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11P01_Units & Measurement_Avanti Module

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P1 – Units & Measurement 1 Sixth Edition P1. Units & Measurement TABLE OF CONTENTS P1. Units & Measurement 1 Synopsis..........................................................................................................................................................................................................3 P1.1 Units and Dimensions ................................................................................................................................................................11 P1.2 Significant Figures and Error Analysis...............................................................................................................................15 Test Practice Problems .........................................................................................................................................................................22 Answer Key ................................................................................................................................................................................................28

P1 – Units & Measurement 2 PRE-REQUISITES 1. Basic Mathematics: a. Percentages b. Ratio and Proportion c. Indices and Radicals d. Algebra in one variable 2. Scientific Notation PRE-TEST Q1. The value of 0.005 in scientific notation is __________. 1000 Q2. Write in decimal form (upto 2 decimal places) I. 5 = ___ II. 4 = ___ III. 10 = ___ 4 5 3 Q3. 20% ������������ 50 is __________. Q4. Find the percent increase / decrease in each of the following cases: I. The length of a rod changes from 4 ������������ to 6 ������������ II. The weight of a person changes from 80 ������������ to 60 ������������ Q5. Compute the value of 68××1100−44. Q6. If a humming bird flaps its wings once every 10−3 second, how many times does it flap its wings in a minute? Q7. Solve the equation ������ + ������ = 15 ; 4������ − ������ = 10 Q8. Simplify the expression √������ × ������6 = SCORE 8 Score 1 point per correct answer. If you score less than ������, please revise the pre-requisite topics.

P1 – Units & Measurement 3 Synopsis Physical Quantities And System Of Units 1. PHYSICAL QUANTITIES All quantities that can be measured are called physical quantities. E.g. Time, length mass, force, work done, etc. 2. MEASUREMENT Measurement is the comparison of a quantity with a standard of the same physical quantity. 3. UNITS All physical quantities are measured w.r.t. standard magnitude of the same physical quantity and these standards are called UNITS. E.g. Second, meter, kilogram, etc. Four basic properties of standard unit are: - i. They must be well defined. ii. They should be easily available and reproducible. iii. They should be invariable e.g. step as a unit of length is not invariable. iv. They should be accepted to all. 4. SET OF FUNDAMENTAL QUANTITIES IN VARIOUS SYSTEM OF UNITS A set of physical quantities which are completely independent of each other and all other physical quantities can be expressed in terms of these physical quantities is called Set of Fundamental Quantities. Physical Quantity Units(������������) Units(������������������) Notations Mass kg (kilogram) ������ ������ Length ������ (meter) ������������ ������ Time ������(second) ������ ������ Temperature ������(kelvin) ℃ ������ Current ������ (ampere) ������ ������ or ������ Luminous intensity ������������ (candela) − ������������ Amount of substance ������������������ − ������������������ Physical Quantity (������������ Unit) Definition Length (������) Mass (������������) The distance travelled by light in vacuum in 1 second is called ������ metre. Time (������) 299.792,458 Electric Current(A) The mass of a cylinder made of platinum-iridium alloy kept at International Bureau of Weights and Measures is defined as ������ kilogram The second is the duration of 9, 192, 631, 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-������������������ atom. If equal currents are maintained in the two parallel infinitely long wires of negligible cross-section, so that the force between them is ������ × ������������−������ newton per metre of the wires, the current in any of the wires is called ������ Ampere. Syn.

P1 – Units & Measurement 4 Thermodynamic The fraction ������ of the thermodynamic temperature of triple point of water is Temperature (K) ������������������.������������ called ������ Kelvin Luminous Intensity (������������) One candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 1012 ������������ and that has a radiant intensity in that direction of 1/683 ������������������������ ������������������ ������������������������������������������������������ Amount of substance (mole) The mole is the amount of a substance that contains as many elementary entities as there are number of atoms in ������. ������������������ ������������ of carbon-������������. Two supplementary units are: i. Plane angle (radian) angle=arc/radius or, ������ = ℓ/������ ii. Solid Angle (steradian) 5. DERIVED PHYSICAL QUANTITIES The physical quantities those can be expressed in terms of fundamental physical quantities are called derived physical quantities. E.g. Speed = distance/time 6. ������������ PREFIXES: Standard prefixes for certain power of 10. Table shows these prefixes: Power of ������������ Prefix Symbol 12 ������ 9 tera ������ 6 giga ������ ������ mega ������ 2 kilo ������ 1 hecto ������������ −1 deka ������ −2 deci ������ −3 centi ������ −6 milli ������ −9 micro ������ −12 nano ������ −15 pico ������ femto

P1 – Units & Measurement 5 Dimensional Equations and Dimensional Analysis 1. DIMENSION The power (exponent) of base quantity that enters into the expression of a physical quantity, is called the dimension of the quantity in that base. Consider the physical quantity “force”. ������������������������������ℎ/������������������������ Force = mass × acceleration = ������������������������ × ������������������������ = mass × length × (������������������������)−2 So the dimensions of force are 1 in mass, 1 in length and −2 in time. Thus [������������������������������] = ������������������−2 Similarly, energy has dimensional formula given by [������������������������������������] = ������������2������−2 i.e. energy has dimensions, 1 in mass, 2 in length and −2 in time. Such an expression for a physical quantity in terms of base quantities is called dimensional formula. 2. DIMENSIONAL EQUATION Whenever the dimension of a physical quantity is equated with its dimensional formula, we get a dimensional equation. 3. PRINCIPLE OF HOMOGENEITY According to this principle, we can multiply physical quantities with same or different dimensional formulae at our convenience, however no such rule applies to addition and substraction, where only like physical quantities can only be added or subtracted. E.g. IF ������ + ������ ⇒ ������ & ������ both represent same physical quantity. Illustration: Calculate the dimensional formula of energy from the equation ������ = 1 ������������2. 2 Sol. Dimensionally, ������ = ������������������������ × (������������������������������������������������)2. 1 Since 2 is a number and has no dimension. ������ 2 Or, [������] = ������ × (������) = ������������2������−2. 4. USES OF DIMENSIONAL ANALYSIS i. To convert units of a physical quantity from one system of units to another: It is based on the fact that, Numerical value × unit = constant So on changing unit, numerical value will also gets changed. If ������1 and ������2 are the numerical values of a given physical quantity and ������1 and ������2 be the units respectively in two different systems of units, then ������2 = ������1 [������������12]������ [������������12]������ [������������12]������ Illustration: A calorie is a unit of heat or energy and it equals about 4.2 ������, where 1 ������ = 1 ������������ ������2/������2. Suppose we employ a system of units in which the unit of mass equals ������ ������������, the unit of length equals ������ metre, the unit of time is ������ second. Show that a calorie has a magnitude 4.2 ������−1 ������−2������2 in terms of the new units. Syn.

P1 – Units & Measurement 6 Sol. 1 ������������������ = 4.2 ������������ ������2������−2 ������������ New system ������1 = 4.2 ������2 =? ������1 = 1 ������������ ������2 = ������ ������������ ������1 = 1 ������ ������2 = ������ ������������������������������ ������1 = 1 ������ ������2 = ������ ������������������������������������ Dimensional formula of energy is [������������2������−2] Comparing with [������������������������������������], we find that ������ = 1, ������ = 2, ������ = −2 Now, ������2 = ������1 [������������12]������ [������������12]������ [������������12]������ = 1 ������������ 1 1 ������ 2 1 ������ −2 = 4.2 ������−1������−2������2 4.2 [������ ������������] [������ ������] [������ ������] ii. To check the dimensional correctness of a given physical relation: It is based on principle of homogeneity, which states that a given physical relation is dimensionally correct if the dimensions of the various terms on either side of the relation are the same. a) Powers are dimensionless b) sin ������ , ������������, cos ������ , log ������ gives dimensionless value and in above expression ������ is dimensionless c) We can add or subtract quantity having same dimensions. Illustration: Let us check the dimensional correctness of the relation ������ = ������ + ������������. Here ′������′ represents the initial velocity, ′������′ represent the final velocity, ′������′ the uniform acceleration and ′������′ the time. Dimensional formula of ′������′ is [������0������������−1] Dimensional formula ′������′ is [������0������������−1] Dimensional formula ′������������′ is [������0������������−2][������] = [������0������������−1] Here dimensions of every term in the given physical relation are the same, hence the given physical relation is dimensionally correct. iii. To establish a relation between different physical quantities: If we know the various factors on which a physical quantity depends, then we can find a relation among different factors by using principle of homogeneity. Illustration: Let us find an expression for the time period ������ of a simple pendulum. The time period ������ may depend upon a) mass ������ of the bob of the pendulum, b) length ������ of pendulum, c) acceleration due to gravity ������ at the place where the pendulum is suspended. Sol. Let a) ������ ∝ ������������ b) ������ ∝ ℓ������ c) ������ ∝ ������������ Combining all the three factors, we get where ������ is a dimensionless constant of proportionality.

P1 – Units & Measurement 7 Writing down the dimensions on either side of equation (������), we get [������] = [������������][������������][������������−2]������ = [������������������������+������������−2������] Comparing dimensions, ������ = 0, ������ + ������ = 0, −2������ = 1 11 ∴ ������ = 0, ������ = − 2 , ������ = 2 From equation (i)������ = ������������0ℓ1/2������−1/2 or ������ = ������ ℓ 1/2 = ������√���ℓ��� (������) 5. LIMITATIONS OF DIMENSIONAL ANALYSIS i. Dimension does not depend on the magnitude. Due to this reason the equation ������ = ������������ + ������������2 is also dimensionally correct. Thus, a dimensionally correct equation need not be actually correct. ii. The numerical constants having no dimensions cannot be deduced by the method of dimensions. iii. It cannot be used to derive relations with more than one term This method is applicable only of relation is of product type. It fails in the case of exponential and trigonometric relations. 6. UNITS AND DIMENSIONS OF SOME PHYSICAL QUANTITIES Quantity ������������ Unit Dimensional Formula Density ������������/������3 ������������−3 Force Newton (������) ������������������−2 Work Joule (������)(= ������-������) ������������2������−2 Energy Joule (������) ������������2������−2 Power Watt (������)(= ������/������) ������������2������−3 Momentum ������������-������/������ ������������������−1 Gravitational constant ������-������2/������������2 ������−1������3������−2 Angular velocity radian/������ ������0������0������−1 Angular acceleration radian/������2 ������0������0������−2 Angular momentum ������������-������2/������ ������������2������−1 Moment of inertia ������������-������2 ������������2������0 Torque ������-������ ������������2������−2 Angular frequency radian/������ ������0������0������−1 Frequency Hertz (������������) ������0������0������−1 Period ������ ������0������0������ Surface Tension ������/������ ������������0������−2 Coefficient of viscosity ������-������/������2 ������������−1������−1 Wavelength ������ ������0������������0 Intensity of wave ������/������2 ������������0������−3 Temperature kelvin (������) ������ Specific heat capacity ������/(������������-������) ������0������2������−2������−1 Syn.

P1 – Units & Measurement 8 Stefan’s constant ������/(������2 − ������4) ������������0������−2������−4 Heat ������������2������−2 Thermal conductivity ������ ������������������−3������−1 Current density ������0������−2������0������ ������/(������-������) Electrical conductivity ������−1������−3������3������2 ������/������2 Electric dipole moment ������0������������������ Electric field ������ℎ������ 1/Ω-������ (= ������ ) ������������������−3������−1 Potential (voltage) Electric flux ������-������ ������������2������−3������−1 Capacitance Electromotive force ������ ������������3������−3������−1 Resistance ������/������ (= ������ ) ������−1������−2������4������2 Permittivity of space ������������2������−3������−1 Permeability of space volt (������) (= ������ ������������2������−3������−2 ������) Magnetic field ������−1������−3������4������2 ������-������ Magnetic flux ������������������−2������−2 Magnetic dipole moment farad (������) Inductance ������������0������−2������−1 Volt (������) ������������2������−2������−1 ������ℎ������ (Ω) ������0������2������0������ ������������2������−2������−2 ������2/������-������2 (= ������ ������) ������/������2 ������������ Tesla (������) = (= ������2 ) Weber (������������) ������-������/������ Henry (������) Significant Figures and Errors in Measurement 7. Significant Figures In the measured value of a physical quantity, the number of digits about the correctness of which we are sure plus the next doubtful digit, are called the significant figures. 8. Rules for Finding Significant Figures i. All non-zeros digits are significant figures, e.g., 4362 ������ has ������ significant figures. ii. All zeros occurring between non-zero digits are significant figures, e.g., 1005 has 4 significant figures. iii. All zeros to the right of the last non-zero digit are not significant, e.g., 6250 has only ������ significant figures. iv. In a digit less than one, all zeros to the right of the decimal point and to the left of a non-zero digit are not significant, e.g., 0.00325 has only ������ significant figures. v. All zeros to the right of a non-zero digit in the decimal part are significant, e.g., 1.4750 has ������ significant digures. 9. Significant Figures in Algebraic Operations i. In Addition or Subtraction -: In addition or subtraction of the numerical values the final result should retain the least decimal place as in the various numerical values. E.g.,

P1 – Units & Measurement 9 If 11 = 4.326 ������ and 12 = 1.50 ������ Then, 11 + 12 = (4.326 + 1.50)������ = 8.826 ������ As 12 has measured upto two decimal places, therefore 11 + 12 = ������. ������������ ������ ii. In Multiplication or Division :- In multiplication or division of the numerical values, the final result should retain the least significant figures as the various numerical values. . E.g., If length 1 = 12.5 ������ and breadth ������ = 4.125 ������. The, area ������ = 1 × ������ = 12.5 × 4.125 = 51.5625 ������2 As 1 has only 3 significant figures, therefore ������ = ������������. ������ ������2 10. Rules of Rounding off Significant Figures i. If the digit to be dropped is less than 5, then the preceding digit is left unchanged. E.g., 1.54 is rounded of to ������. ������. ii. If the digit to be dropped is greater than 5, then the preceding digit is raised by one. E.g., 2.48 is rounded off to ������. ������. iii. If the digit to be dropped is 5 followed by digit other than zero, then the preceding digit is raised by one. E.g., 3.55 is rounded of to ������. ������. iv. If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd and left unchanged if it is even. E.g., 3.750 is rounded off to ������. ������ and 4.650 is rounded off to ������. ������. 11. ORDER-OF MAGNITUDE CALCULATIONS If value of physical quantity ������ satisfy 0.5 × 10������ where,������ ≤ 5 × 10������ ������ is an integer ������ is called order of magnitude Illustration: The diameter of the sun is expressed as 13.9 × 109 ������. Find the order of magnitude of the diameter? Sol. Diameter 13.9 × 109 ������ Diameter 1.39 × 1010 ������ order of magnitude is 10. 12. ERROR The lack in accuracy in the measurement due to the limit of accuracy of the instrument or due to any other cause is called an error. i. Absolute Error The difference between the true value and the measured value of a quantity is called absolute error. If ������1, ������2, ������3, … , ������������ are the measured values of any quantity ������ in an experiment performed ������ times, then the arithmetic mean of these values is called the true value (������������) of the quantity. ������������ = ������1 + ������2 + ������3 + ⋯ ������������ ������ The absolute error in measured values is given by Δ������1 = ������������ − ������1 Δ������2 = ������������ − ������2 Syn.

P1 – Units & Measurement 10 Δ������������ = ������������ − ������������ ii. Mean Absolute Error The arithmetic mean of the magnitude of absolute errors in all the measurement is called mean absolute error. Δ������ = |Δ������1| + |Δ������2| + ⋯ + |Δ������������| ������ iii. Relative Error The ratio of mean absolute error to the true value is called relative error. Mean absolute error ������������ Relative error = Ture value = ������������ iv. Percentage Error The relative error expressed in percentage is called percentage error. Δ������ Percentage error = ������������ × 100 % 13. PROPAGATION OF ERROR i. Error in Addition or Subtraction : Let ������ = ������ + ������ or ������ = ������ − ������ If the measured values of two quantities ������ and ������ are (������ ± Δ������) and (������ ± Δ������), then maximum absolute error in their addition or subtraction. Δ������ = ±(Δ������ + Δ������) ii. Error in Multiplication or Division : Let ������ = ������ × ������ or ������ = ������ ������ If the measured values of ������ and ������ are (������ ± Δ������) and (������ ± Δ������), then maximum relative error Δ������ = ± (Δ������ + Δ������) ������ ������ ������ iii. Error in Raised to a Power : Let ������ = ������������������������/������������ If the measured values of ������, ������ and ������ are (������ ± Δ������) and (������ ± ΔB), then maximum error Δ������ = ������ (Δ������) + ������ (Δ������) + ������ ������ ������ ������ (Δ������) ������

P1 – Units & Measurement 11 P1.1 Units and Dimensions CONCEPTS 1. Various standard system of units for measuring the 7 basic physical quantities. 2. Fundamental and derived units and their relationship. 3. What is a Dimensional Formula? 4. Dimensional analysis along with its applications. PRE-READING You may refer to one of the following sources: Category Book Name Chapter Section 1, 2 1.1-1.5, 2.1-2.5, 2.8, 2.9, 2.10 REQUIRED NCERT Class XI 1 1.3 and 1.4 (Important) ADDITIONAL H C Verma Volume I PRE-READING EXERCISE Q1. What are the ������������ and ������������������ units of length, mass and time? Q2. What is the dimension of length, mass, time and speed? Q3. What is the SI unit used to denote luminous intensity? Q4. Angular momentum is defined as ������������������ where ������ is the mass, ������ is the speed and ������ is the distance of the body from a specific point in space. Write the dimensional formula for angular momentum. Q5. The mass of the earth is 6 × 1024 kg. The average mass of the atoms that make up Earth is 40������ (i.e. atomic mass units). How many atoms are there in earth (1������ ≈ 1.5 × 10−27������������)? IN CLASS EXERCISE LEVEL 1 Q1. Find the dimensional formula of the quantities on the left hand side (LHS) of the following expressions: I. Charge = Current × Time II. Velocity = Straight line distance Time Taken III. Acceleration = Rate of change of velocity IV. Momentum = Mass × Velocity V. Force = Mass × Acceleration VI. Strain = Change in length Original length VII. Stress = Force Area Also find the SI units of these quantities. P1.1

P1 – Units & Measurement 12 Q2. Convert the following units as mentioned I. 2 decimeters = __________ millimeters II. 1 angstrom = __________ kilometers III. 1 ������g = __________ ������������ IV. 1 degree = __________ minutes V. 100 degrees = __________ radians VI. 1 day = __________ seconds VII. 2 nanoseconds = __________ milliseconds VIII. 15 ������������ = __________ ������������ Q3. Let us consider the equation 1 ������������2 = ������������ℎ , where ������ is the mass of the body, ������ is it’s velocity, ������ is the 2 acceleration due to gravity and ℎ is the height. Is the equation dimensionally correct? Q4. A vehicle is moving with a speed of 80 ������������/ℎ. How many meters will it cover in 1 ������? (Give answer to the nearest meter) Q5. The micrometer (1 ������������) is often called as the micron. I. How many microns make up 500 ������������? II. What is the fraction of a millimeter that equals 0.001 ������������ ? Q6. Suppose your hair grows at the rate of 0.17 inches per day. Find the rate at which it grows in nanometers per second. (Find answer to the nearest nanometer) (1 ������������������ℎ = 2.54 ������������) (2.54 × 1.7 ≈ 4.32) LEVEL 2 Q7. The SI unit of energy is J (Joule) which is also ������������ ������2 ������−2; that of speed ������ is ������������−1 and of acceleration ������ ������������ ������������−2. Which of the formulae for kinetic energy (������) given below can you rule out on the basis of dimensional arguments (������ stands for the mass of the body): A) ������ = ������2������3 B) ������ = 1 ������������2 C) ������ = ������������ D) ������ = 3 ������������2 2 16 E) ������ = 1 ������������2 + ������������ 2 Q8. The displacement of a progressive wave is represented by ������ = ������ sin(������������ − ������������), where ������ is distance and ������ is time. Write the dimensional formula of (i) ������ (ii) ������ (iii) ������. Q9. The equation of state for a real gas can be expressed as (������ + ������������2) (������ − ������) = ������������ (where ������ is the pressure (������������������������������), ������������������������ ������ the volume, ������ the absolute temperature and ������, ������ and ������ are constants. What are the dimensions and units of ������, ������ and ������? Also write their SI units. Q10. The velocity of a wave in a string is dependent on the tension (Force) in the string (������) and the mass per unit length of the string ������. Find the expression for velocity of the wave in the string as a function of some power of the tension and the mass per unit length of the string. LEVEL 3 Q11. If momentum (������ = ������������������������ × ������������������������������������������������), area (������) and time (������) are taken to be fundamental quantities, then kinetic energy = 1 ������������2 has the dimensional formula 2 A) (������1������–1������1) B) (������2������1������1) C) (������1������− 1 ������1 ) 1 2 D) (������1������2������–1)

P1 – Units & Measurement 13 Q12. The angle subtended by an asteroid at the eye of two telescopes is measured to be 3′′ each from two observatories placed at diametrically opposite ends of the earth (As shown in the figure given below). If the diameter of earth is 1.3 × 107 ������, compute the distance of the asteroid from the earth in ������������ (Calculate to 1 decimal place). ( 60′′ = 1′ and 60′ = 1°)(1 ������������ = 1.5 × 1011 ������) HOMEWORK LEVEL 1 Q1. Fill in the blanks I. 2 ������/������������������ = __________ ������������/������������������������ II. 5 ������������ ������ ������−2 = __________ ������ ������������ ������−2 III. 4 ������/ ������������ ������ = __________ ������������/������ − ������������ (������������ = mega-joule, ������������ = milli-kelvin) IV. 12 ������������/������ = __________ ������������ − ������������/������������������������������������ V. 2 ������������/������������3 = __________ ������/������3 VI. 3������2 = __________ ������������2 Q2. 15 ������������ /������ = __________ ������ / ������ = __________ ������ /ℎ������������������ = __________ ������������/ℎ������������������ Q3. 4 ������ = __________ c������3 = __________ ������������3 Q4. If one light-year is approximately equal to 9.46 × 1015m, then express 25 light years in centimeters (up to 3 decimal places). Q5. The mass of a solid cube of edge length 5 ������������ is 800 ������. Determine the density ������ (= ������������������������ ) of the cube in SI units. ������������������������������������ Q6. The sun’s angular diameter as measured from some far off planet is 8 × 10−6 ������������������. The diameter of the sun is 1.39 × 109 ������. What is the distance of the planet from the sun? Q7. Surface tension is defined as force exerted per unit length by water surface to the surrounding water and to the surface it contacts. This is responsible for circular shape of bubble and formation of drops on glass. Surface tension of water is 72 ������������������������/������������. Convert this quantity in SI units. (1 dyne = 1 ������– ������������/������2) Q8. The electric field (������) due to a charge is defined as force (������) divided by the magnitude of charge (������). What is the dimensional formula of electric field? (������ℎ������������������������ = ������������������������������������������ × ������������������������) Q9. What is the dimensional formula of electric potential (������) due to a charge ������, if ������ = ������������. Where ������ is the electric field and ������ has a unit of distance. (For dimension of ������, refer question above). Q10. The radius (������) of a circle inscribed in any triangle whose side lengths are ������, ������ and ������ is given by the formulae ������ = 1 [(������−������)(������−������)(������−������)]2 where ������ is an abbreviation for semi-perimeter, ������ = ������+������+������ ������ 2 Check the formula of radius on the circle for dimensional consistency. P1.1

P1 – Units & Measurement 14 LEVEL 2 Q11. The formulae for electric field and electric flux (along with the symbols in brackets) are as given here. ������������������������������������������������ ������������������������������(������) = ������������������������������(������) and ������������������������������������������������ ������������������������(������) = ������������������������������������������������ ������������������������������ (������) × ������������������������ (������) . Find the dimensions of ������ℎ������������������������(������) Electric flux. Q12. The energy density (������) of the electrostatic field is given by ������ = 1 ������0������2, where ������ is the electric field and the value 2 of ������0 = 8.85 × 10−12������2������4������������−1������−3, find the dimensions of ������. Q13. The displacement of a particle is given by ������ = ������������ + ������ where ������ and ������ are constants, and ������ is the time. What are ������2 the dimensions of ������ and ������? Q14. The speed (������) of ripples on the surface of water depends only upon the surface tension (������ = ������������������������������⁄������������������������������ℎ), Density(������) = ������������������������/������������������������������������ and wavelength (������). The speed ������ is proportional to which of the following; (the units of surface tension are ������/������). A) √������������������ B) √������������������ C) ������ D) ������������������ ������������ Q15. Newton’s law of universal gravitation is represented by ������ = ������������������ ������2 Where ������ is the gravitational force, ������ and ������ are masses, and ������ is the distance between the masses. Force has the SI units ������������ − ������/������2. What is the SI unit of the gravitational constant ������? LEVEL 3 Q16. The velocity (speed) of a particle is given by ������ = ������������������2 + ������ where ������ and ������ are constants and ������ is the time. What ������ is the dimension of ������? Q17. If the centripetal force (������) is of the form ������������������������������������ , find the values of ������, ������ and . Q18. The heat produced in a wire carrying an electric current depends on the current, the resistance and the time. Derive an equation relating the quantities using dimensional analysis. The dimensional formula of resistance is [������������2������−2������−3] and it is given that heat is a form of energy. Q19. When a solid sphere moves through a liquid, the liquid opposes the motion with a force ������. The magnitude of ������ depends on the coefficient of viscosity ������ of the liquid, the radius ������ of the sphere and the speed ������ of the sphere. Assuming that ������ is proportional to different powers of these quantities, derive a formula for ������ using the method of dimensions. {Given: Dimension of coefficient of viscosity is [������1������−1������−1]}. Q20. Einstein discovered an important relation in his theory of relativity. However, when he wrote the relation, he forgot to put the constant c, the speed of light in the equation. What are the possible places the constant could be inserted in the equation? 1 ������ = ������������(1 − ������2)2 In the above equation, ������ is the new length, ������������is the original length and ������ is the velocity of the body.

P1 – Units & Measurement 15 P1.2 Significant Figures and Error Analysis CONCEPTS 1. Significant digits and rounding off and its application in arithmetic operations. 2. Different types of errors in measurement of physical quantities. 3. Errors and the terms associated with it (accuracy and precision). 4. Computation of the maximum error resulting from a combination of one or more physical quantities in mathematical operations. PRE-READING You may refer to one of the following sources: Category Book Name Chapter Section 2 2.6 , 2.7 REQUIRED NCERT Class 11 2 2.12,-2.14 ADDITIONAL HC Verma Part 1 (Class 11) PRE-READING EXERCISE Q1. Round off 126.3 to the nearest integer Q2. Round off 12.92 to the nearest integer Q3. The scale used to measure the length of a football ground has a misprint on it. What type of error is this? Q4. A voltmeter is a device used to measure the voltage across a circuit. A voltmeter used in one of the experiments shows an unpredictable fluctuation. What type of error are we encountering in this case? Q5. It is possible to measure the exact length of the table.(T/F) Q6. If the length of a table is 2 ������ and the measurements came to be 2.05, 2.03, 2.02, 1.98 ������ then what is the average error in the measurement? IN CLASS EXERCISE LEVEL 1 Q1. The values of length of a rod in an experiment were measured to be2.48 ������, 2.46 ������, 2.49 ������, 2.50 ������ and 2.48 m. Find the average length, average absolute error, relative error and percentage error. Express the result with an error limit. Q2. Round off the following numbers to 4 significant figures I. 55.324 II. 11.125 III. 29.835 Q3. If the error in measuring a quantity ������ is 2%, compute the percentage error in I. ������2 II. ������5 III. 1 IV. √������ ������ P1.2

P1 – Units & Measurement 16 Q4. 5.74 ������ of a substance occupies1.2 ������������3. Calculate its density=mass/volume to the correct number of significant digits. LEVEL 2 Q5. Listed below are a few physical quantities that depend on three fundamental quantities A, B and C in different ways. The percentage errors in measuring A,B and C are respectively 1%, 3% and 4% respectively. Find the percentage error in the resultant quantities Quantity Percent Error in the quantity 1 I. ������ = ������3������ ������ 3 II. ������ = √������������������ Q6. If displacement of a body is given by ������ = (200 ± 5) ������ and the time taken by it is given by ������ = (20 ± 0.2) seconds, then find the percentage error in the calculation of average velocity (������������������������������ ������������������������������������������������������������������������). ������������������������������ ������������������������ Q7. Indicate the type of systematic error likely to occur in each of the following situations: Situation Type of Error I. The markings of the spring balance between 50 – 55 N have been erased due to corrosion II. Out of habit, you always look at the nearest reading to the right instead of looking at the nearest reading to the left. III. While measuring an angle using a protractor the angle’s vertex is not made to coincide with the central point of the protractor IV. Both your eyes do not perceive the same image of the scale reading due to an eye defect. V. An instrument for measuring the acceleration due to gravity consistently shows ������ = 10.1 ������/������2 VI. You do not stand at the center of a weighing machine when you try to record your weight. LEVEL 3 Q8. Displacement of a particle is given by; ������ = ������������ + 1 ������������2, where ������ is velocity ������ is acceleration, ������ is defined as time 2 taken for motion. The value of the quantities are measured as ������ = 5.0 ������/������, ������ = 10.0 ������ and ������ = 2.0 ������/������2. In an experiment the error in measuring velocity, time and acceleration was 2%, 4% and 2% respectively. Find the value of ������ and percentage error in calculation of displacement(������) to the correct number of significant digits.

P1 – Units & Measurement 17 TO REVISE THE CONCEPTS AND DEFINITIONS LEARNT IN THE CHAPTER, KINDLY SOLVE THE CROSSWORD BELOW HINTS ACROSS 1. I am represented by metre in SI units. How am I represented in FPS system of units?(4) 5. I am a basic property of matter I am independent of temperature, pressure or location of object in space.(4) 6. I define the repeatability of a measuring instrument. Accuracy and me are siblings.(9) 7. I am a derived quantity. I can be added to weight. How many physical dimensions are present in my dimensional formulae?(5) 8. I am a type of error which can only be reduced by changing the measuring instrument.(5,5) 9. I am entirely dependent on relative error. Multiply it by 100 and I arrive.(10,5) DOWN 2. I am unitless. I am not a part of the 7 fundamental quantities but stand independent identity. I am respresented by the symbol Sr(9) 3. I am a device which can measure lengths upto 3 places after the decimal. Who am I?(5,5) 4. I am a method developed by scientists to measure astronomical distances. Recognise me?(8) 10. I am the clock in which any time interval is measured based upon periodic vibrations produced in a Caesium atom(6) P1.2

P1 – Units & Measurement 18 HOMEWORK LEVEL 1 Q1. If the mass of a car is denoted as (200 ± 5) ������������ then: I. What is the absolute error in measuring the car’s mass? II. What are the relative and percentage errors in measuring the car’s mass? Q2. Find the number of significant digits in the following I. 0.0029 II. 1.9 × 106 III. 12.900 IV. 12900 Q3. The masses of two objects are 20 ± 2 ������������ and 10 ± 4 ������������ respectively. Compute error and the value of I. The total mass of the two objects II. The difference between the masses of the two objects Q4. The error in measuring a quantity ������ is 5% , quantity ������ is 2% . Compute the maximum possible error in measuring: I. ������������ II. ������/������ Q5. Identify the more accurate instrument in each of the following cases (Assume that the error is only due to the instrument) : Actual measurement Measurement by instrument 1 Measurement by instrument ������ Which is more accurate? 15 ������������ 14.36 ������������ 15.28 ������������ 100 ������������ 102 ������������ 101 ������������ 18.25 ������������������������������ 18.65 ������������������������������ 17.99 ������������������������������ Q6. Identify the instrument with greater precision in each of the following cases: Instrument 1 Precision Instrument 2 Precision Which is more precise? A 50 ������������ scale with 100 small A 100 ������������ scale with 50 small divisions divisions A weighing machine which A weighing machine which can measure upto a milligram can measure upto a centigram Q7. For each of the following readings presented in the table given below, compute the absolute error in the reading, the relative error in the reading and the percentage error. True Value of Your observation Absolute error Relative error Percentage measurement error 14.6 ������������ 15 ������������ 22 ������ 20 ������ 3000 ������ 3050 ������ I. Is the reading with the maximum absolute error (magnitude wise) also the one with the maximum relative error? II. Will this always be the case?

P1 – Units & Measurement 19 Q8. Find the number of significant digits in I. 0.2620 ������/������������3 II. 2.031 ������/������2 III. 0.0007083 ������ IV. 9.56 × 1026 ������������ V. 55.40 ������ Q9. Round off each of the following numbers to 4 significant figures. I. 14.645 II. 16.324 III. 10.335 Q10. Round off the following numbers to 3 significant figures I. 7.230 II. 9.283 III. 16.24 Q11. Each side of a cube is 7.203 ������ in length. Compute the total surface area and the volume of the cube to appropriate significant figures. (7.2032 = 51.883209, 7.2033 = 373.714754427 ) Q12. Which one of the following instruments to measure length is the most precise? A) A Vernier caliper with 20 divisions on the sliding scale of length 2 ������������. B) A screw gauge of pitch 1 ������������ and 100 divisions on the circular scale. C) An optical instrument that can measure length upto 700 ������������(wavelength of light). Q13. Consider the following measurements of length by 5 different instruments and compute the absolute, relative and percent error for each. Actual Measurement Your observation Absolute error Relative error Percent error 15 ������������ 14. 6 ������������ 200 ������������ 215 ������������ 990 ������������ 1000 ������������ 4.8 ������ 5 ������ 201 ������ 200 ������ I. Which is the reading with the least absolute error? II. Which is the reading with the least relative error? Q14. Consider a standard measuring 15 ������������ ruler that all of you use. I. What is the least count if you use the centimeter side to measure length? II. What is the least count if you use the inch side (in ������������)? ( 1 ������������������ℎ = 2.54 ������������) III. Which side is more precise for length measurements? Q15. Shown below is some information on the measurement of a 10 ������������ long rod using 3 different instruments. Instrument Reading 1 meter scale with 1000 small divisions 10.1 ������������ Vernier Caliper with least count 0.05 ������������ 10.15 ������������ A laser scale capable of measuring lengths as low as 1 ������������ 10.2 ������������ I. Which is the most precise instrument? II. Which is the most accurate instrument? P1.2

P1 – Units & Measurement 20 Q16. State the number of significant figures in II. 0.0007095 ������ I. 77.01 ������/������2 IV. 0.000708300 ������ III. 4.34000 ������ VI. 0.04 ������3 V. 0.390200 ������/������������3 VIII. 0.9720 ������/������������3 VII. 0.019 ������3 X. 6.79 × 1029 ������������ IX. 3.034 ������/������2 XI. 0.001320 XII. 1.2233500 Q17. The sides of a rectangle are (10.5 ± 0.2) ������������ and (5.2 ± 0.1) ������������. Calculate its perimeter with error limits. Q18. The measured values of the resistances of two resistors are (8 ± 0.3) ������ℎ������ and (24 ± 0.5) ������ℎ������. The two resistors are connected in series such that ������������������������������������ = ������1 + ������2. Find the resistance of the combination. Also find the maximum percentage error. Q19. If the errors in measurement of mass and velocity of a body are found to be 3% and 2% respectively then what will be maximum possible error in calculation of kinetic energy? (Kinetic energy is given as K.E.= 1 ������������2). 2 LEVEL 2 Q20. Two quantities ������ and ������ are combined in various ways to result in new quantities. ������ = (50 ± 2) units ������ = (100 ± 4) units Quantity Mean Value Error Value with Error Relative Error 1. ������ + 2������ 2. ������/������ 3. ������������ 4. 3������ – 2������ 5. 2������������ 6. 4������/������ Q21. A physical quantity ������ is related to four observables ������, ������, ������ and ������ as ������ = ������3������2 . The percentage errors of √������������ measurement in ������, ������, ������ and ������ are 1%, 3%, 4% and 2%, respectively. What is the percentage error in the quantity ������? Q22. The coefficient of viscosity (������) of a liquid determined by the method of flow through a capillary tube is given by the formula ������ = ������������4������ where ������ = radius of the capillary tube, ������ = length of the tube, ������ = pressure difference 8������������ between its ends and ������ = volume of liquid flowing per second. Which measurement needs to be made most accurately and why? Q23. The potential difference across a wire is measured with a voltmeter having a least count of 0.2 ������������������������ and the current in the wire is measured with an ammeter having a least count of 0.1 ������������������������������������. The following readings were obtained. Voltmeter reading (������) = 6.4 ������������������������ Ammeter reading (������) = 2.0 ������������������������������������ Find the value of the resistance of the wire with maximum error. Also find the maximum percentage error. Given ������ that resistance ������ = ������ .

P1 – Units & Measurement 21 Q24. In an experiment for determining the density (������) of a rectangular block of a metal, the dimensions of the block are measured with calipers having a least count of 0.01 ������������ and its mass is measured with the beam balance of least count 0.1������. The measured values are: Mass of block(������) = 40.0������; Length of block(������) = 4.0 ������������; Breadth of block (������) = 2.50 ������������ and Thickness of the block(������) = 0.40 ������������ . Find the maximum permissible error in the determination of (������). (������ = ������������������������ ) ������������������������������������ Q25. Listed below are some physical quantities that depend on three fundamental quantities ������, ������ and ������ in different ways. The percent errors in measuring ������, ������ and ������ are respectively 1%, 3% and 4% respectively. Quantity Percent Error in the quantity I. ������ = ������������/������ II. ������ = ������������������ III. ������ = ������2������3������ Q26. The percentage errors in the measurement of the length of a simple pendulum and its time period are 2% and 3% respectively. What is the maximum error in the value of the acceleration due to gravity? (Time period is given by ������ = 2������√������������) Q27. A student measures the value of ������ with the help of a simple pendulum using the formula ������ = 4������������22������. The errors in the measurement of ������ and ������ are Δ������ and Δ������ respectively. Then in which of the following cases is the error in the value of ������ minimum? (Given ������ = 100 ������������ ; ������ = 100 ������) A) Δ������ = 0.5������������ , Δ������ = 0.5������ B) Δ������ = 0.2������������ , Δ������ = 0.2������ C) Δ������ = 0.1������������ , Δ������ = 1.0������ D) Δ������ = 0.1 ������������ , Δ������ = 0.1 ������ P1.2

P1 – Units & Measurement 22 Test Practice Problems Purpose: To practice a mixed bag of questions in a speed based format similar to what you will face in entrance examinations. In most entrance examinations, you will get not more than 3 minutes to attempt a question. Hence, you need to be able to attempt a question in less than 3 minutes, and at the end of 3 minutes skip the question and move to the next one. Approach:  Attempt the Test Practice Problems only when you have the stipulated time available at a stretch.  Start a timer and attempt the section as a test.  DO NOT look at the answer key / solutions after each question.  DO NOT guess a question if you do not know it. Competitive examinations have negative marking.  Solve as much as possible within the stipulated time, and then fill the OMR provided at the end of the TPP.  Fill the table at the end of the TPP and evaluate the number of attempts, and accuracy of attempts, which will help you evaluate your preparedness level for the chapter. No. of questions: 25 TEST PRACTICE PROBLEMS – 1 Time per question: 3 mins Total time: 75 mins Q1. Verify whether each of the following is a correct statement I. Time Period; ������ = ������������ (where ������ is mass, ������ is acceleration due to gravity, ������ is the length) is dimensionally ������2 correct. II. Displacement ������ = ������������ + 1 ������������2 (where ������ = initial speed or velocity, ������ = time, and ������ = acceleration) is 2 dimensionally accurate III. ������2 = ������2 + 2������������ (where ������ = initial speed or velocity, ������ = final speed or velocity, ������ = acceleration, ������ = displacement) is dimensionally incorrect IV. ������ = ������ + ������������ (where ������ = final speed or velocity, ������ = initial speed or velocity, ������ = acceleration, ������ = time) is dimensionally correct A) FTFT B) FTTT C) TTTT D) TTFT Q2. In the expression ������ = 3������������2, ������ and ������ have dimensions of capacitance [������−1������−2������2������2] and magnetic induction [������������−1������−1] respectively. What is the dimension of ������? A) [������−3������−1������3������4] B) [������−3������−2������4������4] C) [������−2������−2������4������4] D) [������−3������−3������4������] Q3. A cube has a side of length 1.2 × 10−2������. Calculate its volume A) 2 × 10−6������3 B) 1.73 × 10−6������3 C) 1.7 × 10−6������3 D) 1.728 × 10−6������3 Q4. A cylindrical wire has mass (0.300 ± 0.003)������, radius (0.500 ± 0.005) ������������ and length (6.00 ± 0.06) ������������ . The maximum percentage error in the measurement of its density is A) 1% B) 2% C) 3% D) 4%

P1 – Units & Measurement 23 Q5. A student uses a simple pendulum of exactly 1������ length to determine ������, the acceleration due to gravity. He uses a stop watch with the least count of 1 ������ for this and records 40 ������ for 20 oscillations, for this observation which of the following statements is/are true A) Error ∆������ is measuring ������, the time period, is 0.1������ B) Error ∆������ is measuring ������, the time period, is 1������ C) Percentage error in the determination of ������ is 5% D) Percentage error in the determination of ������ is 2.5% Q6. A gas bubble, from an explosion under water oscillates with a period ������ proportional to ������������������������������������, where ������ is the static pressure, ������ is the density of water and ������ is the total energy of the explosion. What would the expression for time period be proportional to? (������ = ������������������������������ , ������ = ������������������������ , ������ = ������������������������������ × ������������������������������������������������������������������������, ������������������������������ = ������������������������ × ������������������������������������������������������������������������) ������������������������ ������������������������������������ A) 2√������ 3√������ ������ 5 B) 2√������ 3√������ ������− 5 C) 2√������ 3√������ 5 D) 2√������ 3√������ ������− 5 6 6 6 ������6 Q7. The ������������ and ������������������ units of energy are joule and erg respectively. How many ergs are equal to one joule? A) 103 B) 10−3 C) 107 D) 10−7 Q8. Young’s modulus of steel is 19 × 1010������/������2. Express it in ������������������������/������������2. Here dyne is the CGS unit of force A) 19 × 1010 B) 19 × 1017 C) 19 × 1011 D) 19 × 103 Q9. If velocity([������]), time ([������]) and force ([������]) were chosen as basic quantities then find the dimension of mass A) ������������������−2 B) ������������������ C) ������������2������−2 D) ������������������−1 Q10. When a solid sphere moves through a liquid, the liquid opposes the motion with a force ������; the magnitude of ������ depends on the coefficient of viscosity ������ of the liquid, the radius ������ of the sphere and speed ������ of the sphere. Assuming that ������ is proportional to different powers of these quantities, deduce the formula for ������ using the method of dimensions A) ������ = ������������������2������ B) ������ = ������������������������2 C) ������ = ������������2������������ D) ������ = ������������������������ Q11. The time period of oscillations of a block attached to a spring, undergoing simple harmonic motion is dependent on the mass of the block ( ������ ) and the spring constant (������ ). Assuming that the time period of the block is proportional to some power of mass (������) and the spring constant (������), find the expression for the time period of the block using dimensional analysis. It is given that the dimension of the spring constant is [������1������0������−2] A) ������ = √������������ B) ������ = √������ ������ C) ������ = ������������������������������������������������√������������ D) ������ = ������������������������������������������������ √ ������ ������ Q12. If velocity (������), force (������) and energy (������) are taken as fundamental units, then the dimensional formula for mass will be A) ������−2������0������1 B) ������0������1������2 C) ������1������−2������0 D) ������−2������0������2 Q13. Time period ������ of a simple pendulum may depend on ������, the mass of the bob, ������, the length of the string and ������, the acceleration due to gravity, i.e. ������ = ������������������������������������. What are the values of ������, ������ and ������? A) ������ = 0, ������ = 1 and ������ = − 1 B) ������ = 0, ������ = 1 and ������ = 1 22 22 C) ������ = 1 , ������ = 1 and ������ = − 1 D) ������ = 0, ������ = − 1 and ������ = − 1 22 2 22 T.P.P.

P1 – Units & Measurement 24 Q14. If the time period of oscillation t of a drop of liquid of density ������, radius ������, vibrating under surface tension ������ is given by the formula ������ = √������������������������������������ and ������ = 1, ������ = −1 then what is the value of ������?(������ = ������������������������ , [������] = [������������−2]) ������������������������������������ A) 1 B) 3/2 C) 3 D) 2 Q15. An astronomical unit (AU) is the average distance between the earth and the sun, approximately measured to be 1.5 × 108 ������������ . If the speed of light is a constant 3.0 × 108 ������/������ calculate the speed of light in terms of astronomical units per minute. A) 0.002 B) 0.12 C) 0.2 D) 1.2 Q16. If the error in the measurement of the volume of a sphere is 6% then the error in the measurement of its surface area is A) 2% B) 3% C) 4% D) 6% Q17. If the centripetal force is of the form ������������������������������������ then find the values of ������, ������,and ������ A) ������ = 1, ������ = 0, ������ = −2 B) ������ = 1, ������ = 2, ������ = −1 C) ������ = 1, ������ = 0, ������ = −1 D) ������ = 1, ������ = −2, ������ = −2 Q18. The height of the building is 50 ������������. the same in millimetre is A) 560 ������������ B) 285 ������������ C) 1786.8 ������������ D) 1524 ������������ Q19. The name of the nearest star is proxima centauri. The distance of this star from Earth is 4 × 1016 ������. Find the distance of this star from Earth in mile A) 3.5 × 1013 mile B) 2.5 × 1013 mile C) 5.3 × 1013 mile D) 1.5 × 1013 mile Q20. The radius of hydrogen atom in ground state is 5 × 10−11 ������. Find the radius of hydrogen atom in fermimetre. (1 ������������ = 10−15 ������) A) 5 × 104 ������������ B) 2 × 104 ������������ C) 5 × 102 ������������ D) 5 × 106 ������������ Q21. One nautical mile is 6080 ������������. The same in kilometre is A) 0.9 ������������ B) 0.8 ������������ C) 1.85 ������������ D) None of these Q22. The area of a room is 10 ������2. The same in ������������2 is A) 107.6 feet2 B) 77 feet2 C) 77.6 feet2 D) None of these Q23. The density of iron is 7.87 ������/������������3. If the atoms are spherical and closely packed. The mass of iron atom is 9.27 × 10−26 ������������. What is the volume of an iron atom A) 1.18 × 10−29 ������3 B) 2.63 × 10−29 ������3 C) 1.73 × 10−28 ������3 D) 0.53 × 10−29 ������3 Q24. If ������ = ������ sin ������+������ cos ������, then B) the dimensions of ������ and ������ are not same D) none of the above ������+������ A) the dimensions of ������ and ������ are same C) ������ is dimensionless Q25. The unit of intensity of a wav is /������2 ? What are dimensions of intensity of wave? A) [������ ������−3] B) [������ ������1 ������0 ������−2] C) [������0 ������−1 ������−2] D) None of these

P1 – Units & Measurement 25 TEST PRACTICE PROBLEMS – 2 No. of questions: 18 Total time: 54 mins Time per question: 3 mins Q26. The optical path difference is defined as ∆������ = 2������. What are dimensions of optical path difference? ������ A) [������0������−1������0] B) [������1������1������0] C) [������������0������1] D) [������������−2������] Q27. The dimensions of wavelength is B) [������0������������0] D) None of these A) [������0������0������0] C) [������0������−1������0] Q28. The dimensions of frequency is A) [������−1] B) [������0������0������] C) [������0������−1������−2] D) None of these Q29. The power of lens is ������ = ���1���′, where ′������′ is focal length of the lens. The dimensions of power of lens is A) [������������−2] B) [������0������−1������−1] C) [������0������������0] D) None of these Q30. The radius of nucleus is ������ = ������0������1/3, where ������ is mass number. The dimensions of ������0 is A) [������������������−2] B) [������0������0������−1] C) [������0������������0] D) None of these Q31. If energy of photon is ������ ∝ ℎ������������������������������ Here, ℎ = Planck’s constant ������ = speed of light ������ = wavelength of photon Then the value of ������, ������ and ������ are A) 1,1,1 B) 1, −1,1 C) 1,1, −1 D) None of these Q32. One horse power is equal to A) 746 watt B) 756 watt C) 736 watt D) 766 watt C) Momentum D) Velocity Q33. If ������ = ������������2 Where, ������ = mass of the body ������ = speed of light Which of the following options is most likely to be ������ A) Energy B) Power Q34. One calorie of heat is equivalent to 4.2 ������. One BTU (British thermal unit) is equivalent to 1055 ������. The value of one ������������������ in calorie is A) 251.2 cal B) 200 cal C) 263 cal D) None of these Q35. The value of universal gas constant is ������ = 8.3 ������ − ������������������. Find the value of ������ in ������������������������������������ℎ������������������ ������������������������������ ������������������ ������������������������������������ − ������������������������ ������������������ A) 8.12 ������������������ ������������������������������/������ − ������������������ B) 0.00812 ������������������ ������������������������������/������ − ������������������ C) 81.2 ������������������ ������������������������������/������ − ������������������ D) 0.0812 ������������������ ������������������������������ / ������������������������������������ − ������������������ T.P.P.

P1 – Units & Measurement 26 Q36. Refer the data from above question, find the value of ������ in calorie per °������ per mol A) 2 cal/mol °������ B) 4 cal/mol °������ C) 6 cal/mol °������ D) 8.21 cal/mol °������ Q37. Electron volt is the unit or energy (1 ������������ = 1.6 × 10−19 ������) In ������-atom, the binding energy of electron in first orbit is 13.6 ������������. The same in joule (������) is A) 10 × 10−19 ������ B) 21.76 × 10−19 ������ C) 13.6 × 10−19 ������ D) None of these Q38. 1 ������������ of ������������ pressure is equivalent to one torr and one torr is equivalent to 133.3 ������/������2 . The atmosphere pressure in ������������ of ������������ pressure is A) 70 ������������ B) 760 ������������ C) 3.76 ������������ D) None of these Q39. One bar is equivalent to 105 ������/������2. The atmosphere pressure is 1.013 × 105 ������/������2. The same in bar is A) 1.88 bar B) 1.013 bar C) 2.013 bar D) None of these Q40. 1 revolution is equivalent to 360°. The value of 1 revolution per minute is A) 2 ������ rad/s B) 0.1047 rad/s C) 3.14 rad/s D) None of these Q41. If ������ = velocity of a body ������ = speed of light Then the dimension of ������ is B) ������������������−1 D) None of these ������ A) ������0������0������0 C) ������������2������−2 Q42. The expression for centripetal force depends upon mass of body, speed of the body and the radius of circular path. Find the expression for centripetal force A) ������ = ������������2 B) ������ = ������������2 C) ������ = ������������2 D) ������ = ������2������2 2������3 ������2 ������ 2������ Q43. The maximum static friction on a body is ������ = ������������. Here, ������ = normal reaction force on the body ������ = coefficient of static friction The dimensions of ������ is A) ������������������−2 B) ������0������0������0������−1 C) dimensionless D) None of these

P1 – Units & Measurement 27 DATA ANALYSIS Guide A # of questions Total problems in TPP B # Attempts Total attempts in OMR C # Correct Total questions correct D # Incorrect Out of the ones marked in OMR E # Unattempted ������ − ������ F Percentage attempts ������ ������ × 100 G Percentage Accuracy ������ ������ × 100 Question type # Correct (C) # Incorrect (I) # Unattempted (U) Easy Medium Hard Tip: To begin with, your accuracy must be high, typically > 60%. Percentage attempts should be > 50% As time progresses, your percentage attempts should increase without a reduction in accuracy. Additionally, you should be able to get > 80% Easy questions correct, as they involve basic recall of the concepts and formulae of the chapter. T.P.P.

P1 – Units & Measurement 28 PRE-TEST Answer Key Q1. 5 × 10−6 Q4. I. 50% increase II. 25% decrease Q2. I. 1.25 II. 0.80 Q5. 7.5 × 10−9 III. 3.33 Q6. 6 × 104 Q3. 10 Q7. ������ = 5, ������ = 10 13 Q8. ������ 2 P1.1 UNITS AND DIMENSIONS PRE-READING EXERCISE Q6. 50������������ Q1. SI: meter, kilogram, second, CGS: centimeter, ������ gram, second Q2. Length= [������], Mass= [������], Time=[������], LEVEL 2 Speed = [������0������1������−1] Q7. A, C, E Q3. Candela Q8. I. [������0������1������0] Q4. [������1������2������−1] Q5. 1050 II. [������0������0������−1] III. [������0������−1������0] IN CLASS EXERCISE Q9. ������ = [������1������5������−2];������������������5������−2 ������ = [������0������3������0];������3 ������ = [������1������2������−2������−1]; ������������������2������−2������−1 LEVEL 1 Q10. ������ = ������√������������ Q1. I. [ ������0������0������1������1], ������ − ������ (Also called Coulomb) LEVEL 3 Q11. D II. [������0������1������−1], ������/������ Q12. 3.0������������ III. [������0������1������−2], ������/������2 HOMEWORK IV. [������1������1������−1], ������������ − ������/������ LEVEL 1 Q1. I. 2 × 109 V. [������1������1������−2], ������������ − ������/������2 (Also called Newton) II. 5 × 105 VI. [������0������0������0], Unitless III. 4 × 10−12 IV. 7.20 × 1010 VII. [������1������−1������−2], ������������ , ������/������2 (Also called Pascal) V. 2 × 1012 ������������2 VI. 3 × 1012 Q2. 15000 ������/������ , 5.4 × 107 g /hour , 54000 kg / hour Q2. I. 200 Q3. 4000 ������������3, 4 × 106������������3 II. 10−13 Q4. 2.365 × 1019 ������������ III. 10−9 Q5. 6400 ������������/������3 Q6. 1.7375 × 1014 ������ IV. 60 V. 5������ 9 VI. 86400 VII. 2 × 10−6 VIII. 1.5 × 10−5 Q3. Yes Q4. 22������ Q5. I. 5 × 1011 II. 10−6

P1 – Units & Measurement 29 Q7. 72 × 10−3������/������ Q14. A Q8. [������1������1������−1������−3] Q9. [������1������2������−1������−3] Q15. ������3 Q10. It is dimensionally Consistent ������������������2 LEVEL 2 LEVEL 3 Q11. [������1������3������−1������−3] Q16. [������0������0������−3] Q12. [������1������−1������0������−2] Q17. ������ = 1, ������ = 2 and ������ = −1 Q13. [������] = [������0������1������−1], [������] = [������0������1������2] Q18. ℎ =constant× ������2������������ Q19. ������ =Constant× ������������������ Q20. ������2 as divisor of ������2 P1.2 SIGNIFICANT FIGURES AND ERROR ANALYSIS PRE-READING EXERCISES III. Experimental Q1. 126 IV. Personal Q2. 13 V. Instrumental Q3. Instrumental Error VI. Personal Q4. Random Error Q5. False LEVEL 3 Q6. 0.02 Q8. 1.5 × 102������, 9% IN CLASS EXERCISE CROSSWORD LEVEL 1 Q1. ������������������ = 2.48 ������, ∆������������������������������ = 0.01 ������, ∆������������������������ = 0.004, ∆������������������������������������������������������������������ = 0.4 %, ������������������������ = 2.48 ± 0.01 ������ Q2. I. 55.32 II. 11.12 III. 29.84 Q3. I. 4% II. 10% III. 2% IV. 1% Q4. 4.8 ������/������������3 LEVEL 2 HOMEWORK Q5. I. Percentage Error in ������ = 15.33 % LEVEL 1 II. Percentage Error in ������ = 4% Q1. I. 5������������ Q6. 3.5 % II. 0.025 and 2.5% respectively Q7. I. Instrumental II. Personal Ans.

P1 – Units & Measurement 30 Q2. I. 2 Q12. (C) optical instrument II. 2 III. 5 Q13. Absolute error Relative error Percent error IV. 3 0.4 ������������ 0.02 2% Q3. I. 30 ± 6 ������������ 15 ������������ 0.075 7.5 % II. 10 ± 6 ������������ 10 ������������ 0.01 1.0 % Q4. I. 7% 0.2 ������ 0.04 4% II. 7% 1 ������ 0.005 0.5% Q5. I. Instrument 2 II. Instrument 2 I. The first reading with an error just 0.4 ������������ III. Instrument 2 II. The last reading with a relative error of only Q6. I. Instrument 1 0.005 II. Instrument 1 Q14. I. 1 ������������ Q7. I. No II. 0.1 ������������������ℎ (2.54 ������������) II. No III. ������������ side Absolute error Relative error Percentage error Q15. I. Laser Scale II. Meter Scale 0.4 ������������ 0.027 2.7 % Q16. I. 4 II. 4 2 ������ 0.10 10 % III. 6 50 ������ 0.017 1.7 % IV. 6 V. 6 Q8. I. 4 VI. 1 II. 4 VII. 2 III. 4 VIII.4 IV. 3 IX. 4 V. 4 X. 3 XI. 4 Q9. I. 14.64 XII. 8 II. 16.32 III. 10.34 Q17. 31.4 ± 0.6 ������������ Q10. I. 7.23 Q18. 32Ω, 2.5% II. 9.28 III. 16.2 Q19. 7% Q11. 311.3������2, 373. 7 ������3 LEVEL 2 Q20. Quantity Mean Value Error Value with Error Relative error 1. ������ + 2������ 250 10 250 ± 10 0.04 2. ������/������ 0.50 0.04 0.50 ± 0.04 0.08 3. ������������ 0.08 4. 3������ – 2������ 5.0 × 103 0.4 × 103 (5.0 ± 0.4) × 103 0.28 5. 2������������ −50 14 −50 ± 14 0.08 6. 4������/������ 0.08 1.0 × 104 0.08 × 104 (1.0 ± 0.08) × 104 2.0 0.16 2.0 ± 0.16

P1 – Units & Measurement 31 Q21. 13% Q25. I. 8% II. 8% Q22. Radius (R) III. 15% Q23. 3.2 ± 0.2Ω, 8% Q26. 8% Q24. 0.9 ������������ Q27. D ������3 TEST PRACTICE PROBLEMS Q. No. Ans. Level Mark (C) / (I) / (U) Q. No. Ans. Level Mark (C) / (I) / (U) as appropriate as appropriate Hard Easy Q1. A Medium Q23. A Medium Q2. B Easy Q24. C Medium Q3. C Medium Q25. A Medium Q4. D Hard Q26. A Easy Q5. C Hard Q27. B Easy Q6. B Medium Q28. A Easy Q7. C Easy Q29. B Easy Q8. C Easy Q30. C Easy Q9. D Hard Q31. C Easy Q10. D Hard Q32. A Easy Q11. C Hard Q33. A Medium Q12. A Hard Q34. A Easy Q13. A Hard Q35. D Easy Q14. C Medium Q36. A Medium Q15. B Easy Q37. B Medium Q16. C Medium Q38. B Easy Q17. B Easy Q39. B Easy Q18. D Easy Q40. B Medium Q19. B Easy Q41. A Easy Q20. A Easy Q42. B Medium Q21. C Easy Q43. C Q22. A Ans.


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