JEE-Physics 4 6 . The real time variation of input signals A and B are 4 8 . The following figure shows a logic gate circuit with as shown below. If the inputs are fed into NAND two inputs A and B and output C. The coltage gate, then select the output signal from the following :- waveforms of A, B and C are as shown in second figure given below :- A A I Y At BB AI Logic gate C B t circuit I B (i) C t YY (ii) (1) (2) The iogic circuit gate is : 0 2 4 6 8 t(s) 0 2 4 6 8 t(s) YY (1) OR gate (2) AND gate (3) (4) 0 2 4 6 8 t(s) 0 2 4 6 8 t(s) (3) NAND gate (4) NOR gate 4 7 . The time variations of signals are given as in A, B 4 9 . Select the outputs Y of the combination of gates and C. Point out the true statement from the shown below for inputs A = 1, B = 0 ; A = 1, following :- B = 1 and A = 0, B = 0 respectively :- e A e BY 1.0 t 1.0 B t 0 0 A e 1.0 0C t (1) A, B and C are analogue signals (1) (0 1 1) (2) (0 0 1) (3) (1 0 1) (4) (1 1 1) (2) A and C are analogue, but B is a digital signal (5) (1 0 0) (3) A and C are digital, but B is analogue signal E:\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Electronics\\Eng\\4.Exercise.p65 (4) A, B and C are digital signal BRAIN TEASERES ANSWER-KEY EXERCISE-II Que. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Ans. 1 4 2 4 4 1 1 2 2 3 2 2 2,3 4 1 3 2 3 1 3 Que. 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Ans. 3 3 1 1 4 1 1 4 3 3 1 4 4 3 1 1 2 4 2 3 Que. 41 42 43 44 45 46 47 48 49 Ans. 3 1 1 4 3 2 2 2 5 E 57
JEE-Physics ASSERTION-REASON TYPE QUESTIONS EXERCISE-III 1 . Assertion :- Microwave communication is 8 . Assertion : The television broadcasting becomes weaker with increasing distance. preferred over optical communication. Reason : T he power transmit ted from T.V. Reason :- Microwaves provide large number of transmitter varies inversely as the distance of the channels and band width compared to optical receiver. signals. (1) A (2) B (3) C (4) D (1) A (2) B (3) C (4) D 9 . Assertion : Microwave propagation is better than 2 . Assertion : Diode lasers are used as optical sources the sky wave propagation. in optical communication. Reason : Microwaves have frequencies 100 to Reason : Diode lasers consume less energy. 300 GHz, which have very good directional properties. (1) A (2) B (3) C (4) D (1) A (2) B (3) C (4) D 3 . Assertion : Television signals are received through 1 0 . Assertion : Semiconductors do not obey Ohm's law. sky-wave propagation. Reason : Electric current is determined by the rate Reason : The ionosphere reflects electromagnetic of flow of charge carries. waves of frequencies greater than a certain critical (1) A (2) B (3) C (4) D frequency. 1 1 . Assertion : Germanium is preferred over silicon (1) A (2) B (3) C (4) D for making semicondictor devices. 4 . Assertion:– Electromagnetic waves with Reason : Energy gap for Ge is more than that frequencies smaller than the critical frequency of of Si. ionosphere cannot be used for communication using (1) A (2) B (3) C (4) D sky wave propagation. 1 2 . Assertion : A p-n junction cannot be used at ultra Reason:– The refractive index of the ionosphere high frequencies. becomes very high for frequencies higher than the Reason : Capacitative reactance of a p-n critical frequency. junction increases with increasing frequency. (1) A (2) B (3) C (4) D (1) A (2) B (3) C (4) D 5 . Assertion:– In optical fibre, the diameter of the 1 3 . Assertion : A p-n junction with reverse bias can be used as a photodiode to measure light intensity. core is kept small. Reason : In a reverse bias condition the current Reason:– This smaller diameter of the core ensures is small but it is more sensitive to changes in incident that the fibre should have incident angle more than light intensity. the critical angle required for total internal (1) A (2) B (3) C (4) D reflection. 1 4 . Assertion : NAND or NOR gates are called digital (1) A (2) B (3) C (4) D building blocks. 6 . Assertion : The electromagnetic wave of shorter Reason : The repeated use of NAND or NOR wavelength can travel longer distances on earth's gates can produce all the basic or complex gates. surface than those of longer wavelength. (1) A (2) B (3) C (4) D Reason : Shorter the wavelength, the larger is the 1 5 . Assertion : Two p-n junction diodes placed back velocity of wave propagation. to back, will work as an n-p-n transistor. (1) A (2) B (3) C (4) D Reason : The p-region of two p-n junction 7 . Assertion: The surface wave propagation is used diodes back to back will form the base of n-p-n E:\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Electronics\\Eng\\4.Exercise.p65 for medium wave band and for television transistor. broadcasting. (1) A (2) B (3) C (4) D Reason : The surface waves travel directly from 1 6 . Assertion : When base region has larger width, transmitting antenna to receiver antenna through the collector current decreases. atmosphere. Reason : In transistor, sum of base current and (1) A (2) B (3) C (4) D collector current is equal to emitter current. (1) A (2) B (3) C (4) D 58 E
1 7 . Assertion : For faster action, n-p-n transistor is JEE-Physics used. 2 4 . Assertion : For the same antenna length, power radiated by short wavelength signals would be large. Reason : In n-p-n transistor, the mobility of 1 Reason : Because power radiated 2 . majority charge carries is more. (1) A (2) B (3) C (4) D 1 8 . Assertion : To be used as amplifier, the transistor (1) A (2) B (3) C (4) D in the common emitter configuration is preferred 2 5 . Assertion : The electrical conductivity of earth's to the common base configuration. atmosphere increases with altitude. Reason : In the common emitter, the signal is Reason : The high energy particles (-rays and applied between emitter and base. cosmic rays) coming from outer space while entering (1) A (2) B (3) C (4) D earth's atmosphere cause ionization of the atoms 19 . Assertion : Many channels get allowed when of the gases present in the atmosphere. transmission frequency is high. (1) A (2) B (3) C (4) D Reason : At high frequencies, bandwidth is high. 2 6 . Assertion : Optical communication system is more (1) A (2) B (3) C (4) D economical than other systems of communications. 20 . Assertion : Optical fibres are free from Reason : The information carrying capacity of electromagnetic disturbances. a communication system is directly propotional to Reason : Optical fibres are electrically its bandwidth. insulated. (1) A (2) B (3) C (4) D (1) A (2) B (3) C (4) D 2 7 . Assertion : A communication satellite is essentially 21 . Assertion : Diode lasers are used as optical a repeater in space. sources in optical communication. Reason : It reflects the signals from transmitter Reason : Diode lasers consume less energy. to receiver. (1) A (2) B (3) C (4) D (1) A (2) B (3) C (4) D 22 . Assertion : Electromagnetic waves with frequencies smaller than the critical frequency of ionosphere cannot be used for communication using sky wave propagation. Reason : The refractive index of the ionosphere becomes very high for frequencies higher than the crictical freqeuncy. (1) A (2) B (3) C (4) D 2 3 . Assertion : In optical fibre, the diameter of the core is kept small. Reason : This smaller diameter of the core ensures that the fibre should have incident angle more than the critical angle required for total internal reflection. (1) A (2) B (3) C (4) D Que. 1 2 3 4 5 6 7E:\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Electronics\\Eng\\4.Exercise.p65 8 9 10 11 12 13 14 15 16 17 18 19 20 Ans. 4 2 4 4 1 3 4 3 124411121211 Que. 21 22 23 24 25 26 27 59 Ans. 2 4 1 1 1 1 1 E
JEE-Physics PREVIOUS YEARS' QUESTIONS EXERCISE-IV 7 . Consider telecommunication through optical fibres. 1 . At absolute zero, Si acts as- Which of the following statements is not true ? [AIEEE - 2002] (1) non-metal (2) metal [AIEEE - 2003] (3) insulator (4) none of these (1) Optical fibres can be of graded refractive index (2) Optical fibres are subjected to electromagnetic 2 . The energy band gap is maximum in- interference from outside [AIEEE - 2002] (3) Optical fibres have extremely low transmission (1) metals (2) superconductors loss (3) insulators (4) semiconductors (4) Optical fibres may have homogeneous core with a suitable cladding 3 . The part of a transistor which is most heavily doped to produce large number of majority carriers is- 8 . When npn transistor is used as an amplifier- [AIEEE - 2002] [AIEEE - 2004] (1) emitter (1) electrons move from base to collector (2) base (2) holes move from emitter to base (3) collector (3) electrons move from collector to base (4) can be any of the above three (4) holes move from base to emitter 9. For a transistor amplifier in common emitter 4 . A strip of copper and another of germanium are configuration for load impedance of 1 k (hfe = 50 and hoe = 25 µ A/V), the current gain is- cooled from room temperature to 80 K. The [AIEEE - 2004] resistance of- [AIEEE - 2003] (1) – 5.2 (2) – 15.7 (1) each of these decreases (3) – 24.8 (4) – 48.78 (2) copper strip increases and that of germanium decreases 1 0 . A piece of copper and another of germanium are (3) copper strip decreases and that of germanium cooled from room temperature to 77 K, the increases (4) each of the above increases resistance of- [AIEEE - 2004] (1) each of them increases (2) each of them decreases 5 . The difference in the variation of resistance with (3) copper decreases and germanium increases temperature in a metal and a semiconductor arises (4) copper increases and germanium decreases essentially due to the difference in the- 1 1 . When p-n junction diode is forward biased, then- [AIEEE - 2004] [AIEEE - 2003] (1) the depletion region is reduced and barrier (1) crystal structure height is increased (2) variation of the number of charge carriers with (2) the depletion region is widened and barrier temperature height is reduced (3) type of bonding (3) both the depletion region and barrier height E:\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Electronics\\Eng\\4.Exercise.p65 (4) variation of scattering mechanism with are reduced temperature (4) both the depletion region and barrier height are increased 6 . In the middle of the depletion layer of reverse 1 2 . The electrical conductivity of a semiconductor biased p-n junction, the- [AIEEE - 2003] increases when electromagnetic radiation of (1) electric field is zero wavelength shorter than 2480 nm, is incident on (2) potential is maximum it. The band gap in (eV) for the semiconductor is- (3) electric field is maximum (1) 1.1 eV [AIEEE - 2005] (4) potential is zero (3) 0.5 eV (2) 2.5 eV (4) 0.7 eV 60 E
1 3 . In a common base amplifier, the phase difference JEE-Physics between the input signal voltage and output voltage is- [AIEEE - 2005] 20 . If the lattice constant of this semiconductor is decreased, then which of the following is correct ? [AIEEE - 2006] (2) (3) zero (1) (4) 4 2 1 4 . In a full wave rectifier circuit operating from 50 Conduction Ec band width Ev Hz mains frequency, the fundamental frequency Band gap in the ripple would be- [AIEEE - 2005] Valence Eg band width (1) 50 Hz (2) 25 Hz (3) 100 Hz (4) 70.7 Hz 1 5 . In a common-base mode of a transistor, the collector current is 5.488 mA for an emitter current of 5.60 mA. The value of the base current (1) All Ec, Eg, Ev increase (2) Ec and Ev increase, but Eg decreases amplification factor () will be- [AIEEE - 2006] (3) Ec and Ev decrease, but Eg increases (4) All Ec, Eg, Ev decrease (1) 49 (2) 50 (3) 51 (4) 48 2 1 . Carbon, silicon and germanium have four valence 1 6 . A solid which is not transparent to visible light and electrons each. At room temperature which one of the following statements is most appropriate ? whose conductivity increases with temperature is [AIEEE - 2007] formed by- [AIEEE - 2006] (1) The number of free conduction electrons is (1) ionic binding significant in C but small in Si and Ge (2) covalent binding (2) The number of free conduction electrons is negligible small in all the three (3) Van der Waal's binding (3) The number of free electrons for conduction (4) metallic binding is significant in all the three 1 7 . If the ratio of the concentration of electrons to that (4) The number of free electrons for conduction is significant only in Si and Ge but small in C 7 of holes in a semiconductor is 5 and the ratio of 2 2 . If in a p-n junction diode, a square input signal of 10 V is applied as shown : [AIEEE - 2007] 7 currents is , then what is the ratio of their drift 4 velocities ? [AIEEE - 2006] 5 4 5 4 (1) 8 (2) 5 (3) (4) 7 4 1 8 . The circuit has two oppsitely connected ideal diodes in parallel. What is the current flowing in the circuit [AIEEE - 2006] 4 D1 D2 12V 3 2 (1) 1.71 A (2) 2.00 A 5V RL (3) 2.31 A (4) 1.33 A -5V 1 9 . In the following , which one of the diodes is reverse biased ? [AIEEE - 2006] E:\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Electronics\\Eng\\4.Exercise.p65 +10V –12V Then, the output singnal across R will be :- L R (2) R 5V 10V (1) –10V (1) (2) +5V +5V R (3) R (4) 61 –10V E (3) (4) -10V -5V
JEE-Physics 2 6 . The logic circuit shown below has the input waveform 'A' and 'B' as shown. Pick out the correct output 2 3 . A working transitor with its three legs marked P, waveform :- Q and R is tested using a multimeter. No conduction is found betwen P and Q. By connecting A the common (negative) terminal of the multimeter Y to R and the other (positive) terminal to P or Q, some resistance is seen on the multimeter. Which B of the following is true for the transistor? Input A [AIEEE - 2008] Input B (1) It is an npn transistor with R as base (2) It is a pnp transistor with R as collector (3) It is a pnp transistor with R as emitter (4) It is an npn transistor with R as collector 2 4 . In the circuit below, A A Output is :- [AIEEE - 2009] (1) B C (2) and B represent two inputs and C represents (3) the output. The circuit (4) 2 7 . The combination of gates shown below yields:- represents [AIEEE - 2010] (1) NOR gate [AIEEE - 2008] (2) AND gate (3) NAND gate (4) OR gate 2 5 . A p-n junction (D) shown in the figure can act as a rectifier. An alternating current source (V) is connected in the circuit. D (1) NAND gate (2) OR gate v~ R (3) NOT gate (4) XOR gate The current (I) in the resistor (R) can be shown 28. This question has Statement-1 and by :- Statement-2. Of the four choices given after the [AIEEE - 2009] statements, choose the one that best describes the I two statements. (1) Statement-1: t Sky wave signals are used for long distance radio I communication. These signals are in general, less (2) stable than ground wave signals. t Statement-2 : The state of ionosphere varies from hour to hour, E:\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Electronics\\Eng\\4.Exercise.p65 I day to day and season to season. [AIEEE - 2011] (1) Statement-1 is true, Statement-2 is true and (3) Statement-2 is not the correct explanation t of Statement-1. I (2) Statement-1 is false, Statement-2 is true (4) (3) Statement-1 is true, Statement-2 is false t (4) Statement-1 is true, Statement-2 is true and Statement-2 is the correct explanation of statement-1. 62 E
JEE-Physics 2 9 . The ouput of an OR gate is connected to both the 3 2 . A diode detector is used to detect an amplitude modulated wave of 60% modulation by using a inputs of a NAND gate. The combination will serve condenser of capacity 250 pico farad in parallel with a load resistance 100 kilo ohm. Find the maximum as a : [AIEEE - 2011] modulated frequency which could be detected by it. [JEE(Main) - 2013] (1) OR gate (2) NOT gate (3) NOR gate (4) AND gate 3 0 . Which of the following four alternatives is not correct (1) 10.62 MHz (2) 10.62 kHz We need modulation :- [AIEEE - 2011] (3) 5.31 MHz (4) 5.31 kHz (1) To increase the selectivity 3 3 . The I-V characteristic of an LED is [JEE(Main) - 2013] (2) To reduce the time lag between transmission and Red reception of the information signal Yellow Green Blue I (R) (Y) (G) (B) (3) to reduce the size of antenna (4) To reduce the fractional band width, that is the (1) ratio of the signal band width to the centre frequency O 3 1 . Truth table for system of four NAND gates as shown V I V in figure is :- [AIEEE - 2012] (B) A (G) Y (Y) (R) B (2) O AB Y AB Y I V 001 000 (3) 010 011 O (1) 1 0 0 (2) 1 0 1 111 110 AB Y AB Y V O I 000 001 (R) 010 011 (4) (Y) (3) 1 0 1 (4) 1 0 0 (G) (B) 111 110 E:\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Electronics\\Eng\\4.Exercise.p65 PREVIOUS YEARS QUESTIONS ANSWER-KEY EXERCISE-IV Que. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Ans. 3 3 1 3 2 3 2 4 4 3 3 3 3 3 1 2 3 2 4 3 Que. 21 22 23 24 25 26 27 28 29 30 31 32 33 Ans. 4 1 1 4 1 3 2 1 3 2 2 2 1 E 63
JEE-Physics CHECK YOUR GRASP EXERCISE–01 d2x 1 . The equation of motion of a particle of mass 1 g is dt2 + 2x = 0 where x is displacement(in m)from mean position . The frequency of oscillation is (in Hz) : 1 (B) 2 (C) 5 10 1 (A) (D) 5 10 2 2 . Two bodies performing S.H.M. have same amplitude and frequency. Their phases at a certain instant are as shown in the figure. The phase difference between them is 0.5 A (-x) + (+x) 0 AA (-x) + (+x) 0.5 A 0 11 5 3 (A) (B) (C) (D) 6 3 5 3. The figure shows the displacement time graph of a particle executing S.H.M. x (in mm) If the time period of oscillation is 2 s the equation of motion of its SHM is 10 (A) x = 10sin(t+ /3) 5 (B) x = 10sin t O1 t(s) (C) x = 10sin(t+ /6) (D) x = 10 sin (2pt+p/6) 4 . Two particle executes S.H.M. of same amplitude and frequency along the same straight line. They pass one another when going in opposite directions, each time their displacement is half of their amplitude. The phase difference between them is :– (A) 30° (B) 60° (C) 90° (D) 120° 5 . A small mass executes linear S.H.M. about O with amplitude 'a' and period 'T'. Its displacement from O at time T/8 after passing through O is (A) a/8 a (C) a/2 a (B) 2 2 (D) 2 6 . Two particles A and B perform SHM along the same straight line with the same amplitude 'a', same frequency \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 'f' and same equilibrium position 'O'. The greatest distance between them is found to be 3a/2. At some instant of time they have the same displacement from mean position. What is this displacement? (A) a/2 (B) a 7 /4 (C) 3 /a2 (D) 3a/4 7 . A particle executes S.H.M. along a straight line with mean position x = 0, period 20 s and amplitude 5 cm. The shortest time taken by the particle to go from x = 4 cm to x = 3cm is (A) 4 s (B) 7 s (C) 5 s (D) 6 s 8 . A particle performing S.H.M. is found at its equilibrium at t = 1 s and it is found to have a speed of 0.25 m/s at t = 2 s. If the period of oscillation is 6s Calculate amplitude of oscillation 3 3 6 3 (A) 2 m (B) 4 m (C) m (D) 8 m 42 E
JEE-Physics 9 . A particle executes S.H.M. in a straight line. In the first second starting from rest it travels a distance 'a' and in the next second a distance 'b' in the same direction. The amplitude of S.H.M. will be 2a2 (B) a b (C) 2a b (D) a / b (A) 3a b 1 0 . A particle is subjected to two mutually perpendicular simple harmonic motions such that its x and y coordinates are given by: x = 2 sin t; y = 2 sin t . The path of the particle will be 4 (A) an ellipse (B) a straight line (C) a parabola (D) a circle 1 1 . The period of a particle is 8s. AT t = 0 it is at the mean position. The ratio of distance covered by the particle in first second and second will be- 2 1 1 1 (D) 2 1 (A) (B) 2 (C) 2 1 2 1 2 . A particle executes SHM with time period T and amplitude A. The maximum possible average velocity in time T/4 is 2A 4A 8A 4 2A (A) (B) (C) (D) T T T T 1 3 . The time taken by a particle performing S.H.M. to pass from point A to B where its velocities are same is 2 seconds . After another 2 seconds it returns to B . The time period of oscillation is (in seconds) : (A) 2 (B) 4 (C) 6 (D) 8 1 4 . The P.E. of an oscillating particle at rest position is 15 J and its average K.E. is 5 J. The total energy of particle at any instant will be- (A) 10 J (B) 20 J (C) 25 J (D) 5 J 1 5 . Block A in the figure is released from the rest when the extension in the spring is x0. The maximum downward displacement of the block. k \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 Mg Mg AM 2Mg (A) 2k x 0 (B) 2k x0 2Mg (D) k x0 (C) k x0 1 6 . A system is shown in the figure. The time period for small oscillations of the two blocks will be :- k 2k mm 3m 3m 3m 3m (A) 2 (B) 2 (C) 2 (D) 2 k 2k 4k 8k E 43
JEE-Physics 1 7 . A block of mass 0.9 kg attached to a spring of force constant K is compressed by 2 cm and the block 1 is at a distance 2 cm from the wall. When the block is released, it makes elastic collision with the wall and its period of motion is 0.2 s. The value of K is (take 2=10) K (A) 100 Nm–1 (B) 10 Nm–1 1/2cm (D) 1 Nm–1 (C) 0.1 Nm–1 1 8 . The length of a spring is when a force of 4 N is applied on it and the length is when 5 N force is applied. Then the length of spring when 9 N force is applied is- (A) 5 – 4 (B) – (C) 5 – 4 (D) 9 ( – ) 1 9 . A horizontal spring is connected to a mass M. It executes simple harmonic motion. When the mass M passes through its mean position, an object of mass m is put on it and the two move together. The ratio of frequencies before and after will be- (A) 1 m 1/2 (B) 1 m M 1/2 M M M (C) M m (D) M m 2 0 . A pendulum is suspended in a lift and its period of oscillation when the lift is stationary is T . What must 0 be the acceleration of the lift for the period of oscillation of the pendulum to be T /2 ? 0 (A) 2g downward (B) 2g upward (C) 3g downward (D) 3g upward 2 1 . Two simple pendulums, having periods of 2s and 3s respectively, pass through the mean position simultaneously at a particular instant. They may be in phase after an interval of : (A) 5s (B) 3s (C) 1s (D) none of the above 2 2 . Time period of small oscillation (in a vertical plane normal to the plane of strings) of the bob in the arrangement shown will be 450 (A) 2 g (B) 2 2g 2 2 (C) 2 g (D) 2 g 2 3 . The frequency of a simple pendulum is n oscillations per minute while that of another is (n + 1) oscillations per minute. The ratio of length of first pendulum to the length of second is- \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 n (B) FGHn n1IJK 2 GFH n 1IKJ 2 HFG n 1IJK 2 (A) n 1 n (C) (D) n 2 4 . A system of two identical rods (L-shaped) of mass m and length are resting P on a peg P as shown in the figure. If the system is displaced in its plane by a (D) 3 3g small angle , find the period of oscillations (A) 2 2 (B) 2 2 2 (C) 2 2 3g 3g 3g 44 E
JEE-Physics 2 5 . The distance of point of a compound pendulum from its centre of gravity is , the time period of oscillation relative to this point is T. If g = 2, the relation between and T will be :– T2 T2 T2 T2 2 (B) 2 + +k2 (C) 2 – 2+ – (A) – 4 + k2 = 0 4 = 0 4 – k2 =0 (D) 4 k2 =0 2 6 . A man of mass 60 kg standing on a plateform executing S.H.M. in the vertical plane . The displacement from the mean position varies as y = 0.5 sin (2 ft) . The minimum value of f, for which the man will feel weightlessness at the highest point is : (y is in metres) g (B) 4 g 2g (D) 2 2g (A) (C) 4 2 2 7 . A heavy brass-sphere is hung from a spiral spring and it executes vertical vibrations with period T. The ball is now immersed in nonviscous liquid with a density one-tenth that of brass. When set into vertical vibrations with the sphere remaining inside the liquid all the time, the period will be- 9 (B) T 10 (C) Unchanged (D) T 9 (A) 10 T 9 10 2 8 . A moving particle of mass has one-dimensional potential energy U(x) = ax2 + bx4, where 'a' and 'b' are positive constants. The angular frequency of small oscillations about the minima of the potential energy is equal to a a 2a a (A) (B) 2 (C) (D) 2b m m 2m 2 9 . A particle performs S.H.M. of amplitude A with angular frequency w along a straight line. When it is at a distance 3 1 m2A2 due to A from mean position, its kinetic energy gets increased by an amout 22 an impulsive force. Then its new amplitude becomes- (A) 5 A (B) 3 A (C) 2 A (D) 5 A 2 2 3 0 . A particle executes SHM on a line 8 cm long. Its K.E. and P.E. will be equal when its distance from the mean position is :– (A) 4 cm (B) 2 cm (C) 2 2 cm (D) 2 cm \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 3 1 . The total energy of a vibrating particle in SHM is E. If its amplitude and time period are doubled, its total energy will be :– (A) 16E (B) 8E (C) 4E (D) E 3 2 . The distance between the point of suspension and the centre of gravity of a compound pendulum is and the radius of gyration about the horizontal axis through the centre of gravity is k, then its time period will be k 2 k2 k2 2k (A) 2 g (B) 2 g (C) 2 g (D) 2 g 3 3 . Displacement of a particle is x = 3 sin 2t + 4cos 2t, the amplitude and the maximum velocity will be :– (A) 5, 10 (B) 3, 2 (C) 4, 2 (D) 3, 8 E 45
JEE-Physics 3 4 . The graph shows the variation of displacement of a particle executing S.H.M. with time. We inference from this graph that :– y t T 2 3T 4 TT 4 3T T (A) the force is zero at time 4 (B) the velocity is maximum at time 2 (C) the acceleration is maximum at time T T (D) the P.E. is equal to half of total energy at time 2 13 3 5 . The phase of a particle in SHM at time t is 6 . The following inference is drawn from this a (A) the particle is at x = and moving in + X-direction 2 a (B) the particle is at x = and moving in –X-direction 2 a (C) the particle is at x = – and moving in + X-direction 2 a (D) the particle is at x = – and moving in –X-direction 2 3 6 . The time period of an oscillator is 8 sec. The phase difference from t = 2 sec to t = 4 sec will be :– (A) (D) 2 (B) (C) 2 4 3 7 . Some springs are combined in series and parallel arrangement as shown in the figure and a mass m is suspended from them. The ratio of their frequencies will be :– k k k kk \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 m m (A) 1 : 1 (B) 2 : 1 (C) 3 : 2 (D) 4 : 1 g 3 8 . The acceleration due to gravity at height R above the surface of the earth is . The periodic time of a simple 4 pendulum in an artificial satellite at this height will be :– 2l l (C) zero (D) infinity (A) T = 2 g (B) T = 2 2g 3 9 . The magnitude of average acceleration in half time period in a simple harmonic motion is 2A2 A2 A2 (D) zero (A) (B) (C) 2 2 46 E
JEE-Physics 4 0 . A particle performs S.H.M. with time period T. The time taken by the particle to move from half the amplitude to the maximum displacement is T T T T (A) (B) (C) (D) 2 4 6 8 4 1 . A particle of mass m executing SHM makes f oscillation per second. The difference of its kinetic energy when at the centre, and when at a distance x from the centre is (A) 2f2 x2m (B) 22f2 x2m (C) 1 2f2x2m (D) f2x2m 2 4 2 . Acceleration a and time period T of a body in S.H.M. is given by a curve shown below. Then corresponding graph between kinetic energy KE and time t is correctly represented by a T t KE KE KE KE (A) (B) (C) (D) T T T T t t t t 4 3 . A particle is performing S.H.M. with acceleration a = 8 2 4 2 x where x is coordinate of the particle w.r.t. the origin.The parameters are in S.I. units. The particle is at rest at x= 2 at t=0. (A) coordinate of the particle w.r.t. origin at any time t is 2 4 cos2 t (B) coordinate of the particle w.r.t. origin at any time t is 2 + 4 sin2t (C) coordinate of the particle w.r.t. origin at any time t is 4 + 2 cos2t (D) the coordinate cannot be found because mass of the particle is not given. 4 4 . An oscillation is described by the equation x=A sin 21t where A changes with time according to the law A= A0 (1+cos 22t) where A0 is constant. Find the ratio of frequencies of harmonic oscillations forming oscillation (A) 1 : 2 : 1 2 (B) 1 : 1 2 : 1 2 (C) 1 : 2 : 2 1 (D) 1 : 2 : 1 2 4 5 . Vertical displacement of a plank with a body of mass 'm' on it is var ying according to law y = sint + 3 cost. The minimum value of for which the mass just breaks off the plank and the moment it occurs first after t = 0 are given by : (y is positive vertically upwards) g 2 g 2 g 2 \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 (A) , (B) , (C) , (D) 2g , 2g 2 6g 2 3 g 2 3 g CHECK YOUR GRASP ANSWER KEY EXERCISE –1 Que. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Ans. A C C D D B C A A A C D D C A C A A A D Que. 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Ans. B B C B A C B B C C D B A B A B C D A C Que. 41 42 43 44 45 Ans. B A A B A E 47
JEE-Physics EXERCISE–02 BRAIN TEASERS [MCQs] MCQs with one or more than one correct answer 1 . A mass M is performing linear simple harmonic motion, then correct graph for acceleration 'a' and corresponding linear velocity 'v' is v2 v2 v2 v2 (A) (B) (C) (D) a2 a2 a2 a2 2 . A uniform cylinder of mass m and length l having area of cross-section a is suspended lengthwise with the help of a massless spring of constant k. The cylinder is half submerged in a liquid of density . A small push and release makes it vibrate with small amplitude. The frequency of oscillation is k m 1k 1 kag 1 m ag 1 k ag (A) (B) (C) (D) 2 m 2 m 2 k 2 m 3 . Two identical springs are fixed at one end and masses 1kg and 4kg are suspended at their other ends . They are both stretched down from their mean position and let go simultaneously. If they are in the same phase after every 4 seconds then the springs constant k is N (B) 2 N N (D) given data is insufficient (A) m (C) 2 m m 4 . A cylindrical block of density is partially immersed in a liquid of density 3 . The plane surface of the block remains parallel to the surface of the liquid. The height of the block is 60 cm. The block performs SHM when displaced from its mean position. [Use g = 9.8 m/s2] (A) the maximum amplitude is 20 cm. (B) the maximum amplitude is 40 cm (C) the time period will be /7 seconds (D) none 5 . A mass of 0.2kg is attached to the lower end of a massless spring of force-constant 200 N/m, the upper end of \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 which is fixed to a rigid support. Which of the following statements is/are true? (A) In equilibrium, the spring will be stretched by 1cm. (B) If the mass is raised till the spring is unstretched state and then released, it will go down by 2cm before moving upwards. (C) The frequency of oscillation will be nearly 5 Hz. (D) If the system is taken to the moon, the frequency of oscillation will be the same as on the earth. 6 . A horizontal plank has a rectangular block placed on it. The plank starts oscillating vertically and simple harmonically with an amplitude of 40 cm. The block just loses contact with the plank when the latter is at momentary rest. Then : 2 (A) the period of oscillation is 5 (B) the block weighs double of its weight, when the plank is at one of the positions of momentary rest 48 E
JEE-Physics (C) the block weighs 0.5 times its weight on the plank halfway up (D) the block weighs 1.5 times its weight on the plank halfway down 7 . A particle is subjected to two simple harmonic motions along x and y directions according to, x = 3sin100t; y = 4sin100t. (A) Motion of particle will be on ellipse traversing it in clockwise direction (B) Motion of particle will be on a straight line with slope 4/3 (C) Motion will be simple harmonic motion with amplitude 5 (D) Phase difference between two motions is /2 8. A particle moves in the x-y plane according to the equation, = (ˆi 2ˆj)A cos t . The motion of the particle r is- (B) on an ellipse (C) periodic (D) simple harmonic (A) on a straight line 9 . Two block A and B each of mass m are connected by a massless spring of natural length L and spring constant k. The blocks are initially resting on a smooth horizontal floor with the spring at its natural length as shown in fig. A third identical block C, also of mass m, moving on the floor with a speed v along the line joining A and B, and collides elastically with A. Then- mv mL m C AB [A] The kinetic energy of the A-B system, at maximum compression of the spring, is zero. mv2 [B] The kinetic energy of A-B system, at maximum compression of the spring is 4 [C] The maximum compression of the spring is v m k [D] The maximum compression of the spring is v m 2k 1 0 . A solid cylinder of mass M attached to a massless spring of force constant k is placed on a horizontal surface in such a way that cylinder can roll without slipping. If the system is released from the stretched position of the spring, then the period will be- \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 (A) 2 M (B) 2 3M (C) 2 M (D) 2 2M k 2k 2k 3k 11. A ball is suspended by a thread of length L at the point O on the wall PQ which is inclined to Q the vertical through an angle . The thread with the ball is now displaced through a small angle E away from the vertical and the wall. If < , then the time period of oscillation of the pendulum O will be- L (B) 2 L 2 sin 1 C (A) 2 g g A B P (C) 2 L sin 1 g 2 (D) None of the above 49
JEE-Physics 1 2 . A cage of mass M hangs from a light spring of force constant k. A body of mass m falls from height h inside the cage and sticks to its floor. The amplitude of oscillations of the cage will be- 2mgh 1/2 k 1/2 k (A) k (B) 2mgh mg mg1/2 (C) (D) k k 1 3 . In the above problem, the frequency of oscillations of the cage will be- 1 k 1 / 2 1 k 1 / 2 1 k 1 / 2 1 m 1 / 2 (A) 2 m (B) 2 M (C) 2 M m (D) 2 k 1 4 . The amplitude of a particle executing SHM about O is 10 cm . Then : (A) when the kinetic energy is 0.64 times of its max. kinetic energy its displacement is 6 cm from O (B) when the displacement is 5 cm from O its kinetic energy is 0.75 times its maximum kinetic energy (C) Its total energy of SHM at any point is equal to its maximum kinetic energy (D) Its speed is half the maximum speed when its displacement is half the maximum displacement 1 5 . The angular frequency of a spring block system is 0 . This system is suspended from the ceiling of an elevator moving downwards with a constant speed v . The block is at rest relative to the elevator. Lift is suddenly stopped. 0 Assuming the downwards as a positive direction, choose the wrong statement : v0 (B) the initial phase of the block is (A) the amplitude of the block is 0 v0 0 t (C) the equation of motion for the block is 0 sin (D) the maximum speed of the block is v 0 1 6 . The displacement of a particle varies according to the relation x = 3 sin 100t + 8 cos2 50t . Which of the following is/are correct about this motion . (A) the motion of the particle is not S.H.M. (B) the amplitude of the S.H.M. of the particle is 5 units (C) the amplitude of the resultant S.H. M. is 73 units (D) the maximum displacement of the particle from the origin is 9 units . 1 7 . Two blocks of masses 3 kg and 6 kg rest on a horizontal smooth surface. The 3 kg block is attached to a spring \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 with a force constant k = 900 Nm-1 which is compressed 2 m from beyond the equilibrium position. The 6 kg block is at rest at 1 m from mean position. 3 kg mass strikes the 6 kg mass and the two stick together. 1m 2m 6kg 3kg equilibrium E position (A) velocity of the combined masses immediately after the collision is 10 ms-1 50
JEE-Physics (B) velocity of the combined masses immediately after the collision is 5 ms-1 (C) amplitude of the resulting oscillation is 2 m (D) amplitude of the resulting oscillation is 5 /2m. 1 8 . A disc of mass 3 m and a disc of mass m are connected by massless spring of stiffness k. The heavier disc is placed on the ground with the spring vertical and lighter disc on top. From its equilibrium position, the upper disc is pushed down by a distance and released. Then (A) if >3mg/k, the lower disc will bounce up (B) if =2 mg/k, maximum normal reaction from ground on lower disc = 6 mg (C) if =2 mg/k, maximum normal reaction from ground on lower disc = 4 mg (D) if >4mg/k, the lower disc will bounce up 1 9 . The displacement-time graph of a particle executing SHM is shown. y 0 T/2 T/4 3T/4 T t Which of the following statements is/are true? (A) The velocity is maximum at t = T/2 (B) The acceleration is maximum at t = T (C) The force is zero at t = 3T/4 (D) The potential energy equals the oscillation energy at t = T/2. BRAIN TEASURES\\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65ANSWER KEY EXERCISE –2 Q u e. 1 2 3 45 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Ans. B D B C ABC B BD AC BD BCD AC ABCD ABCD BC CD BD B A A E 51
JEE-Physics EXERCISE–03 MISCELLANEOUS TYPE QUESTIONS TRUE / FALSE 1 . Any physical system, when disturbed, will execute simple harmonic motion. 2 . The motion of earth around the sun is periodic and oscillatory. 3 . If the displacement, velocity and acceleration at a particular instant of particle describing SHM are respectively 1cm, 1cm/s and 1cm/s2 respectively then period of oscillation is also 1sec. 4 . A pendulum clock, in a lift that descends at a constant velocity shows correct time. 5 . The velocity of a particle in S.H.M. increases as the particle moves towards mean position and decreases as it moves towards extreme positions. 6 . The phase of an oscillator is determined by its displacement and veloicity at time t =0. 7 . The amplitude of a pendulum, oscillating in air, decreases with time. FILL IN THE BLANKS 1 . An object of mass 0.2 kg executes simple harmonic oscillational along the x-axis with a frequency of (25/ ) Hz. At the position x = 0.04, the object has kinetic energy of 0.5 J and potential energy 0.4 J. The amplitude of oscillations is_________ m. 2 . A second pendulum A (time period 2 second) and another simple pendulum B of slightly less length than A are made to oscillate at t = 0 in same phase. If they are again in the same phase first time, after 18 seconds, then the time period of B is_________ 3 . Suppose the mass m is attached to a long uniform spring of length L and observed to oscillate at a frequency f . o Now the spring is cut into two pieces of lengths xL and (1-x)L. Mass m is divided into two pieces in this same ratio with m =xm and m =(1-x)m. The larger mass is attached to the shorter spring and the smaller mass to the 12 larger spring. The frequency of oscillation for each of the two spring is _________ 4 . If velocity of a particle moving along a straight line changes sinusoidally with time as shown in the given graph,its average velocity over time interval t = 0 to t = 2(2n - 1) seconds, n being any +ve integer, will be_________ v 4m/s 2s 6s t -4m/s 4s 8s 5 . A block is kept on a platform which is oscillating simple harmonically along a horizontal line with angular \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 frequency . If the coefficient of friction between the block and the plate form is µ then the maximum amplitude of SHM of the platform for which the block remains in constant with the platform is _________. 6 . A particle performs SHM. If velocity at x is v and x is v then amplitude is _________and angular frequency 11 22 is _________ 7 . Two particles perform SHM with the same amplitude and same frequency about the same mean position and along the same line. If the maximum distance between them during the motion is A (A is the amplitude of either) then the phase difference between their SHM is _________ 52 E
JEE-Physics MATCH THE COLUMN 1 . In y = Asinwt + A sin t 2 3 Column-I Column-II (p) is periodic but not SHM (A) Motion (q) is SHM (r) A (B) Amplitude (s) / 3 (t) A/2 (C) Initial phase (u) None (D) Maximum velocity 2 . A particle of mass 2 kg is moving on a straight line under the action of force F = (8 – 2x) N. The particle is released at rest from x = 6 m. For the subsequent motion (All the values in the right column are in their S.I. units.) Column- I Column-II (A) Equilibrium position is at x = (p) /4 (B) Amplitude of S.H.M is (q) /2 (C) Time taken to go directly from (r) 4 x = 2 m to x = 4 m (D) Energy of S.H.M. is (s) 6 (E) Phase constant of S.H.M. assuming (t) 2 equation of the form Asin(t + ) 3 . A block is executing SHM on a rough horiozntal surface under the action of an external variable force. The force is plotted against the position x of the particle from the mean position. Fext I II Ox III IV Column I Column II (A) x positive, v positive (p) I (B) x positive, v negative (q) II (C) x negative, v positive (r) III (D) x negative, v negative (s) IV \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 ASSERTION & REASON In each of the following questions, a statement of Assertion (A) is given followed by a corresponding statement of Reason (R) just below it . Of the statements mark the correct answer as 1 . Statement–1 : The motion of a simple pendulum is simple harmonic only for a << . a and Statement–2 : Motion of a simple pendulum is SHM for small angular displacement. E 53
JEE-Physics \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 (A) Statement–1 is True, Statement–2 is True ; Statement–2 is a 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 . Statement–1 : Pendulum clocks go slow in summer and fast in winter. and Statement–2 : The length of the pendulum used in clock increases in summer. (A) Statement–1 is True, Statement–2 is True ; Statement–2 is a 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. 3 . Statement–1 : SHM is not a periodic motion. and Statement–2 : Periodic motion does not repeat its position after certain interval of time. (A) Statement–1 is True, Statement–2 is True ; Statement–2 is a 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. 4 . Statement–1 : In compound pendulum, if suspension point and centre of oscillation are mutually interchange, then no change in time period is obtained. and Statement–2: Length of equivalent simple pendulum remains same in both the case. (A) Statement–1 is True, Statement–2 is True ; Statement–2 is a 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. 5 . Statement–1 :Any oscillatory motion cannot be treated as simple harmonic. and Statement–2:Even under larger amplitude restoring force should be proportional to displacement for being classified as SHM. (A) Statement–1 is True, Statement–2 is True ; Statement–2 is a 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. 6 . Statement–1 : When a girl sitting on a swing stands up, the periodic time of the swing will increase. and Statement–2: In standing position of a girl, the length of the swing will decrease. (A) Statement–1 is True, Statement–2 is True ; Statement–2 is a 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. 7 . Statement–1: Mechanical energy of a particle executing SHM is E. Maximum KE of particle may be greater than E. and Statement–2 :Potential energy of a system may be negative. (A) Statement–1 is True, Statement–2 is True ; Statement–2 is a 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. 54 E
JEE-Physics COMPREHENSION BASED QUESTIONS Comprehension # 1 Angular frequency in SHM is given by = k . Maximum acceleration in SHM is 2A and maximum value of friction m between two bodies in contact is µN, where N is the normal reaction between the bodies. 1 . In the figure shown, what can be the maximum amplitude of the system so that there is no slipping between any of the blocks ? k=54N/m 1kg µ=0.6 2kg µ=0.4 3kg smooth (A) 2m (B) 3 m (C) 4 m (D) 10 m 7 4 9 3 2 . Now the value of k, the force constant is increased then the maximum amplitude calculated in above question will :- (A) remain same (B) increase (C) decrease (D) data in insufficient Comprehension # 2 In case of pure rolling a = R, where a is the linear acceleration and the angular acceleration. A disc of mass m and radius R is attached with a spring of force constant k at its centre as shown in figure. At x = 0, spring is unstretched. The disc is moved to x = A and then released. There is no slipping between disc and ground. Let f be the force of friction on the disc from the ground. A k +x x=0 x=A 1 . f versus t (time) graph will be as : fff f (A) t (B) t (C) t (D) t 2 . In the problem if k = 10 N/m, m = 2 kg, R = 1 m and A = 2 m. Find linear speed of the disc at mean position: 40 (B) 20 m/s 10 50 (A) m/s (C) m/s (D) m/s 3 3 3 \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 Comprehension # 3 A 2 kg block moving with 10 m/s strikes a spring of constant 2 N/m attached to 2kg block at rest kept on a smooth floor. 10 m/s 2kg 2kg 1. The time for which rear moving block remain in contact with spring will be 2. (A) 2 s 1 (C) 1 s 1 (B) s (D) s E 2 2 In the above question, the velocity of the rear 2 kg block after it separates from the spring will be (A) 0 m/s (B) 5 m/s (C) 10 m/s (D) 7.5 m/s 55
JEE-Physics Comprehension #4 An object of mass m is moving in uniform circular motion in xy-plane. The circle has radius R and object is moving clockwise around the circle with speed v. The motion is projected onto the x-axis where it appears as simple harmonic motion according to x(t) = Rcos(t + ). The motion starts from point of coordinates (0, R). 1 . In this projection is- (B) m2/R (C) R/v (D) None of these (A) v/R 2 . In this projection is- (B) (C) 3/2 (D) 0 (A) /2 Comprehension #5 Two particles A and B are performing SHM along x and y-axis respectively with equal amplitude and frequency of 2 cm and 1 Hz respectively. Equilibrium positions of the particles A and B are at the coordinates (3 cm, 0) and (0, 4 cm) respectively. At t = 0, B is at its equilibrium position and moving towards the origin, while A is nearest to the origin and moving away from the origin. 1 . Equation of motion of particle A can be written as- (A) x = (2 cm) cos 2t (B) x = (3 cm) – (2 cm) cos 2t (C) x = (2 cm) sin 2t (D) x = (3 cm) – (2 cm) sin 2t 2 . Equation of motion of particle B can be written as- (A) y = (2 cm) cos 2t (B) y = (4 cm) – (2 cm) cos 2t (C) y = (2 cm) sin 2t (D) y = (4 cm) – (2 cm) sin 2t 3 . Minimum and maximum distance between A and B during the motion is- (A) 5 cm and 61 cm (B) 3 cm and 7 cm (C) 1 cm and 5 cm (D) 9 cm and 16 cm MISCELLANEOUS TYPE QUESTION ANSWER KEY EXERCISE –3 True / False 1. F 2. F 3. F 4. T 5. T 6. T 7. T 2. 1.8 s f0 8 Fill in the Blanks 1. 0.06 3. x 1 x 4. 2n 1 m/s g v 2 x 2 v 2 x 2 v 2 v 2 7. , 5 \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 5. 2 1 2 2 1 1 2 33 6. , v 2 v 2 x 2 x 2 2. (A) -r, (B) -t (C)-q, (D) -r, (E) -q 1 2 2 1 Ma t c h t h e C o l u m n 1. (A) -q, (B) -r, (C) -s, (D) -u 3. (A) -q, (B) -s, (C) -p, (D) r Assertion – Reason 1. A 2. A 3. D 4. A 5. B 6. D 7. A Comprehension Based Questions Comprehension #1 : 1. C 2. C 2. A Comprehension #2 : 1. C 2. A 2. C Comprehension #3: 1. C 2. D Comprehension #4 : 1. A 56 Comprehension #5 : 1. B 3. B E
JEE-Physics EXERCISE–04 [A] CONCEPTUAL SUBJECTIVE EXERCISE 1 . A particle executing simple harmonic motion completes 1200 oscillations per minute and passes through the mean position with a velocity of 3.14ms–1. Determine the maximum displacement of the particle from its mean position. Also obtain the displacement equation of the particle if its displacement be zero at the instant t = 0. 2 . Find the resulting amplitude A' and the phase of the vibrations S = A cos(t) + A cos t + A t + A t 3 = A' cos t . 2 2 cos cos 2 2 8 3 . A particle is executing SHM given by x = A sin(t + ). The initial displacement of particle is 1 cm and its initial velocity is cm/sec. Find the amplitude of motion and initial phase of the particle. 4 . The shortest distance travelled by a particle executing S.H.M. from mean position in 2 seconds is equal F I3 GH KJto 2 times of its amplitude. Determine its time period. 5 . Two particles A and B execute SHM along the same line with the same amplitude a and same frequency about same equilibrium position O. If the phase difference between them is = 2 sin-1 (0.9), then find the maximum distance between the two. 6 . A body executing S.H.M. has its velocity 10 cm/s and 7 cm/s when its displacement from the mean position are 3 cm and 4 cm respectively. Calculate the length of the path. 7 . The particle executing SHM in a straight line has velocities 8 m/s, 7 m/s, 4 m/s at three points distance one metre from each other. What will be the maximum velocity of the particle? 8 . A particle is oscillating in a straight line about a centre of force O , towards which when at a distance x the force is mn2x where m is the mass , n a constant . The amplitude is a = 15 cm . When at a distance a3 from O the 2 particle receives a blow in the direction of motion which generates a velocity na . If the velocity is away from O, find the new amplitude. 9 . The displacement of a particle varies with time as x = (12 sint – 16 sin3t) cm. If its motion is SHM, find its maximum acceleration. 1 0 . A point particle of mass 0.1 Kg is executing SHM with amplitude of 0.1m. When the particle passes through the mean position, its kinetic energy is 8 × 10–3 J. Obtain the equation of motion of this particle if the initial phase of oscillation is 45°. 1 1 . A body of mass 1 kg suspended by an ideal spring oscillates up and down. The amplitude of oscillation is 1 metre and the periodic time is 1.57 second. Determine. (i) Maximum speed of body. (ii) Maximum kinetic energy. \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 (iii) Total energy. (iv) Force constant of the spring. 1 2 . Potential Energy (U) of a body of unit mass moving in a one-dimension conservative force field is given by, U = (x2 - 4x + 3). All units are in S.I. (i) Find the equilibrium position of the body. (ii) Show that oscillations of the body about this equilibrium position is simple harmonic motion & find it time period. (iii) Find the amplitude of oscillations if speed of the body at equilibrium posiiton is 2 6 m/s 1 3 . A body of mass 1 kg is suspended from a weightless spring having force constant 600 N/m. Another body of mass 0.5 kg moving vertically upwards hits the suspended body with a velocity of 3.0 m/s and get embedded in it. Find the frequency of oscillations and amplitude of motion. E 57
JEE-Physics 1 4 . In the figure shown, the block A collides with the block B and after collision they stick together. Calculate the amplitude of resultant vibration. k 1 5 . A block of mass 1kg hangs without vibrating at the end of a spring with a force constant 1 N/m attached to the ceiling of an elevator. The elevator is rising with an upward acceleration of g/4. The acceleration of the elevator suddenly ceases. What is the amplitude of the resulting oscillations? 1 6 . A small ring of mass m is connected by a string of length to a small heavy bob of mass m . The 12 ring is free to move (slide) along a fixed smooth horizontal wire . The bob is given a small displacement from its equilibrium position at right angles to string . Find period of small oscillations. m1 m2 1 7 . Calculate the time period of a uniform square plate of side 'a' if it is suspended through a corner. 1 8 . Two identical rods each of mass m and length L, are rigidly joined and then suspended in a vertical plane so as to oscillate freely about an axis normal to the plane of paper passing through 'S' (point of suspension). Find the time period of such small oscillations. s 1 9 . A half ring of mass m , radius R is hanged at its one end in verticle plane and is free to oscillate in its plane. Find oscillations frequency of the half ring. R CONCEPTUAL SUBJECTIVE EXERCISE ANSWER KEY EXERCISE–4(A) \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 1. 0.025 m, y =0.025 sin 40t m 3 5 1 4. 12 s 2. 8 A , tan-1 2 3. 2 cm, rad 9. 362 4 5. 1.8 a 6. 9.52 cm 7. 65 m/s 8. 15 3 cm 10. y = 0.1 sin 4 t 11. (i) 4 ms-1 (ii) 8J (iii) 8J (iv) 16 N/m 4 10 5 37 M 15. 2.45 m 12. (i) 2m (ii) 2 s (iii) 2 3 m 13. Hz, cm 14. u 6 2k 16. 2 m1 17. 2 2 2a 18. 2 17L 1 4 3g 18g 19. 2 g 1 2 m1 m2 g 2R E 58
JEE-Physics EXERCISE–04 [B] BRAIN STORMING SUBJECTIVE EXERCISE 1 . In the arrangement as shown in fig., pulleys are small and springs are ideal. k , k , k and k are force 123 4 constants of the springs. Calculate period of small vertical oscillations of block of mass m. k2 k4 m k1 k3 2 . A mass M is in static equilibrium on a massless vertical spring as shown in the figure. A ball of mass m dropped from certain height sticks to the mass M after colliding with it. The oscillations they perform reach to height 'a' above the original level of scales & depth 'b' below it. m a M b B (i) find the constant of force of the spring (ii) find the oscillation frequency. (iii) what is the height above the initial level from which the mass m was dropped? 3 . Two non-viscous, incompressible and immiscible liquids of densities and 1.5 are poured into the two limbs of a circular tube of radius R and small cross-section kept fixed in a vertical plane as shown in figure. Each liquid occupies one-fourth the circumference of the tube. \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 R O 4. (i) Find the angle that the radius to the interface makes with the vertical in equilibrium position. (ii) If the whole liquid column is given a small displacement from its equilibrium position, show that the resulting E oscillations are simple harmonic. Find the time period of these oscillations. A weightless rigid rod with a load at the end is hinged at point A to the wall so that it can rotate in all directions. The rod is kept in the horizontal position by a vertical inextensible thread of length , fixed at its midpoint. The load receives a momentum in the direction perpendicular to the plane of the figure. Determine the period T of small oscillations of the system. 59
JEE-Physics B A 5 . One rope of a swing is fixed above the other rope by b. The distance between the poles of the swing is a. The lengths and of the ropes are such that 2 + 2 = a2 + b2 (Fig.) Determine the period T of 12 12 small oscillations of the swing, neglecting the height of the swining person in comparison with the above lengths. a b 1 2 6 . A massless rod rigidly fixed at O . A string carrying a mass m at one end is attached to point A on the rod so that OA = a . At another point B (OB = b) of the rod , a horizontal spring of force constant k is attached as shown . Find the period of small vertical oscillations of mass m around its equilibrium position k B A \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 m O 7 . A lift operator hung an exact pendulum clock on the lift wall in a lift in a building to know the end of the working day. The lift moves with an upward & downward accelerations during the same time (according to a stationary clock), the magnitudes of the acceleration remaining unchanged. Will the operator work for more or less than required time . 8 . In the figure shown, the spring are connected to the rod at one end and at the midpoint. The rod is hinged at its lower end. k k g 60 E
JEE-Physics (i) Find the minimum value of k for rotational SHM of the rod (Mass m, length L) (ii) If k = mg/L then find the angular frequency of oscillations of the rod. 9 . Two identical balls A and B each of mass 0.1 kg are attached to two identical massless springs. The spring mass system is constrained to move inside a rigid smooth pipe in the form of a circle as in fig. The pipe is fixed in a horizontal plane. The centres of the ball can move in a circle of radius 0.06 m. Each spring has a natural length 0.06 m and force constant 0.1 N/m. Initially both the balls are displaced by an angle of = / radian with respect to diameter PQ of the circle and released from rest A B P Q (i) calculate the frequency of oscillation of the ball B. (ii) what is the total energy of the system (iii) find the speed of the ball A when A and B are at the two ends of the diameter PQ. 1 0 . A rod of mass M and length L is hinged at its one end and carries a block of mass m at its other end. A spring of force constant k is installed at distance a from the hinge and another of force constant k at a distance b as 12 shown in the figure. If the whole arrangement rests on a smooth horizontal table top. Find the frequency of vibration. k2 b a k1 M m \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 BRAIN STORMING SUBJECTIVE EXERCISE ANSWER KEY EXERCISE–4(B) 1. 1 2mg 1 2mg M m ab 3. tan 1 1 , 1.803R 1 1 1 1 2. (i) m b a 5 (ii) 2 g 4 k1 k2 k3 k 4 ba (ii) 2 b aM m (iii) 2 2 5. 2 12 6. a m 4x g g 8. (i) k= 2mg 3 k 4. g 2 7. ga (ii) m g bk 5L 2 a g a where x is the total distance travelled 1 1 k1a2 k2b2 9. (i) Hz, (ii) 4 2 105 J (iii) 2 102 ms1 10. 2 L2 (m M ) 3 E 61
JEE-Physics PREVIOUS YEAR QUESTIONS EXERCISE–05(A) 1 . If a spring has time period T, and is cut into n equal parts, then the time period of each part will be -[AIEEE-2002] (1) T n T (3) nT (4) T (2) n 2 . In a simple harmonic oscillator, at the mean position- [AIEEE-2002] (1) kinetic energy is minimum, potential energy is maximum (2) both kinetic and potential energies are maximum (3) kinetic energy is maximum, potential energy is minimum (4) both kinetic and potential energies are minimum 3 . A child swinging on a swing in sitting position, stands up, then the time period of the swing will- [AIEEE-2002] (1) increase (2) decrease (3) remain same (4) increase if the child is long and decrease if the child is short 4 . A mass M is suspended from a spring of negligible mass. The spring is pulled a little and then released so that the mass executes SHM of time period T. If the mass is increased by m, the time period becomes 5T/3, then m the ratio of is- [AIEEE-2003] M 3 25 16 5 (1) (2) (3) (4) 3 5 9 9 5 . The displacement of a particle varies according to the relation x = 4(cos t + sin t). The amplitude of the particle is- [AIEEE-2003] (1) – 4 (2) 4 (3) 4 2 (4) 8 6 . A body executes simple harmonic motion. The potential energy (PE), the kinetic energy (KE) and total energy (TE) are measured as function of displacement x. Which of the following statements is true ? [AIEEE-2003] (1) KE is maximum when x = 0 (2) TE is zero when x = 0 (3) KE is maximum when x is maximum (4) PE is maximum when x = 0 7 . Two particles A and B of equal masses are suspended from two massless springs of spring constants k and k , 12 respectively. If the maximum velocities, during oscillations are equal, the ratio of amplitudes of A and B is- (1) k1 / k2 (2) k /k (3) k2 / k1 (4) k /k [AIEEE-2003] 12 21 8 . The bob of a simple pendulum executes simple harmonic motion in water with a period t, while the period of oscillation of the bob is t in air. Neglecting frictional force of water and given that the density of the bob is (4/ 0 3) × 1000 kg/m3. What relationship between t and t is true? [AIEEE-2004] 0 (1) t = t (2) t = t /2 (3) t = 2t (4) t = 4t 00 00 9 . A particle at the end of a spring executes simple harmonic motion with a period t , while the corresponding 1 \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 period for another spring is t. If the period of oscillation with the two springs in series is T, then- [AIEEE-2004] 2 (1) T = t + t (2) T2 = t12 + t22 (3) T–1 = t11 + t21 (4) T–2 = t12 + t12 12 1 0 . The total energy of a particle, executing simple harmonic motion is- [AIEEE-2004] (1) x (2) x2 (3) independent of x (4) x1/2 1 1 . A particle of mass m is attached to a spring (of spring constant k) and has a natural angular frequency 0. An external force F(t) proportional to cost( 0) is applied to the oscillator. The time displacement of the oscillator will be proportional to- [AIEEE-2004] m 1 1 m (1) (2) m (20 2 ) (3) m (20 2 ) (4) (20 2 ) 20 2 62 E
JEE-Physics 1 2 . A particle of mass 0.3 kg is subjected to a force F = –kx with k = 15 N/m. What will be its initial acceleration, if it is released from a point 20 cm away from the origin ? [AIEEE-2005] (1) 3 m/s2 (2) 15 m/s2 (3) 5 m/s2 (4) 10 m/s2 1 3 . The function sin2(t) represents- [AIEEE-2005] (1) a periodic, but not simple harmonic, motion with a period 2/ (2) a periodic, but not simple harmonic, motion with a period / (3) a simple harmonic motion with a period 2/ (4) a simple harmonic motion with a period / 14. Two simple harmonic motions are represented by the equations y= 0.1 sin 100t and y = 0.1 cost. 1 3 2 The phase difference of the velocity of particle 1, with respect to the velocity of particle 2 is- [AIEEE-2005] – – (1) 6 (2) 3 (3) 3 (4) 6 1 5 . If a simple harmonic motion is represented by d2 x + x = 0, its time period is [AIEEE-2005] dt2 2 2 (3) 2 (4) 2 (1) (2) 1 6 . The bob of a simple pendulum is a spherical hollow ball filled with water. A plugged hole near the bottom of the oscillating bob gets suddenly unplugged. During observation, till water is coming out, the time period of oscillation would- [AIEEE-2005] (1) first increase and then decrease to the original value (2) first decrease and then increase to the original value (3) remain unchanged (4) increase towards a saturation value 1 7 . The maximum velocity of a particle, executing simple harmonic motion with an amplitude 7 mm,is 4.4 m/s. The period of oscillation is- [AIEEE-2006] (1) 0.01 s (2) 10s (3) 0.1 s (4) 100 s 1 8 . Starting from the origin a body oscillates simple harmonically with a period of 2s. After what time will its kinetic energy be 75% of the total energy ? [AIEEE-2006] 1 1 1 1 (1) 6 s (2) s (3) 3 s (4) s 4 12 1 9 . The displacement of an object attached to a spring and executing simple harmonic motion is given by x = 2 × 10–2 cos t metre. The time at which the maximum speed first occurs is- [AIEEE-2007] (1) 0.5 s (2) 0.75 s (3) 0.125 s (4) 0.25 s 20. A point mass oscillates along the x-axis according to the law x = x cos(wt – /4) If the acceleration of the 0 particle is written as- a = A cos (t + ), then- [AIEEE-2007] \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 (1) A = x0, = – /4 (2) A = x02 , = /4 (3) A = x02, = –/4 (4) A = x02, = 3/4 2 1 . A 2 kg block slides on a horizontal floor with a speed of 4 m/s. It strikes a uncompressed spring and compress 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 2 2 . Two springs, of force constants k and k , are connected to a mass m as shown. The frequency of oscillation of 12 the mass is f. If both k1 and k2 are made four times their original values, the frequency of oscillation becomes k1 m k2 [AIEEE-2007] (1) f/2 (2) f/4 (3) 4f (4) 2f E 63
JEE-Physics 2 3 . A particle of mass m executes simple harmonic motion with amplitude a and frequency . The average kinetic energy during its motion from the position of equilibrium to the end is- [AIEEE-2007] (1) 2ma22 1 (3) 42ma22 (4) 22ma22 (2) ma22 4 2 4 . If x, v and a denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period T, then, which of the following does not change with time ? [AIEEE-2009] (1) aT + 2v aT (3) a2T2 + 42v2 aT (2) (4) v x 2 5 . A mass M, attached to a horizontal spring, executes S.H.M. with amplitude A1. When the mass M passes through its mean position then a smaller mass m is placed over it and both of them move together with amplitude A2. The ratio of A1 is :- [AIEEE-2011] A 2 M 1/2 M m 1/2 M M m (1) M m (2) M (3) M m (4) M 2 6 . Two particles are executing simple harmonic motion of the same amplitude A and frequency along the x- axis. Their mean position is separated by distance X0(X0 > A). If the maximum separation between them is (X0 + A), the phase difference between their motion is :- [AIEEE-2011] (1) 4 (2) 6 (3) 2 (4) 3 2 7 . A wooden cube (density of wood 'd') of side '' floats in a liquid of density '' with its upper and lower surfaces horizontal. If the cube is pushed slightly down and released, it performs simple harmonic motion of period 'T'. Then, 'T' is equal to :- [AIEEE-2011] (1) 2 (2) 2 d (3) 2 d ( d)g g dg (4) 2 ( d)g 2 8 . If a spring of stiffness 'k' is cut into two parts 'A' and 'B' of length A : B = 2 : 3, then the stiffness of spring 'A' is given by :- [AIEEE-2011] 5 3k 2k (4) k (1) k (2) (3) 2 5 5 2 9 . If a simple pendulum has significant amplitude (up to a factor of 1/e of original) only in the period between t = 0 s to t = s, then may be called the average life of the pendulum. When the spherical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity, with 'b' as the constant of proportionality, the average life time of the pendulum is (assuming damping is small) in seconds :- [AIEEE-2012] (1) 2 0.693 (3) b (4) 1 (2) bb b 3 0 . An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass M. The piston \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 and the cylinder have equal cross sectional area A. When the piston is in equilibrium, the volume of the gas is V0 and its pressure is P0. The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency. [AIEEE-2013] (1) 1 AP0 (2) 1 V0MP0 1 A 2 P0 1 M V0 2 V0M 2 A2 (3) 2 M V0 (4) 2 A P0 ANSWER-KEY Que. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Ans. 2 3 2 3 3 1 3 3 2 3 2 4 2 1 2 1 1 1 1 4 Que. 21 22 23 24 25 26 27 28 29 30 Ans. 1 4 1 3,4 2 4 2 1 1 3 64 E
JEE-Physics EXERCISE–05(B) PREVIOUS YEAR QUESTIONS MCQ's one correct answers 1 . A particle of mass m is executing oscillation about the origin on the x-axis. Its potential energy is U(x) = k|x|3, where k is a positive constant. If the amplitude of oscillation is a, then its time period T is :- [IIT-JEE 1998] (A) proportional to 1 (B) independent of a (C) proportional to a (D) proportional to a3/2 a 2 . A spring of force constant k is cut into two pieces such that one piece is double the length of the other. Then the long piece will have a force constant of :- [IIT-JEE 1999] (A) 2k (B) 3k (C) 3k (D) 6k 3 2 3 . A particle free to move along the x-axis has potential energy given by : U(x) = k[1 – exp (–x2)] for – x + , where k is a positive constant of appropriate dimensions. Then :- [IIT-JEE 1999] (A) at points away from the origin, the particle is in unstable equilibrium (B) for any finite non-zero value of x, there is a force directed away from the origin (C) if its total mechanical energy is k , it has its minimum kinetic energy at the origin 2 (D) for small displacements from x = 0, the motion is simple harmonic. 4 . The period of oscillation of simple pendulum of length L suspended from the roof of the vehicle which moves without friction, down an inclined plane of inclination , is given by :- [IIT-JEE 2000] L L L L (A) 2 g cos (B) 2 g sin (C) 2 g (D) 2 g tan 5 . A particle executes simple harmonic motion between x = –A and x = +A. The time taken for it to go from AA [IIT-JEE 2001] O to 2 is T1 and to go from 2 to A is T2, then :- (D) T1 = 2T2 (A) T1 < T2 (B) T1 > T2 (C) T1 = T2 6 . For a particle executing SHM the displacement x given by x = Acost. Identify the graph which represents the variation of potential energy (PE) as a function of time t and displacement x :- [IIT-JEE 2003] PE I II PE III IV (t) x \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 (A) I, III (B) II, IV (C) II, III (D) I, IV 7 . A block P of mass m is placed on a frictionless horizontal surface. Another block Q of same mass is kept on P and connected to the wall with the help of a spring of spring constant k as shown in the figure, s is the coefficient of friction between P and Q. The block move together performing SHM of amplutude A. The maximum value of the friction force between P and Q is [IIT-JEE 2004] (A) kA kA k Q S (D) smg (B) 2 P smooth E (C) zero 65
JEE-Physics 8 . A simple pendulum has time period T1. The point of suspension is now moved upward according to the relation y = Kt2, (K = 1 m/s2) where y is the vertical displacement. The time period now becomes T . The 2 ratio of T12 is : (g = 10 m/s2) [IIT-JEE 2005] T22 6 5 (C) 1 4 (A) (B) (D) 5 6 5 9 . A block (B) is attached to two unstretched springs S and S with spring constants k and 4k, respectively (see 12 figure I). The other ends are attached to identical supports M1 and M2 not attached to the walls. The springs and supports have negligible mass. .There is no friction anywhere. The block B is displaced towards wall 1 by a small distance x (figure II) and released. [IIT-JEE 2008] 2 S1 M1 1 M2 S2 I 2 x1 M2 S2 S1 M1 II x The block returns and moves a maximum distance y towards wall 2. displacements x and y are measured with y respect to the equilibrium position of the block B. The ratio is x (A) 4 (B) 2 1 1 (C) (D) 2 4 1 0 . The x-t graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the particle at t = 4 / 3 s is [IIT-JEE 2009] x(cm) 1 8 12 t(s) \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p6504 –1 (A) 3 2cm / s2 (B) – 2 cm/s2 (C) 2 cm/s2 (D) – 3 2cm/s2 32 32 32 32 1 1 . The mass M shown in the figure oscillates in simple harmonic motion with amplitude A. The amplitude of the point P is : [IIT-JEE 2009] k1 k2 M P k1A k2A k1A k2A (A) k2 (B) k1 (C) k1 k2 (D) k1 k2 66 E
JEE-Physics 1 2 . A uniform rod of length L and mass M is pivoted at the centre. Its two ends are attached to two springs of equal spring constant k. The springs are fixed to rigid supports as shown in the figure, and the rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle in one direction and released. The frequency of oscillation is :- [IIT-JEE 2009] 1 2k 1k 1 6k 1 24k (A) (B) (C) (D) 2 M 2 M 2 M 2 M MCQ's one or more than one correct answers 1 . Three simple harmonic motions in the same direction having the same amplitude and same period are superposed. If each differ in phase from the next by 45°, then the :- [IIT-JEE 1999] (A) resultant amplitude is (1 + 2)a (B) phase of the resultant motion relative to the first is 90° (C) energy associated with the resulting motion is (3 + 22) times the energy associated with any single motion (D) resulting motion is not simple harmonic 2 . Function : x = Asin2t + Bcos2t + C sint cost represents SHM :- [IIT-JEE 2006] (A) for any value of A, B and C (except C = 0) (B) if A = –B, C = 2B, amplitude = |B2| (C) if A = B; C = 0 (D) if A = B; C = 2B, amplitude = |B| 3 . A particle of mass m is attached to one end of a mass-less spring of force constant k, lying on a frictionless horizontal plane. The other end of the spring is fixed. The particle starts moving horizontally from its equilibrium position at time t = 0 with an initial velocity u0. When the speed of the particle is 0.5 u0, it collides elastically with a rigid wall. After this collision :- [IIT-JEE 2013] (A) the speed of the particle when it returns to its equilibrium position is u0 \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 m (B) the time at which the particle passes through the equilibrium position for the first time is t= k (C) the time at which the maximum compression of the spring occurs is t 4 m 3k (D) the time at which the particle passes through the equilibrium position for the second time is t 5 m 3k E 67
JEE-Physics Match the column 1 . Column I describes some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situations in column I with the characteristics in column II. [IIT-JEE 2007] Column I Column II (A) The object moves on the x-axis under a conservative force (p) The object executes a simple in such a way that its \"speed\"and \"positive\" satisfy harmonic motion. v=c c2 x2 , where c and c are positive constants. 1 1 2 (B) The object moves on the x-axis in such a way that its velocity (q) The object does not change its and its displacement from the origin satisfy v = –kx, direction. where k is a positive constant. (C) The object is attached to one end of a mass-less spring of a (r) The kinetic energy of the object given spring constant. The other end of the spring is keeps on decreasing attached to the ceiling of an elevator. Initially everything is at rest. The elevator starts going upwards with a constant acceleration 'a'. The motion of the object is observed from the elevator during the period it maintains this acceleration. (D) The object is projected from the earth's surface vertically (s) The object can change its direction only once. GMe upwards with a speed 2 Re , where M is the mass of e the earth and R is the radius of the earth. Neglect forces e from objects other than the earth. 2 . Column I gives a list of possible set of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column II. Match the set of parameters given Column I with the graphs given in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 × 4 matrix given in the ORS. Column I [IIT-JEE 2008] Column II y (A) Potential energy of a simple pendulum (y axis) as a (p) function of displacement (x axis) (B) Displacement (y axis) as a function of time (x axis) o x \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 for a one dimensional motion at zero or constant y x acceleration when the body is moving along the positive x-direction. (q) (C) Range of a projectile (y axis) as a function of o its velocity (x axis) when projected at a fixed angle. y (r) ox y (D) The square of the time period (y axis) of a simple (s) x pendulum as a function of its length (x axis) o E 68
JEE-Physics Paragraph A uniform thin cylindrical disk of mass M and radius R is attached to two identical massless springs of spring constant k which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance d from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstrethced length of each spring is L. The disk is initially at its equilibrium position with its centre of mass (CM) at a distance L from the wall. The disk rolls without slipping with velocity [IIT-JEE 2008] V V ˆi . The coefficient of friction is µ. 00 y d 2d V0 R x 1 . The net external force acting on the disk when its centre of mass is at displacement x with respect to its equilibrium position is (A) – kx (B) – 2kx 2kx 4kx (C) (D) 3 3 2 . The centre of mass of the disk undergoes simple harmonic motion with angular frequency equal to k 2k 2k 4k (A) (B) (C) (D) M M 3M 3M 3 . The maximum value of V0 for which the disk will roll without slipping is M M 3M 5M (A) µg (B) µg (C) µg (D) µg k 2k k 2k \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 Subjective Questions 1 . Two masses m and m connected by a light spring of natural length is compressed completely and tied 12 0 by a string. This system while moving with a veloicty v along +ve x-axis pass through the origin at t=0. 0 At this position the string snaps. Position of mass m at time t is given by the equation 1 x (t) = vt -A ( 1- c o s t ) . Calculate:(i) position of the particle m as a functionof time. (ii) in terms of A. 1 0 20 [IIT-JEE 2003] 2. A solid sphere of radius R is floating in a liquid of density with half of its volume submerged. If the sphere E is slightly pushed and released, it starts performing simple harmonic motion. Find the frequency of these oscillations. [IIT-JEE 2004] 69
JEE-Physics 3 . A mass m is undergoing SHM in the vertical direction about the mean position y with amplitude A and 0 angular frequency . At a distance y from the mean position, the mass detaches from the spring. Assume that the spring contracts and does not obstruct the motion of m. Find the distance y. (measured from the mean position) such that the height h attained by the block is maximum (A2 > g). [IIT-JEE 2005] y m PREVIOUS YEARS QUESTIONS ANSWER KEY EXERCISE –5 MCQ's with one correct answer 1. A 2. B 3. D 4. A 5. A \\\\node6\\E_NODE6 (E)\\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-6\\SHM\\Eng\\Theory.p65 6.A 7. B 8. A 9. C 10. D 11. D 12. C MCQ's one or more than one correct 1. A,C 2. A,B,D 3. A,D Match the column 1. (A) -p, (B) -q,r (C) -p (D) -q,r 2. (A)-p,(B)-q,r,s(C)-s,(D)-q Comprehension Based questions 1. D 2. D 3. C Subjective Questions (i) m1 1 cos t (ii) m1 1 A 2. 1 3g 3. mg g 1. vt + A m2 m 2 2 2R a 0 k 2 70 E
JEE-Physics EXERCISE–01 CHECK YOUR GRASP SELECT THE CORRECT ALTERNATIVE (ONLY ONE CORRECT ANSWER) 1 . The centre of mass of a non uniform rod of length L whose mass per unit length varies as = kx2/L (where k is a constant and x is the distance measured from one end) is at the following distance from the same end. (A) 3L/4 (B) L/4 (C) 2L/3 (D) L/3 2 . Centre of mass of two uniform rods of same length but made up of different materials & kept as shown, if the meeting point is the origin of co–ordinates y L L x (A) (L/2,L/2) (B) (2L/3,L/2) (C) (L/3,L/3) (D) (L/3,L/6) 3 . A uniform wire of length is bent into the shape of 'V' as shown. The distance of its centre of mass from the vertex A is B A 600 C (A) / 2 3 3 (D) None of these (B) (C) 4 8 4 . Three man A, B & C of mass 40 kg, 50 kg & 60 kg are standing on a plank of mass 90 kg, which is kept on a smooth horizontal plane. If A & C exchange their positions then mass B will shift 40kg 50kg 60kg A B C NODE6 E : \\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-3\\Centre of mass & collision\\Eng\\Exercise.p65 (A) 1/3 m towards left (B) 1/3m towards right (C) will not move w.r.t. ground (D) 5/3 m towards left 5. Considering a system having two masses m and m in which first mass is pushed towards centre of mass by a 12 E distance a, the distance required to be moved for second mass to keep centre of mass at same position is m1 m2 a (A) m1 a (C) m2 a (D) m 2m1 a m2 (B) m1m2 m1 m1 m a 2 51
JEE-Physics 6 . An isolated particle of mass m is moving in horizontal plane (x–y), along the x–axis, at a certain height m 3m above the ground. It suddenly explodes into two fragment of masses 4 and 4 . An instant later, the smaller fragment is at y = +15 cm. The larger fragment at this instant is at :– (A) y = –5 cm (B) y = +20 cm (C) y = +5 cm (D) y = –20 cm 7 . Two particles A and B initially at rest, move towards each other under the mutual force of attraction. At the instant when the speed of A is v and the speed of B is 2v, the speed of the centre of mass of the system is:– (A) 3v (B) v (C) 1.5v (D) zero 8 . The velocity of centre of mass of the system as shown in the figure y 1kg 2m/s x’ x 2 kg y’ 300 2m/s 2 2 3 ˆi 1 ˆj 2 2 3 ˆi 2 ˆj 3 3 3 3 (A) (B) (C) 4ˆi (D) None of these 9 . The figure shows the positions and velocities of two particles. If the particles move under the mutual attraction of each other, then the position of centre of mass at t =1 s is (A) x=5m 5m/s 3m/s (D) x=2m 1kg 1kg x=2m x=8m (B) x=6m (C) x=3m 1 0 . A particle of mass 2m is connected by an inextensible string of length 1.2 m to a ring of mass m which is free to slide on a horizontal smooth rod. Initially the ring and the particle are at the same level with the string taut. Both are then released simultaneously. The distance in meters moved by the ring when the string becomes vertical is (A) 0 (B) 0.4 (C) 0.8 (D) 1.2 1 1 . A ball of mass 1 kg drops vertically on to the floor with a speed of 25 m/s. It rebounds with an initial NODE6 E : \\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-3\\Centre of mass & collision\\Eng\\Exercise.p65 velocity of 10 m/s. What impulse acts on the ball during contact? (A) 35kg m/s downwards (B) 35 kg m/s upwards (C) 30 kg m/s downwards (D) 30kg m/s upwards 1 2 . A particle of mass m is made to move with uniform speed v along the perimeter of a regular hexagon, 0 inscribed in a circle of radius R. The magnitude of impulse applied at each corner of the hexagon is (A) 2 mv s i n/ 6 (B) mv0sin/6 (C)m v si n/ 3 (D) 2 m v s i n/ 3 0 0 0 1 3 . Two balls of same mass are dropped from the same height h, on to the floor. The first ball bounces to a height h/4 ,after the collision & the second ball to a height h/16. The impulse applied by the first & second ball on the floor are I and I respectively. Then 12 (A) 5I = 6I (B) 6I = 5I (C) I = 2I (D) 2I = I 12 12 12 12 52 E
JEE-Physics 1 4 . An impulse I changes the velocity of a particle from v1 to v2 . Kinetic energy gained by the particle is 1 1 (A) 2 ·(v1 v2 ) (B) 2 ·(v1 v2 ) (C) ·(v1 v2 ) (D) ·(v1 v2 ) 1 5 . A particle of mass 4m which is at rest explodes into masses m, m & 2m. Two of the fragments of masses m and 2m are found to move with equal speeds v each in opposite directions. The total mechanical energy released in the process of explosion is (A) mv2 (B) 2mv2 (C) 1/2 mv2 (D) 4mv2 1 6 . Two blocks A(3 kg) and B(2 kg) resting on a smooth horizontal surface is connected by a spring of stiffness 480N/m. Initially the spring is undeformed and a velocity of 2 m/s is imparted to A along the line of the spring away from B. The maximum extension in meters of the spring during subsequent motion is 1 1 1 (D) 0.15 (A) (B) (C) 10 2 10 2 15 1 7 . A cannon of mass 5m (including a shell of mass m) is at rest on a smooth horizontal ground, fires the shell with its barrel at an angle with the horizontal at a velocity u relative to cannon. Find the horizontal distance of the point where shell strikes the ground from the initial position of the cannon: 4u2 sin 2 u2 sin 2 3u2 sin 2 8u2 sin 2 (A) 5g (B) 5g (C) 5g (D) 5g 1 8 . A shell is fired from a cannon with a velocity v (m/s) at an angle with the horizontal direction. At the highest point in its path it explodes into two pieces of equal mass. One of the pieces retraces its path to the cannon and the speed (m/s) of the other piece immediately after the explosion is :– (A) 3vcos (B) 2vcos 3 3 (C) 2 vcos (D) 2 vcos 1 9 . A ball hits the floor and rebounds after an inelastic collision. In this case :– (A) the momentum of the ball just after the collision is the same as that just before the collision (B) the mechanical energy of the ball remains the same in the collision (C) the total momentum of the ball and the earth is conserved (D) the total energy of the ball and the earth is conserved 2 0 . Three balls A, B and C (mA = mC = 4mB) are placed on a smooth horizontal surface. Ball B collides with ball C with an initial velocity v as shown in the figure. Total number of collisions between the balls will be (All collisions are elastic) v ABC NODE6 E : \\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-3\\Centre of mass & collision\\Eng\\Exercise.p65 (A) One (B) Two (C) Three (D) Four 2 1 . A body of mass 1 kg strikes elastically with another body at rest and continues to move in the same direction with one fourth of the initial velocity. The mass of the other body is – (A) 0.6 kg (B) 2.4 kg (C) 3 kg (D) 4 kg 22. A small bucket of mass M kg is attached to a long inextensible cord of length L m . The bucket is released from rest when the cord is in a horizontal position. At its lowest position, the bucket scoops up m kg of E water and swings up to a height h. The height h in meters is M 2 M M m 2 M m M M (A) M m L (B) M m L (C) L (D) L 53
JEE-Physics 2 3 . A particle moving horizontally collides with a fixed plane inclined at 60o to the horizontal. If it bounces vertically, the coefficient of restitution is: 1 2 1 (D) None of these (A) (B) (C) 3 3 3 2 4 . A ball of mass 2m impinges directly on a ball of mass m, which is at rest. If the velocity with which the larger ball impinges be equal to the velocity of the smaller mass after impact then the coefficient of restitution 1 3 1 2 (A) (B) (C) (D) 3 4 2 5 2 5 . A body of mass 2kg is projected upward from the surface of the ground at t = 0 with a velocity of 20m/s. One second later a body B, also of mass 2kg, is dropped from a height of 20m. If they collide elastically, then velocities just after collision are (A) vA = 5 m/s downward, vB = 5 m/s upward (B) vA = 10 m/s downward, vB= 5 m/s upward (C) v = 10 m/s upward, v = 10 m/s downward (D) both move downward with velocity 5 m/s AB 2 6 . A ball of mass 1 kg strikes a heavy platform, elastically, moving upwards with a velocity 1kg 10 m/s 5 m/s of 5m/s. The speed of the ball just before the collision is 10m/s downwards. Then the impulse imparted by the platform on the ball is (A) 15 N–s (B) 10 N–s (C) 20 N–s (D) 30 N–s 2 7 . Two particles of mass m, constrained to move along the circumference of a smooth v0 v0 circular hoop of equal mass m, are initially located at opposite ends of a diameter and given equal velocities v shown in the figure. The entire arrangement is located 0 in gravity free space. Their velocity just before collision is 1 3 2 7 (A) 3 v0 (B) 2 v0 (C) 3 v0 (D) 3 v0 2 8 . Two objects move in the same direction in a straight line. One moves with a constant velocity v . The other 1 starts at rest and has constant acceleration a. They collide when the second object has velocity 2v . The 1 distance between the two objects when the second one starts moving is (A) zero (B) v12 (C) v 2 (D) 2 v 2 2a 1 1 a a 2 9 . A uniform rope of linear mass density and length is coiled on a smooth horizontal surface. One end is pulled up with constant velocity v. Then the average power applied NODE6 E : \\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-3\\Centre of mass & collision\\Eng\\Exercise.p65 by the external agent in pulling the entire rope just off the ground is : (A) 1 v2 2g (B) gv (C) 1 v3 vg (D) gv 1 v 3 22 22 2 CHECK YOUR GRASP ANSWER KEY EXERCISE –1 Que. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Ans. A D C B A A D B B C B A A A B A A A C B Que. 21 22 23 24 25 26 27 28 29 Ans. A A C C A D D A C 54 E
JEE-Physics EXERCISE–02 BRAIN TEASERS SELECT THE CORRECT ALTERNATIVES (ONE OR MORE THEN ONE CORRECT ANSWERS) 1 . Two particles A and B start moving due to their mutual interaction only. If at any time 't', aA and aB are their respective accelerations, v A and vB are their respective velocities, and upto that time WA and WB are the work done on A and B respectively by the mutual force, mA and mB are their masses respectively, then which of the following is always correct. (C) WA + WB = 0 (A) v A vB 0 (B) m A v A mB vB 0 (D) aA aB 0 2 . On a smooth carom board, a coin moving in negative y–direction with y a speed of 3 m/s is being hit at the point (4, 6) by a striker moving along (4,6) u x negative x–axis. The line joining centres of the coin and the striker just coin before the collision is parallel to x–axis. After collision the coin goes into the hole located at the origin. Masses of the striker and the coin are equal. 3m/s Considering the collision to be elastic, the initial and final speeds of the striker in m/s will be– (0,0) (A) (1.2, 0) (B) (2, 0) (C) (3, 0) (D) None of these 3. A balloon having mass 'm' is filled with gas and is held in hands of a boy. Then suddenly it gets released 4. and gas starts coming out of it with a constant rate. The velocity of the ejected gas is 2m/s with respect 5. to the balloon. Find out the velocity of the balloon when the mass of gas is reduced to half. 6. (A) n 2 (B) 2n 4 (C) 2n 2 (D) None of these E Two men 'A' and 'B' are standing on a plank. 'B' is at the middle of 40kg 60kg A B the plank and 'A' is at the left end of the plank. Surface of the plank is smooth. System is initially at rest and masses are as shown in figure. A and B starts moving such that the position of 'B' remains fixed with 40kg smooth respect to ground then 'A' meets 'B'. Then the point where A meets B 120cm is located at– (A) the middle of the plank (B) 30 cm from the left end of the plank (C) the right end of the plank (D) None of these A gun which fires small balls of mass 20 gm is firing 20 NODE6 E : \\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-3\\Centre of mass & collision\\Eng\\Exercise.p65 balls per second on the smooth horizontal table surface DC ABCD. If the collision is perfectly elastic and balls are striking AB at the centre of table with a speed 5 m/s at an angle of 60° with the vertical just before collision, then force exerted by one of the leg on ground is (assume total weight of the table is 0.2 kg and g = 10 m/ s2) : (A) 0.5 N (B) 1 N (C) 0.25 N (D) 0.75 N A ball is bouncing down a set of stairs. The coefficient of restitution is e. The height of each step is d and the ball bounces one step at each bounce. After each bounce the ball rebounds to a height h above the next lower step. Neglect width of each step in comparison to h and assume the impacts to be effectively head on. Which of the following relation is correct ? (A) h = 1 – e2 h h1 h1 d (B) d = 1 – e (C) d = 1 e2 (D) d = 1 e 55
JEE-Physics 7 . The diagram shows the velocity–time graph for two masses v(m/s) S R and S that collided elastically. Which of the following 1.2 statements is true ? R I. R and S moved in the same direction after the collision 0.8 II. The velocities of R and S were equal at the mid time 0.4 of the collision. III. The mass of R was greater than mass of S. 1 2 3 4 t(s) (A) I only (B) II only (C) I and II only (D) I, II and III 8 . A system of two blocks A and B are connected by an inextensible massless strings as shown. The pulley is massless and frictionless. Initially the system is at rest when, a bullet of mass 'm' moving with a velocity 'u' as shown hits the block 'B' and gets embedded into it. The m impulse imparted by tension force to the block of mass 3m is– u 5mu 4mu 2mu 3mu mB (A) (B) 5 (C) 5 (D) 5 A 3m 4 9 . A piece of paper (shown in figure1) is in form of a square. Two corners of this square are folded to make it appear like O 2a O Fig.2 2a figure.2 . Both corners are put together at centre of square 'O'. If O is taken to be (0,0), the centre of mass of new system will Fig.1 be at (A) a , 0 (B) a , 0 (C) a , 0 (D) a , 0 8 6 12 12 1 0 . An arrow sign is made by cutting and rejoining a quarter part of a square plate of side 'L' as shown. The distance OC, where 'C' is the centre of mass of the arrow, is L L 3L (A) (B) (C) (D) None of these 3 48 1 1 . A block of mass M is tied to one end of a massless rope. The other end of the rope is in the hands of a man of mass 2M as shown M 2M in the figure. The block and the man are resting on a rough wedge 2m M of mass M as shown in the figure. The whole system is resting on a smooth horizontal surface. The man pulls the rope. Pulley is massless and frictionless. What is the displacement of the wedge when the block meets the pulley. (Man does not leave his position during the pull) NODE6 E : \\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-3\\Centre of mass & collision\\Eng\\Exercise.p65 (A) 0.5m (B) 1m (C) Zero (D) 2/3 m 1 2 . A continuous stream of particles of mass m and velocity v, is emitted from a source at a rate of n per second. The particles travel along a straight line, collide with a body of mass M and get embedded in the body. If the mass M was originally at rest, its velocity when it has received N particles will be mvn mvN mv Nm M (A) (B) (C) (D) Nm n Nm M Nm M mv 1 3 . Three particles start from origin at the same time with a velocity 2ms–1 along positive x–axis, the second with a velocity 6ms–1 along negative y–axis. Find the velocity of the third particle along x=y line so that the three particles may always lie in a straight line (A) 3 3 (B) 3 2 (C) 3 2 (D) 2 2 56 E
JEE-Physics 1 4 . A bead can slide on a smooth straight wire and a particle of mass m 2m attached to the bead by a light string of length L. The particle is held m in contact with the wire and with the string taut and is then let fall. If the bead has mass 2m then when the string makes an angle with the wire, the bead will have slipped a distance (A) L (1 cos ) (B) L (1 cos ) (C) L (1 cos ) (D) L (1 cos ) 2 3 6 1 5 . A body of mass M moves in outer space with velocity v. It is desired to break the body into two parts so that the mass of one part is one–tenth of the total mass. After the explosion, the heavier part comes to rest while the lighter part continues to move in the original direction of motion. The velocity of the small part will be (A) v v v (D) 10 v (B) 2 (C) 10 1 6 . A ball of mass m is released from A inside a smooth wedge of mass m as shown A in the figure. What is the speed of the wedge when the ball reaches point B ? 450 B 1/2 5gR 1/2 3 gR (B) 2gR (C) 2 3 (D) gR smooth (A) 3 2 2 17.. A uniform metallic spherical shell is suspended from ceiling. It has two holes A and B A at top and bottom respectively. Which of the following is/are true: sand (A) If B is closed and sand is poured from A, centre of mass first rises and then falls B (B) If shell is completely filled with sand and B is opened then centre of mass falls initially (C) If shell is slightly filled with sand and B is opened, then centre of mass falls. (D) None of these 1 8 . If both the blocks as shown in the given arrangement are given together a 1kg µ=0.1 2 kg µ=0.2 horizontal velocity towards right. If a be the subsequent acceleration of cm the centre of mass of the system of blocks then a equals cm (A) 0 m/ s2 5 7 (D) 2 m/s2 (B) m/s2 (C) m/s2 3 3 1 9 . A bead of mass m and diameter d is sliding back and forth with velocity v on a wire held between two rigid walls of length L. Assume that the collisions with the wall are perfectly elastic and there is no friction. The average force that the bouncing bead exerts on the one of the walls is NODE6 E : \\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-3\\Centre of mass & collision\\Eng\\Exercise.p65 mv2 mv2 2mv2 2mv2 (A) (B) (C) (D) L d L d L d L d 2 0 . A set of n–identical cubical blocks lies at rest parallel to each other along a line on a smooth horizontal surface. The separation between the near surfaces of any two adjacent blocks is L. The block at one end is given a speed v towards the next one at time t = 0. All collisions are completely inelastic, then L (A) The last block starts moving at t = n (n–1) 2v L (B) The last block starts moving at t = (n–1) v v (C) The centre of mass of the system will have a final speed n (D) The centre of mass of the system will have a final speed v. E 57
JEE-Physics 2 1 . The Fig. shows a string of equally placed beads of mass m, separated by distance d. The beads are free to slide without friction on a thin wire. A constant force F acts on the first bead initially at rest till it makes collision with the second bead. The second bead then collides with the third and so on. Suppose that all collisions are elastic, Fd (A) Speed of the first bead immediately before and immediately after its collision with the second bead is 2Fd 1 2 3 4 and zero respectively. m (B) Speed of the first bead immediately before and immediately after its collision with the secondbead is 2Fd 1 2Fd and respectively. m 2m (C) Speed of the second bead immediately after its collision with third bead is zero. (D) The average speed of the first bead is 1 2Fd . 2m 2 2 . Two persons A and B of weight 80 kg and 50 kg respectively are standing at opposite ends of a boat of mass 70 kg and length 2m at rest. When they interchange their positions then displacement of the centre of mass of the boat will be: (A) 60 cm towards left (B) 30 cm towards right A B (C) 30 cm towards left (D) stationary 2 3 . In a one dimensional collision between two identical particles A and B, B is stationary and A has momentum p before impact. During impact, B gives impulse J to A. (A) The total momentum of the 'A plus B' system is p before and after the impact, and (p–1) during the impact. (B) During the impact A gives impulse J to B 2J (C) The coefficient of restitution is 1 p (D) The coefficient of restitution is J 1 p 2 4 . Two balls of same mass are dropped from the same height onto the floor. The first ball bounces upwards from the floor elastically. The second ball sticks to the floor. The first applies an impulse to the floor of I 1 and the second applies an impulse I . The impulses obey 2 (A) I = 2I (B) I = I1 (C) I = 4I (D) I = I1 21 22 21 24 2 5 . A small ball falling vertically downward with constant velocity 4m/s strikes 4m/s NODE6 E : \\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-3\\Centre of mass & collision\\Eng\\Exercise.p65 elastically a massive inclined cart moving with velocity 4m/s horizontally as 4m/s shown. The velocity of the rebound of the ball is 45° (A) 4 2 m/s (B) 4 3 m/s (C) 4 m/s (D) 4 5 m/s 2 6 . A particle of mass 4m which is at rest explodes into four equal fragments. All v 4 fragments scattered in the same horizontal plane. Three fragments are found to 900 move with velocity v each as shown in the fig. The total energy released in the process 1350 of explosion is (A) mv2 (3– 2 ) (B) mv2 (3– 2 )/2 v (C) 2mv2 (D mv2 (1+ 2 )/2 v E 58
JEE-Physics 2 7 . The fig. shows the velocity as a function of the time for an object with mass 10 kg being pushed along a frictionless surface by external force. At t= 3s, the force V stops pushing and the object moves freely. It then collides head on and sticks to another object of mass 25 kg. 15 (A) External force acting on the system is 50 N 5 (B) Velocity of the 2nd particle just before the collision is 1 m/s 34 6t (C) Before collision both bodies are moving in the same direction (D) Before collision, bodies are moving in opposite direction with respect to each other 2 8 . A particle of mass m = 0.1 kg is released from rest from a point A of a wedge A of mass M = 2.4 kg free to slide on a frictionless horizontal plane. The particle slides m down the smooth face AB of the wedge. When the velocity of the wedge is 0.2 m/s the velocity of the particle in m/s relative to the wedge is V v M 600 B (A) 4.8 (B) 5 (C) 7.5 (D) 10 m=4kg 2 9 . A ball of mass 1 kg is suspended by an inextensible string 1 m long attached AO B 300 to a point O of a smooth horizontal bar resting on a fixed smooth supports A and B. The ball is released from rest from the position when the string makes an angle of 30° with the vertical. The mass of the bar is 4 kg. The displacement in meters of the bar when the string makes the maximum angle on the other m=1kg side of the vertical is (A) 0 (B) 0.2 (C) 0.25 (D) 0.5 m 3 0 . Find the distance between centre of gravity and centre of mass of a two particle system attached to the ends of a light rod. Each particle has same mass. Length m of the rod is R, where R is the radius of earth (A) R (B) R/2 (C) zero (D) R/4 3 1 . After scaling a wall of 3 m height a man of weight W drops himself to the ground. If his body comes to a complete stop in 0.15 s. After his feet touch the ground, calculate the average impulsive force in the vertical direction exerted by ground on his feet. (A) 5W (B) 5.21W (C) 3W (D) 6 W 3 2 . An open water tight railway wagon of mass 5 x 103 kg coasts at an initial velocity 1.2 m/s without friction on a railway track. Rain drops fall vertically downwards into the wagon. The velocity of the wagon after it has collected 103 kg of water will be NODE6 E : \\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-3\\Centre of mass & collision\\Eng\\Exercise.p65 (A) 0.5 m/s (B) 2m/s (C) 1 m/s (D)1.5 m/s 3 3 . Three blocks A, B and C each of mass m are placed on a surface as shown in the figure. Blocks B and C are initially at rest. Block A is moving to the right with speed v. It collides with block B and sticks to it. The A–B combination collides elastically with block C. Which of the following statement is (are) true about the velocity, of block B and C. mm m (A) Velocity of the block C after collision is 2/3 v towards right ABC v (B) Velocity of the A–B combination after collision is 3 towards left 2 (C) Velocity of the A–B combination after collision is v towards left 3 v (D) Velocity of the block C after collision is towards right. 3 E 59
JEE-Physics 3 4 . Two masses A and B of mass M and 2M respectively are connected MA ukˆ 2M B by a compressed ideal spring. The system is placed on a horizontal X frictionless table and given a velocity ukˆ in the z–direction as shown ukˆ ˆi in the figure. The spring is then released. In the subsequent motion the line from B to A always points along the ˆi unit vector. At some instant Z of time mass B has a x–component of velocity as v xˆi . The velocity kˆ ukˆ vA of mass A at that instant is (A) v xˆi ukˆ (B) – v xˆi ukˆ (C) –2 v xˆi ukˆ (D) 2 v xˆi ukˆ 3 5 . A disk A of radius r moving on perfectly smooth surface at a speed v A B undergoes an elastic collision with an identical stationary disk B. Find the velocity of the disk B after collision if the impact parameter is r/2 as v shown in the figure r/2 (A) 15 v v v 3v 4 (B) (C) (D) 4 2 2 3 6 . A spherical ball of mass 1 kg moving with a uniform speed of 1 m/s collides symmetrically with two identical spherical balls of mass 1 kg each at rest touching each other. If the two balls move with 0.5m/s in two directions at the same angle of 60° with the direction of the first ball, the loss of kinetic energy on account of the collision is : (A) 0.125 J (B) 0.5J (C) 1.0 J (D) 0.75J 3 7 . A smooth sphere A of mass m collides elastically with an identical sphere B at rest. The velocity of A before collision is 8 m/s in a direction making 60° with the line of centres at the time of impact. (A) The sphere A comes to rest after collision. (B) The sphere B will move with a speed of 8 m/s after collision. (C) The directions of motion A and B after collision are at right angles. (D) The speed of B after collision is 4 m/s. 3 8 . A particle moving with kinetic energy = 3J makes an elastic head–on collision with a stationary particle which has twice its mass. During the impact, (A) the minimum kinetic energy of the system is 1J. (B) the maximum elastic potential energy of the system is 2J. (C) momentum and total energy are conserved at every instant. (D) the ratio of kinetic energy to potential energy of the system first decreases and then increases. 3 9 . Two blocks A and B each of mass m, are connected by a massless v A B spring of natural length L and spring constant k. The blocks are initially C resting on a smooth horizontal floor with the spring at its natural length, NODE6 E : \\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-3\\Centre of mass & collision\\Eng\\Exercise.p65 as shown in fig. A third identical block C, also of mass m, moves on the floor with a speed v along the line joining A and B, and collides elastically with A. Then :– (A) the kinetic energy of the A–B system, at maximum compression of the spring, is zero mv2 (B) the kinetic energy of the A–B system, at maximum compression of the spring, is 4 m (C) the maximum compression of the spring is v k m E (D) the maximum compression of the spring is v 2k 60
JEE-Physics 4 0 . Assuming potential energy 'U' at ground level to be zero. Solid sphere Solid Cube Solid Cone Solid Cylinder P Q QS D DD DD D U=0 DD All objects are made up of same material. UP = Potential energy of solid sphere UQ = Potential energy of solid cube UR = Potential energy of solid cone US = Potential energy of solid cylinder (A) US > UP (B) UQ > US (C) UP > UQ (D) US > UR 4 1 . A bag of mass M hangs by a long thread and a bullet (mass m) comes horizontally with velocity v and gets caught in the bag. Then for the combined system (bag + bullet) : (A) Momentum is mMv/(M + m) (B) kinetic energy is (1/2) Mv2 (C) Momentum is mv (D) kinetic energy is m2v2/2(M + m) 4 2 . A ball moving with a velocity v hits a massive wall moving towards the ball with a velocity u. An elastic impact lasts for a time t. m(u v) (A) The average elastic force acting on the ball is t 2m(u v) (B) The average elastic force acting on the ball is t (C) The kinetic energy of the ball increases by 2mu(u + v) (D) The kinetic energy of the ball remains the same after the collision NODE6 E : \\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-3\\Centre of mass & collision\\Eng\\Exercise.p65 BRAIN TEASERS ANSWER KEY EXERCISE –2 Que. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Ans. B B C C B C D D D B A B B C D Que. 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Ans . A B D A A,C A,C C B,C B D A A,B,C D B C Que. 31 32 33 34 35 36 37 38 39 40 41 42 Ans . B C A C A A C,D A,B,C,D B,D A,B,D C,D B,C E 61
JEE-Physics EXERCISE–03 MISCELLANEOUS TYPE QUESTIONS TRUE / FALSE 1 . In an elastic collision of two bodies, the momentum and energy of each body is conserved. 2 . In an inelastic collision, the final kinetic energy is always less than the initial kinetic energy of the system. 3 . Two particles of mass 1 kg and 3 kg move towards each other under their mutual force of attraction. No other force acts on them. When the relative velocity of approach of the two particles is 2 m/s, their centre of mass has a velocity of 0.5 m/s. When the relative velocity of approach becomes 3 m/s, the velocity of the centre of mass is 0.75 m/s. FILL IN THE BLANKS 1 . A particle of mass 4m which is at rest explodes into three fragments. Two of the fragments each of mass m are found to move with a speed v each in mutually perpendicular directions. The total energy released in the process of explosion is ................ 800 C 2 . The magnitude of the force (in newtons) acting on a body Force (N) 600 varies with time t (in microseconds) as shown in the figure AB, 400 BC and CD are straight line segments. The magnitude of the 200 A B total impulse of the force on the body from t = 4 µs to EF D 0 2 4 6 8 10 12 14 16 t = 16 µs is ........................... N–s. Time (µs) 3 . A ball is dropped from height h to the ground. If the coefficient of restitution is 0.8, the height to which ball goes up after it rebounds third time is ............... 4 . A ball of mass 1 g is released down an inclined plane, just describes a circle of radius 10 cm in the vertical plane, on reaching the bottom, the minimum height of the inclined plane is .................cm. 5 . The rate of change of total momentum of a many–particle system is proportional to the .............. on the system. 6 . In an inelastic collision of two bodies, the quantities which do not change after the collision are the ............... of the system of two bodies. 7 . A block of mass m moving with a velocity v, enters a region where it starts colliding with the stationary dust particles. If the density of dust particles is & all colliding particle stick to its front surface of cross–sectional area A. The velocity of block after it has covered a distance x in this region is _________. 8 . Two spheres of masses 3 kg and 2 kg collide directly. Their relative velocity before collision is 15 m/s and after collision is 5 m/s. The total loss of K.E. in joules due to collision is ______. MATCH THE COLUMN 1 . A particle of mass m, kinetic energy K and momentum p collides head on elastically with another particle of mass 2 m at rest. After collision : Column I Column II (A) Momentum of first particle (p) 3/4 p NODE6 E : \\Data\\2014\\Kota\\JEE-Advanced\\SMP\\Phy\\Unit No-3\\Centre of mass & collision\\Eng\\Exercise.p65 (B) Momentum of second particle (q) – K/9 (C) Kinetic energy of first particle (r) – p/3 (D) Kinetic energy of second particle (s) 8K 9 (t) None 2 . Two balls of mass m and 2m each have momentum 2p and p in the direction m 2p 2m p shown in figure. During collision they exert an impulse of magnitude p on each other. E Column I Column II (A) After collision momentum of m (p) 2p (B) After collision momentum of 2m (q) p (C) Coefficient of restitution between them (r) 1 (s) None 62
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