTUAL SUBJECTIVE EXERCISE ANSWER KEY EXERCISE–4(A) 1. v = v = 2 gy gx 2. 9.52 cal 3. 10 kg, 0.098 J 4.–1J CF mRbt 5. 500 N/m 6. mRbt, 4a 8. 5 ms–1 2 7. 15 3 m 9. m = 10. 9 10N 3 12. N g 1 sin 2 12 14. × 11. tan–1 2h 33 13. 4 5 ms–1 15. 6mg 19r 16. (i) 22r, (ii) h= 27 (iii) g E 51
JEE-Physics EXERCISE–04 (B) BRAIN STORMING SUBJECTIVE EXERCISE 1 . A particle of mass m is hanging with the help of an elastic string of unstretched length a and force constant mg a . The other end is fixed to a peg on vertical wall. String is given an additional extension of 2a in vertical downward direction by pulling the mass and released from rest. Find the maximum height reached by it during its subsequent motion above point of release. (Neglect interaction with peg if any) 2 . A 0.5 kg block slides from the point A (see figure) on a horizontal track with an initial speed of 3 m/s towards a weightless horizontal spring of length 1 m and force constant 2 N/m. The part AB of the track is frictionless and the part BC has the coefficients of static and A BD C kinetic friction as 0.22 and 0.2 respectively. If the distance AB and BD are 2m and 2.14 m respectively, find the total distance through which the block moves before it comes to rest completely. (Take g = 10 m/s2). 3 . A body of mass 2 kg is moving under the influence of a central force whose potential energy is given by U(r) = 2r3 J. If the body is moving in a circular orbit of radius 5m, then find its energy. 4 . A square plate is firmly attached to a frictionless horizontal plane. Da C One end of a taut cord is attached to point A of the plate and the a other end is attached to a sphere of mass m. In the process, the 4a A B cord gets wrapped around the plate. The sphere is given an initial velocity v on the horizontal plane perpendicular to the cord which 900 v0 0 causes it to make a complete circle of the plate and return to point A. Find the velocity of the sphere when it hits point A again after moving in a circle on the horizontal plane. Also find the time taken by the sphere to complete the circle. 5 . A small bead of mass m is free to slide on a fixed smooth vertical wire, as indicated in the diagram. One end of a light elastic string, of unstretched length a and force constant 2mg/a is attached to B. The OaR aA string passes through a smooth fixed ring R and the other end of the string is attached to the fixed point A, AR being horizontal. The point O on the wire is at same horizontal level as R, and AR = RO =a. B (i) In the equilibrium position, find OB. (ii) The bead B is raised to a point C of the wire above O, where OC =a, and is released from rest. Find the speed of the bead as it passes O, and find the greatest depth below O of the bead in the subsequent motion. 6 . A ring of mass m slides on a smooth vertical rod. A light string m a node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 is attached to the ring and is passing over a smooth peg distant =0 M a from the rod, and at the other end of the string is a mass M (M>m). The ring is held on a level with the peg and released. Show that M 2mMa it first comes to rest after falling a distance M2 m2 m 7 . A string, with one end fixed on a rigid wall, passing over a fixed E frictionless pulley at a distance of 2m from the wall, has a point mass M = 2kg attached to it at a distance of 1m from the wall. A mass m = 0.5 kg attached at the free end is held at rest so that the string is horizontal between the wall and the pulley and vertical beyond the pulley. What will be the speed with which the mass M will hit the wall when the mass m is released? 52
JEE-Physics 8 . In figure two identical springs, each with a relaxed length of 50 B 6m cm and a spring constant of 500 N/m, are connected by a short A 5m cord of length 10 cm. The upper string is attached to the ceiling, a box that weighs 100N hangs from the lower spring. Two additional O cords, each 85 cm long, are also tied to the assembly; they are L/8 A limp (i.e. slack). (i) If the short cord is cut, so that the box then hangs from the springs L and the two longer cords, does the box move up or down? uB (ii) How far does the box move before coming to rest again? h W m 9 . 'A' block of mass m is held at rest on a smooth horizontal floor. k B A light frictionless, small pulley is fixed at a height of 6 m from the floor. A light inextensible string of length 16m, connected with A passes over the pulley and another identical block B is hung from the string. Initial height of B is 5m from the floor as shown in figure. When the system is released from rest, B starts to move vertically downwards and A slides on the floor towards right. (i) If at an instant string makes an angle with horizontal, calculate relation between velocity u of A and v of B (ii) Calculate v when B strikes the floor. 1 0 . A particle is suspended vertically from a point O by an inextensible L massless string of length L. A vertical line AB is at a distance 8 from O as shown in figure. The object is given a horizontal velocity u. At some point, its motion ceases to be circular and eventually the object passes through the line AB. At the instant of crossing AB, its velocity is horizontal. Find u. 1 1 . In the figure shown the pulley is smooth. The spring and the string are light. The block 'B' slides down from the along the fixed rough wedge of inclination . Assuming that the block reaches the end of the wedge. Find the speed of the block at the end. Take the coefficient of friction between the block and the wedge to be µ and the spring was relaxed when the blockwas released from the top of the wedge. node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 1 2 . As shown in the figure a person is pulling a mass 'm' from ground on a fixed rough hemispherical surface upto the top of the hemisphere with the help of a light inextensible string. Find the work done by tension in the string if radius of hemisphere is R and friction co–efficient is µ. Assume that the block is pulled with negligible velocity. m E 53
JEE-Physics 1 3 . Starting from rest, a particle rotates in a circle of radius R = 2 m with an angular acceleration = rad/sec2. 4 Calculate the magnitude of average velocity of the particle over the time it rotates quarter circle. 1 4 . A stone weighing 0.5 kg tied to a rope of length 0.5m revolves along a circular path in a vertical plane. The tension of the rope at the bottom point of the circle is 45 newtons. To what height will the stone rise if the rope breaks at moment the velocity is directed upwards? (g=10 m/s2) BRAIN STORMING SUBJECTIVE EXERCISE ANSWER KEY EXERCISE–4(B) 9a 5 a node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 1. 2 . 4.24 m 3. 625 J 4. v=v , v0 2 7 . 3.71 ms–1 0 a 8 . up, 20 cm 5 . (i) 2 (ii) 2 ga , 2a 40 gL 2 3 3 2 1 k h 2 m g h ms–1 1 0 . u = m mgh 2 sin cot 9 . u = v sec , v = 41 2 1 1 . v= 1 2 . mgR (1+ ) 13. 1 ms–1 14. 1.5 m 54 E
JEE-Physics EXERCISE–05 [A] PREVIOUS YEAR QUESTIONS 1 . A spring of force constant 800 N/m has an extension of 5 cm. The work done in extending it from 5 cm to 15 cm is- [AIEEE - 2002] (1) 16 J (2) 8 J (3) 32 J (4) 24 J 2 . A spring of spring constant 5 × 103 N/m is stretched initially by 5 cm from the unstretched position. Then the work required to stretch it further by another 5 cm is- [AIEEE - 2003] (1) 12.50 N-m (2) 18.75 N-m (3) 25.00 N-m (4) 6.25 N-m 3 . A body is moved along a straight line by a machine delivering a constant power. The distance moved by the body in time t is proportional to- [AIEEE - 2003] (1) t3/4 (2) t3/2 (3) t1/4 (4) t1/2 4 . A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement x is proportional to- [AIEEE - 2004] (1) x2 (2) ex (3) x (4) logex 5 . A body of mass m accelerates uniformly from rest to v1 in time t1. The instantaneous power delivered to the body as a function of time t is- [AIEEE - 2004] m v1 t (2) mv12 t (3) m v1 t2 (4) mv12 t (1) t1 t12 t1 t1 6 . A uniform chain of length 2 m is kept on a table such that a length of 60 cm hangs freely from the edge of the table. The total mass of the chain is 4 kg. What is the work done in pulling the entire chain on the table [AIEEE - 2004] (1) 7.2 J (2) 3.6 J (3) 120 J (4) 1200 J 7 . A bullet fired into a fixed target loses half of its velocity after penetrating 3cm. How much further it will penetrate before coming to rest, assuming that it faces constant resistance to motion ? [AIEEE - 2005] (1) 3.0 cm (2) 2.0 cm (3) 1.5 cm (4) 1.0 cm 8 . The block of mass M moving on the frictionless horizontal surface collides with the spring of spring constant k and compresses it by length L. The maximum momentum of the block after collision is- [AIEEE - 2005] M node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 (1) Mk L (2) kL2 (3) zero (4) ML2 2M k 9 . A particle of mass 100 g is thrown vertically upwards with a speed of 5 m/s. The work done by the force of gravity during the time the particle goes up is- [AIEEE - 2006] (1) – 0.5 J (2) –1.25 J (3) 1.25 J (4) 0.5 J x4 x2 1 0 . The potential energy of a 1 kg particle free to move along the x-axis is given by V(x) = J. The total 4 2 mechanical energy of the particle is 2 J. Then, the maximum speed (in m/s) is- [AIEEE - 2006] (1) 3/ 2 (2) 2 (3) 1/ 2 (4) 2 E 55
JEE-Physics 1 1 . A mass of M kg is suspended by a weightless string. The horizontal force that is required to displace it until the string makes an angle of 45° with the initial vertical direction is- [AIEEE - 2006] (1) Mg( 2 + 1) (2) Mg 2 Mg (4) Mg ( 2 – 1) (3) 2 1 2 . A 2 kg block slides on a horizontal floor with a speed of 4 m/s. It strikes an uncompressed spring, and compresses it till the block is motionless. The kinetic friction force is 15 N and spring constant is 10, 000 N/m. The spring compresses by :- [AIEEE - 2007] (1) 5.5 cm (2) 2.5 cm (3) 11.0 cm (4) 8.5 cm 1 3 . An athlete in the Olympic games covers distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range :- [AIEEE - 2008] (1) 200 J – 500 J (2) 2 × 105 J – 3 × 105 J (3) 2 × 104 J – 3 × 104 J (4) 2 × 103 J – 5 × 103 J 1 4 . The potential energy function for the force between two atoms in a diatomic molecule is approximately given ab by U(x) = x12 – x6 , where a and b are constant and x is the distance between the atoms. if the dissociation energy of the molecule is D = [U(x = ) – Uat equilibrium], D is : [AIEEE - 2010] b2 b2 b2 b2 (1) (2) (3) (4) 6a 2a 12a 4a 1 5 . At time t = 0s particle starts moving along the x-axis. If its kinetic energy increases uniformly with time 't', the net force acting on it must be proportional to :- [AIEEE - 2011] (1) t (2) constant (3) t 1 (4) t 16. This question has Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. If two springs S1 and S2 of force constants k1 and k2, respectively, are stretched by the same force, it is found that more work is done on spring S1 than on spring S2. [AIEEE - 2012] Statement-1: If stretched by the same amount, work done on S1, will be more than that on S2 Statement-2 : k1 < k2. (1) Statement-1 is true, Statement-2 is true and Statement-2 is not the correct explanation of Statement-1. (2) Statement-1 is false, Statement-2 is true (3) Statement-1 is true, Statement-2 is false (4) Statement-1 is true, Statement-2 is true and Statement-2 is the correct explanation of statement-1. ANSWER-KEY node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 Que. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 E Ans. 2 2 2 1 2 2 4 1 2 1 4 1 4 4 4 2 56
JEE-Physics EXERCISE–05 [B] PREVIOUS YEAR QUESTIONS MCQs with one correct answer 1 . A wind–powered generator converts wind energy into electric energy. Assume that the generator converts a fixed fraction of the wind energy intercepted by its blades into electrical energy. For wind speed v, the electrical power output will be proportional to :– [IIT-JEE 2000] (A) v (B) v2 (C) v3 (D) v4 2 . A particle, which is constrained to move along x–axis, is subjected to a force in the same direction which varies with the distance x of the particle from the origin as F(x) = –kx + ax3. Here, k and a are positive constant. For x 0, the functional form of the potential energy U(x) of the particle is :–[IIT-JEE 2002] U(x) U(x) U(x) U(x) X (A) X (B) X (C) X (D) 3 . An ideal spring with spring–constant k is hung from the ceiling and a block of mass M is attached to its lower end. The mass is released with the spring initially unstretched. Then the maximum extension in the spring is :– [IIT-JEE 2002] 4Mg 2Mg Mg Mg (A) (B) (C) (D) k k k 2k 4 . If W , W and W represent the work done in moving a particle from A to B 12 3 B along three different paths 1, 2 and 3 respectively (as shown) in the gravitational 1 2 3 field of a point mass m. Find the correct relation between W , W and W :– 12 3 (A) W > W > W (B) W = W = W A 123 123 (C) W < W < W (D) W > W > W [IIT-JEE 2003] 123 213 5 . A particle is placed at the origin and a force F = kx is acting on it (where k is positive constant). If U(0)=0, the graph of U(x) versus x will be (where U is the potential energy function) :– [IIT-JEE 2004] U(x) U(x) U(x) (A) x (B) U(x) x (C) x (D) x node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 6 . A bob of mass M is suspended by a massless string of length L. The horizontal velocity v at position A is just sufficient to make it reach the point B. The angle at which the speed of the bob is half of that A, is [IIT-JEE 2008] B V L A 3 (D) 3 (A) (B) (C) 4 4 42 24 E 57
JEE-Physics 7 . A ball of mass (m) 0.5 kg is attached to the end of a string having length (L) 0.5 m. The ball is rotated on a horizontal circular path about vertical axis. The maximum tension that the string can bear is 324 N. The maximum possible value of angular velocity of ball (in radian/s) is [IIT-JEE-2011] L m (A) 9 (B) 18 (C) 27 (D) 36 8 . Consider a disc rotating in the horizontal plane with a constant angular speed about its centre O. The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles P and Q are simultaneously projected at an angle towards R. The velocity of projection is in the y-z plane and is same for both pebbles with respect 1 to the disc. Assume that (i) they land back on the disc before the disc has completed 8 rotation, (ii) their range is less than half the disc radius, and (iii) remains constant throughout. Then [IIT-JEE-2012] R y Q x O P (A) P lands in the shaded region and Q in the unshaded region. (B) P lands in the unshaded region and Q in the shaded region. (C) Both P and Q land in the unshaded region. (D) Both P and Q land in the shaded region. MCQs with one or more than one correct answer 9 . A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle. The motion of the particle takes place in a plane. It follows that :– [IIT-JEE 1987] (A) its velocity is constant (B) its acceleration is constant (C) its kinetic energy is constant (D) it moves in a circular path ASSERTION & REASON 1 0 . Statement–I : A block of mass m starts moving on a rough horizontal surface with a velocity v. It stops node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65 due to friction between the block and the surface after moving through a certain distance. The surface is now tilted to an angle of 30° with the horizontal and the same block is made to go up on the surface with the same initial velocity v. The decrease in the mechanical energy in the second situation is smaller than that in the first situation. [IIT-JEE 2007] and Statement–II : The coefficient of friction between the block and the surface decreases with the increase in the angle of inclination. (A) statement–I is true, statement–II is true; statement–II is a correct explanation for statement–I (B) statement–I is true, statement–II is true, statement–II is NOT a correct explanation for statement–I (C) statement–I is true, statement–II is false (D) statement–I is false, statement–II is true 58 E
JEE-Physics SUBJECTIVE QUESTIONS 1 1 . A spherical ball of mass m is kept at the highest point in the space between two fixed, concentric spheres A and B (see figure). The smaller sphere A has a radius R and the space between the two spheres has a width d. The ball has a diameter very slightly less then d. All surface are frictionless. Then ball is given a gentle push (towards the right in the figure). The angle made by the radius vector of the ball with the upward vertical is denoted by . [IIT-JEE 2002] Sphere B dRO Sphere A (i) Express the total normal reaction force exerted by the spheres on the ball as a function of angle . (ii) Let N and N denote the magnitudes of the normal reaction forces on the ball exerted by the spheres AB A and B, respectively. Sketch the variations of N and N as function of cos in the range 0 A B by drawing two separate graphs in your answer book, taking cos on the horizontal axis. 1 2 . A light inextensible string that goes over a smooth fixed pulley as shown in the figure connects two blocks of masses 0.36 kg and 0.72 kg. Taking g = 10 m/s2, find the work done (in joules) by the string on the block of mass 0.36 kg during the first second after the system is released from rest. [IIT-JEE 2009] 1 3 . A block of mass 0.18 kg is attached to a spring of force-constant 2 N/m. The coefficient of friction between the block and the floor is 0.1. Initially the block is at rest and the spring is un-stretched. An impulse is given to the block as shown in the figure. The block slides a distance of 0.06 m and comes to rest for the first time. The initial velocity of the block in m/s is V= N/10. Then N is [IIT-JEE-2011] node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Work Power Energy\\Eng\\Exercise.p65ANSWER KEY \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ JEE-[ADVANCED] : PREVIOUS YEAR QUESTIONS EXERCISE –5 [B] MCQ's with one correct answer 1. (C) 2. (D) 3. (B) 4. (B) 5. (A) 6. (D) 7. (D) 8. (C) MCQ's with one or more than 9. (C, D) one correct answers Assertion – Reason 10. (C) Subjective Questions 2 E 11. (i) N = mg(3cos – 2) (ii) For cos–1 3 , N = 0, B 2 N = mg(3cos – 2) and for cos–1 3 , A N = 0, N = mg(2–3cos) A B 12. 8J 13. 4 59
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