-1- This book is purely based on NCERT syllabus for CBSE & STATE Class XI. Class XI. Our Mission is Your Success

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1 -1- This book is purely based on NCERT syllabus for CBSE & STATE Class XI Part - 1 A REAL MENTOR Class XI Our Mission is Your Success Education Solutions Pvt. Ltd. Opp. St.Joseph s High School, Angamaly, Ernakulam , Kerala Ph: , info@radicesolutions.com web site :

2 -2- Title: Science Companion PART - 1 For Class XI Prepared by: "" "" "" "" Seby Paul Chithra Balakrishnan Ayana Abbas Hema Jose Published and Distributed by: Education Solutions Pvt. Ltd. Opp. St.Joseph s High School Angamaly, Ernakulam , Kerala Ph: , info@radicesolutions.com Web site: Programme Coordinator: Polachan Paul Layout & Design: Sini. S Printed at: Premier Printers, Angamaly Price: ` 260/- COPYRIGHT All rights reserved. No part of this publication or related recorded material, may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior permission of Radice Education Solutions Pvt. Ltd, Angamaly.

3 -3- Dear Parent/Student It is very important to recognize the fact that students differ from one another. Our current education system makes it difficult to address such differences, it is still based on one size fit all approach. When students are not presented with learning experience that is appropriate for their abilities, they lose motivation and sometimes even their interest in learning. Some students need a little more practice and mentoring to become high performers in. provides opportunity to practise more and also mentor every student with specific action plan to improve their weak areas. We provide this book in for the class XI students to develop confidence in the subject and to prepare the students in facing the examinations with ease. Here more conceptual questions are included which we hope will improve the ability of the students to face a wide variety of questions. We welcome suggestions for improving the quality of the book. Yours sincerely Seby Paul M.Tech Chairman, Solutions Pvt. Ltd.

4 part - 1 includes: -4- Physical World Units and Measurement Motion in a straight line Motion in a plane Laws of Motion Work, Energy and Power System of particles and rotational motion Gravitation VALUE BASED QUESTIONS SELF ASSESSMENT Tests Each chapter includes: topic wise detailed description Important derivations Worked out problems Very short answer questions Short answer questions Numerical problems HOTS questions SCERT previous year questions Entrance exam corner

5 -5-1 Physical World The word 'science' comes from Latin word which means 'to know'. The knowledge which humans have gained through observations and experiments, when organised systematically is called science. The sciences which deal with non living things are called physical sciences. For example: physics, chemistry, geology, geography etc. is a basic discipline in the category of 'Natural Science'. is the branch of science which is devoted to the study of nature and natural phenomena. has basically two domains of interest: macroscopic and microscopic. The macrosopic domain includes ordinary lab experimental phenomena, other terrestrial phenomena and astronomical phenomena. The microscopic domain includes, atomic, molecular and nuclear phenomena. Classical physics and modern physics are the names of the two domains which respectively deal with macroscopic and microscopic objects. The sub divisions are given below. Classical Mechanics Thermodynamics Electrodynamics Optics Modern Atomic and Molecular physics Nuclear and Particle physics Quantum Mechanics Condensed matter physics Fundamental forces in Nature has revealed that all the forces occurring in different contents arise from a small number of fundamental forces in nature. These are, (i) Gravitational force (ii) Weak nuclear force (iii) Electromagnetic force (iv) Strong nuclear force (i) Gravitational force It is the force of mutual attraction between any two objects by virtue of their masses. It is a universal force because everywhere in this universe, this force is present between two objects. The gravitational mutual force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This is Newton's universal law of gravitation. Gravitational force is long range. It does not depend on the intervening medium. This is the weakest fundamental force. (ii) Electromagnetic force Electric force or electrostatic force is the force between two charged particles at rest. The basic law is Coulomb's Law. It is attractive for unlike charges and repulsive for like charges. A moving electric charge in a magnetic field experiences a force called electromagnetic force. This is also a long range force and does not require any intervening medium. Electric force is nearly times greater than the gravitational force. This force dominates in atomic and molecular phenomena.

6 (iii) Weak nuclear force -6- This force appears only in certain nuclear processes such as the β-decay. In β-decay, the nucleus emits one electron and a neutrino. Weak nuclear force is greater than gravitational force but it is short range, with range within the nucleus. It is much smaller than strong nuclear force. (iv) Strong Nuclear force This is the force between protons and neutrons inside the nucleus. It binds them. This is the strongest fundamental force, 100 times greater than the electromagnetic force. It is charge independent and is seen between proton and neutron, proton and proton and neutron and neutron. Steam engine Technology Nuclear reactor Radio and Television Computers Lasers Production of ultra high magnetic field Rocket propulsion Electric generator Hydroelectric power Aeroplane Particle accelerators Sonar Optical fibres Non-reflecting coatings Electron microscope Photocell Scientific principle(s) Law of thermodynamics Controlled nuclear fission Generation, propagation and detection of electromagnetic waves Digital logic Light amplification by stimulated emission of radiation Superconductivity fields Newton's laws of motion Faraday's law of electromagnetic induction Conversion of gravitational potential energy into electrical energy Bernoulli's principle in fluid dynamics Motion of charged particles in electromagnetic fields Reflection of ultrasonic waves Total internal reflection of light Thin film optical interference Wave nature of electrons Photoelectric effect Fusion test reactor (Tokamak) Magnetic confinement of plasma Giant Metrewave Radio Telescope (GMRT) Bose-Einstein condensate Detection of cosmic radio waves Trapping and cooling of atoms by laser beams and magnetic fields.

7 Physical World -7- Nature of physical laws is the study of nature and natural phenomena. The keen observations and experiments lead the physicists to certain facts. An attempt is made to explain these facts on the basis of certain laws. A remarkable fact is that some special physical quantities remain constant in time. These are called the conserved quantities of nature. In classical physics, we often deal with the following conservation laws. (a) Law of conservation of energy The sum total of energy of all kinds in this universe remain constant. Energy can be transformed from one form to the other or transported from one place to the other, but the total amount of energy never changes. (b) Law of conservation of linear momentum In the absence of an external force, the linear momentum of a system remains unchanged. (c) Law of conservation of angular momentum In the total external torque acting on a system is zero, angular momentum of the system remains constant. (d) Law of conservation of charge It means that charges are neither created nor destroyed but are simply transferred from one body to another.

8 -8-2 Units and Measurements is a branch of natural science which deals with the physical world and principles governing its behaviour. Units A quantitative description of any physical phenomena always involves certain measurable quantities in terms of which the laws of physics are invariably expressed. Such quantities like force, velocity, time are called physical quantities. Measurement of any physical quantity involves comparison with a certain basic, widely accepted reference standard called Unit. A measurement is said to be complete only if we have the following informations. (i) Name of the unit (For eg: kg) (ii) How many times the unit is contained in that measurement (For eg: 2). This part of the measurement is also known as numerical value. Thus a measurement is represented as, 2kg. For example: You want to puchase a cloth of length 5 metre. Here '5' is numerical value and 'metre' is unit. A unit must possess following characteristics, (i) (ii) It should not change with time. It should be easily available for comparison with various measurements. (iii) It should not change with varying physical conditions such as temperature, pressure etc.. (iv) It should be convenient in size. Fundamental units: The unit of mass, length and time are taken as the fundamental units. This is because they cannot be derived from one another. Derived units: The units of physical quantities which can be expressed in terms of the units of mass, length and time are called derived units. System of Units: Following systems of measurements are commonly in use, (a) C.G.S system This system is based on centimetre as unit of length, the gram as the unit of mass and the second as unit of time. (b) F.P.S system In this system, the basic units are foot, pound and second. (c) M.K.S system This system was based upon metre, kilogram and second as its fundamental quantities.

9 Units and Measurements Illustration 1-9- Find the fundamental quantities in terms of which density can be expressed. Ans. We know, density Mass Volume Also volume (length) 3 Mass density (length) 3 Hence density is a derived quantity which is depend on mass and length. SI units (International system of units) The system of units which is at present internationally accepted for measurement. It is based on the following seven fundamental units and two supplimentary units. No Physical quantity Unit Symbol 1 Length Metre m 2 Mass Kilogram kg 3 Time Second S 4 Electric Current ampere A 5 Temperature Kelvin K 6 Amount of substance mole mol 7 Luminous intensity Candela Cd Two supplementary quantities are, (i) Radian : One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle. O r dθ ds (ii) If an arc of length ds subtends an angle dθ at the centre O of a circle of radius r as shown in figure, then dθ ds radian r Steradian : One steradian is the solid angle substended at the centre of a sphere, by that surface of the sphere, which equal in area to the square of radius of the sphere. If an area da of a spherical surface subtends a solid angle d Ω at the centre of the sphere O of radius r as shown in the figure, the d Ω da r 2 steradian O r dω da Conversion factors To convert a physical quantity from one set of units to the other, the required multiplication factor is called conversion factor.

10 -10- Magnitude of a physical quantity numerical value (η) unit (µ), while conversion from one set of units to the other, the magnitude of the quantity must remain same. η 1 µ 1 η 2 µ 2 Illustration 2: i.e., η The acceleration due to gravity is 9.8 ms -2. Give its value in ft s µ Ans. As 1m 3.2 ft DIMENSIONS 9.8 ms ft s ft s -2 All physical quantities can be expressed in terms of the seven fundamental units. So dimensions of a physical quantity are the powers to which the base quantities represent. Dimensions and dimensional analysis Dimensions of physical quantities The unit of mass, length and time are represented by the capital letters M, L and T. Area length breadth L L L 2 Here 2 is known as dimension in length. Therefore area is used to possess two dimensions in length. Similarly volume length breadth height L L L L 3 So volume has three dimensions in length. Therefore area and volume are independent of mass and time, i.e., they have zero dimensions in mass and time. So area M 0 L 2 T 0, volume is M 0 L 3 T 0 Eg: Find out the dimensional formula for velocity. We know that velocity distance time L T L1 T 1 M 0 L 1 T 1 Therefore velocity has zero dimension in mass, one dimension in length and 1 dimension in time. Dimension raised to represent that quantity. For example, Acceleration Dimension of acceleration, [Acceleration] But [Velocity] [Acceleration] Velocity Time [Velocity] [Time] [Length] [Time] LT 1 T 2 LT LT 1 T

11 Units and Measurements -11-

12 -12-

13 Units and Measurements Illustration 3: -13- If a physical quantity is represented as Z IFV2 where I is moment of inertia, F force, V velocity, W work 3 WL and L length. Find the dimension of Z. Ans. We have [I] ML 2 [F] MLT -2 [V] LT -1 [W] ML 2 T -2 [Z] [Z] ML 2 MLT -2 (LT -1 ) 2 ML 2 T -2 L 3 M 2 L 5 T -4-2 MT ML 5-2 T Hence the quantity may be surface tension, force gradient or spring constant. Applications of dimensional analysis: 1. To check the dimensional correctness of a given physical relation. This is based on the 'Principle of homogeneity' of dimensions. According to this principle the dimensions of each term on both sides of an equation must be the same i.e., [L. H. S] [R. H. S] This principle is based on the fact that only quantities of same kind can be added or subtracted. Illustration 4 : Ans. Consider the formula, F MV 2 r 2 Dimensionally, [F] MLT -2 [R H S] M(LT -1 ) 2 L 2 MT -2 Here [L H S] [R H S], this formula is not correct dimensionally. 2. To convert a physical quantity from one system of units to the other. Ans. This is based on the fact that magntiude of a physical quantity remains same whatever system is used for measurement. i.e., magnitude numerical value (η) unit (u) i.e., η 1 u 1 η 2 u 2 So if a quantity is represented by [M a L b T c ] then, Illustration 5: η 2 η[ M 1 M 2 ] a [ L 1 L 2 ] b [ T 1 T 2 ] c Convert 1N into its C G S unit dyne. Ans. Newton is the unit of force having dimension MLT -2. In SI system, M 1kg, L 1m, T 1 sec. In Cgs system, M 1gm, L 1 cm, T 1 sec. From dimensional formula of force we have, 1 newton 1 Kg 1m (1s) -2

14 -14- But 1 Kg 1000g, 1m 100 cm 1N 1000 gm 100cm (1s) gm 1cm 1s 2 By definition, 1 dyne 1 gm 1 cm 1s 2 1 newton 10 5 dynes 3. Deducing relationship among the physical quantities. If one knows the quantities on which a particular physical quantity depends, method of dimension may be helpful in the derivaton of the relation. Illustration 6: The time period of a simple pendulum depends upon (i) mass m of the bob (ii) length l of the string and (iii) acceleration due to gravity g. Deduce its expression. Ans. Given T m a l b g c θ d i.e., T km a l b g c θ d (1) [k is dimensionless constant] Taking dimensions on both sides, M 0 L 0 T 1 M a L b [LT 2 ] c, θ has no dimension. i.e., M 0 L 0 T 1 M a L b+c T 2c Equating the powers of M, L, and T on both sides. a 0, b + c 0, 2c 1 c -1 2 b 1 2 and d 0 Substituting these values in (1), T K m 0 l ½ g ½ θ 0 i.e., T K l ½ g K l ½ g Illustration 7: Find the dimension of Planck's constant h. Ans. We have, E hν E Energy h [h] E ν h planck's constant [E] [ν] ML2 T 2 T 1 ν frequency Illustration 8: [h] ML 2 T 1 Check the dimensional accuracy of the relation. λ h mν h Ans. λ mν

15 Units and Measurements [λ] L -15- [h] ML 2 T 1 [m] M [v] LT 1 h mν ML 2 T 1 MLT 1 L since [λ] [ h ], the above equation is correct. mν Limitations of Dimensions Numerical constants have no dimensions. The method of dimensions cannot be used to derive relation other than product of power functions. SIGNIFICANT FIGURES Larger the number of significant figures obtained in a measurement greater is the accuracy of the measurement. The digit given in reporting the result of an experiment are called significant figures. Significant figures in a physical quantity indicate the number of digits in which we have confidence with respect of their accuracy. For example the length of a rod measured is 3.52 cm. Here there are 3 significant figures. The digit 3 and 5 are reliable and last 2 is uncertain. Zeroes are written at the right end of a measurement. Rules to find the significant digits 1. All non-zero digits are significant. Eg: 1234 has 4 significant figures. 2. All zeroes occuring between two non-zero digits are significant. Eg: has 6 significant digits. 3. If the number is less than 1, the zeroes on the right of decimal point, but to the left of the first non-zero digit are not significant. Eg: has 5 significant digits. 4. In a number without a decimal point the terminal or trailing zeroes are not significant. Eg: has four significant digits. 5. In a number with a decimal point the trailing zeroes are significant. Eg: cm is the length of a rod, then it has 4 significant figures. Rounding off : 1. If the digit to be dropped is less than 5, then the preceding digit to left is unchanged. Eg: 8.22 is rounded off to If the digit to be dropped is more than 5, then the preceding digit is raised by one. Eg: х 6.87 is roudned off to If the digit to be dropped is 5 followed by digit other than zero, then the preceding digit is raised by one. Eg : is rounded off to If the digit to be dropped is 5 or 5 followed by zero then preceding digit to the left is unchanged, if it is even. Eg : rounding off to 5.2.

16 If the digit to be dropped is 5 or 5 followed by zero, then the preceding digit is raised by one, if it is odd. Eg : is rounded off to 3.8. Illustration 9: Write down the number of significant figures in the following (a) 6928 (b) (c) (d) 1200 Ans. (a) 4 (b) 2 (c) 3 (d) 2 Illustration 10: Round off to four significant figures. (a) (b) Ans. (a) (b) Illustration 11: A thin wire has a length of 21.9 cm and radius 0.49 mm. Calculate the volume of the wire to correct significant figures. Ans. Given, l 21.9 cm, r 0.49 mm r cm Volume of wire, V πr 2 l 22 7 (0.049)2 (21.9) i.e., v 0.14 cm 3 ERRORS The result of every measurement contains some uncertainity, which is called error. The difference between the true value and the measured value of a quantity is known as the error of the measurement. various types of errors. Errors may arise from different sources and are usually classified as, 1. Systematic or controllable errors : Systematic errors are the errors whose causes are known. They can be either positive or negative. Due to knowing causes these errors can be minimised. Systematic errors can further be classified into the following categories. Instrumental errors : These errors are due to imperfect design. These can be reduced by using more accurate instruments. Environmental errors : These errors are due to the changes in external environmental conditions such as temperature, pressure, humidity etc. Observational errors : These errors arise due to improper setting of the apparatus or carelessness in taking observations. 2. Random errors : These errors are due to unknown causes. Therefore they occur irregularly and are variable in magnitude and sign. 3. Gross errors : Gross errors arise due to human carelessness and mistakes in reading the instruments or calculating and recording the measurement results.

17 Units and Measurements -17- Absolute error of n measurements, a mean a 1 + a a n η Mean absolute error, a mean a 1 + a a n η Relative error a mean a mean Percentage error, δ a Illustration 12: a mean a mean 100 The period of oscillation of a simple pendulum in an experiment is recorded as 2.63s, 2.56s, 2.42s, 2.71s and 2.80s respectively. Find its percentage error. Ans. Absolute error, a mean s a a 1 a mean a 1 a a 2 a mean a 2 a a 3 a mean a 3 a a Combination of errors a mean Percentage error a mean a % mean 0.11 sec Error in sum or difference. Error in product or division If X A + B or A B, then X A ± B If X AB or X A B A X A + B B Error in power of a quantity If X A n Illustration 13: then X n A A If two times a and b are given as, a 25.4 s ± 0.1 sec and b 16.5 sec ± 0.1 sec. Then find a + b. Ans. Let X a + b

18 sec Maximum volume of X ( ) + ( ) 42.1 sec X (41.9 ± 0.2) sec important derivations 1. (i) Convert one atmospheric pressure(10 +5 Nm -2 ) into dyne cm -2. (ii) The centripetal force (F) acting on a body may depend upon mass of the body (m), radius of the circle (r) and frequency of revolution(ν). Derive the formula dimensionally. Ans. (i) Dimension of pressure n ML -1 T -2 here n n 2 n 1 [ M 1 M 2 ] a [ L 1 L 2 ] b [ T 1 T 2 (here a 1, b -1, c -2) ] c 10 5 [ 10 5 [ M 1 M 2 ] 1 [ 1kg 1g ]1 [ L 1 L 2 ] -1 [ T 1 T 2 ] -2 1m 1cm ]-1 [ 1s 1s ] [100] -1 (1) atm 10 6 dyne /cm 2 (ii) Here F α m x r y ν z Taking dimensions of F MLT -2 i.e., х 1, y 1, z -2 F m r ν 2 OR F k m r ν 2, k is a constant 2. Explain parallax method for measuring distance of a nearby star. Ans. Large distance such as the distance of a planet or star from earth can be measured by the parallax method. Parallax is the name given to change in the position of an object with respect to the background when the object is seen from two different positions. The distance between the two positions is called the basis. The angle subtended by an object on the basis depends upon following two factors, length of the basis and distance of the object from the basis. Let θ be the angle subtended by a star on the diameter of earth's orbit around the sun. then, θ d s or s d θ knowing d and θ, value of s can be calculated.

19 Units and Measurements -19- Star To a Distant star θ 1 θ 2 Distant star θ 1 θ 1 E 1 d Sun E 2 Let E 1 be the position of earth at any time. Angle θ 1 between the star and another much distant star is determined. After six months, the position of earth E 2 will be diametrically opposite. The angle θ 2 is again noted. From figure it is clear that θ θ 1 + θ 2, knowing value of s is calculated. 3. Derive an expression for the velocity of sound in a material medium, the temperature of the medium remaining constant. Ans. The velocity of sound in a material medium may depend upon, (i) the elasticity of the medium E. (ii) the density of the medium (ρ) Velocity, v KE a ρ b Taking dimensions on both sides M 0 L 1 T 1 [M 1 L 1 T 2 ][ML 3 ] b Equating powers of M, L and T on both sides a + b 0 a 1 2 -a - 3b 1 or b V K ½ E -½ ` or v K E ρ Worked out problems 1. A body moves with a velocity of 36 Kmh 1. What is the value of the velocity in ms 1? Ans. v 36 Kmh 1 We know that 1 km 1000 m 1 hr sec v m sec 10 ms 1 2. Calculate the angle of (a) 1 o, (b) 1, (c) 1 in radian.

20 Ans. (a) 1 o π 180 rad rad. (b) 1 I 1 60 π rad. (c) 1 II 1 o π rad. 3. It is estimated that per minute, each cm 2 of earth receives about 2 calories of heat energy from the sun and is called solar constant. Express in SI unit. Ans. S 2 calories min. cm joule 60 sec m Wm drops of olive oil of radius 0.40 mm spread into circular film of radius 15.2 cm on the surface of water. Estimate the size of an oil molecule. Ans. Here n 20, r 0.40 mm m R 15.2 cm m Let d be the diameter of oil molecule thickness of layer. Volume of olive oil area thickness of layer η 4 3 πr3 d πr 2 d 4 3 η r3 R ( ) 3 ( ) 2 d m 5. Show dimensionally that the relation t 2π ( l g ) is incorrect; where l is length, t is time period of simple pendulum, g is acceleration due to gravity. Find correct form of relation dimensionally. Ans. Given, t 2π ( l g ) -20- [L. H. S] T [R. H. S] [ l ] [ g ] L T 2 LT 2 2π is a constant Here [L. H. S] [R. H. S], hence equation is incorrect. Let t l a g b t k. l 0 g b Taking dimensions on both sides, T L a (LT 2 ) b M o L o T 1 L a L b T 2b L a+b T 2b Equating dimensions of M, L and T. a + b 0, 2b 1 b ½ t l g a + ½ 0 a ½ 6. The heat produced in a wire carrying an electric current depends on the current, the resistance and time. Find the relation between these quantities. The dimensional formula of resistance is ML 2 A 2 T 3 and heat is a form of energy. Ans. Let H I a R b t c H K I a R b t c Taking dimensions on both sides, ML 2 T 2 A a (ML 2 A 2 T 3 ). T c Equating dimensions of M, L, T and A. ML 2 T 2 M b. l 2b. A a 2b. T 3b+c i.e., b 1 2b 2 b 1 a 2b 0 a 2b 2 3b + c 2 c 2 + 3b 1 H I 2 Rt 7. Two resistors of resistances R ± 3 ohm and R ± 4 ohm are connected (a) in series (b) in parallel. Find the equivalent resistance of the

21 Units and Measurements (a) series combination, -21- [Energy] ML2 T 2 (b) parallel combination. Use relation R R 1 + R 2 for series 1 R 1 R R 2 for parallel, R R 2 R 1 R R 2 R 2 2 Ans. Here R 1 (100 ± 3) ohm R 2 (200 ± 4) ohm (a) Series combination R R 1 + R R ±( R 1 + R 2 ) ±(3 + 4) ±7 ohm R 300 ± 7ohm (b) Parallel combination 1 R 1 R 1 R 1 R R 1 R ohm 3 R 1 R + R R 2 R 1 R 1( R1 R 1 ) 2 + R 2 ( R1 R 2 ) 2 3 ( ( 200 ) )2 Hence R 1 (66.7 ± 1.8) ohm 8. The SI and CGS units of energy are joule and erg respectively. How many ergs are equal to one joule? Ans. Dimensionally, Energy Mass (velocity) 2 Mass ( length time )2 Thus 1Joule 1kg 1m 2 1s 2 1 erg 1g 1cm 2. 1s 2 1 Joule 1 erg 1000g 1g 1 Kg 1 g. ( 1m 1cm )2. ( 1s 1s )2. [ 100cm. 1cm ( 1s ]2 1s )2 1 Joule 10 7 erg 9. Young's modulus of steel is Nm 2. Express it in dyne cm 2. Here dyne is the C.G.S unit of force. Ans. The unit of Young's modulus is Nm 2 [Y] [Force] [distance] 2 MLT 2 ML 1 T 2 L 2 So 1 Nm 2 1kg (1m) 1 (1s) 2 1 dyne cm 2 1g. (1 cm) 1 (1s) 2 1N/M 2 1dyne/cm 2 1 Kg 1 g. ( 1m 1cm )2. ( 1s 1s ) dyne/cm N/M dyne/cm If velocity, time and force were chosen as basic quantities, find the dimension of mass. Ans. Dimensionally, Force mass acceleration mass velocity time mass force time [FTV 1 ] velocity 11. Test dimensionally if the equation v 2 u 2 + 2as is correct. Ans. [L.H.S] [ν 2 ] [LT 1 ] 2 L 2 T 2 [R.H.S] [u 2 ] (LT 1 ) 2 L 2 T 2

22 [2as] [a][s] [L.H.S] [R.H.S] L T 2 L L2 T 2 Thus the equation is correct. 12. If the centripetal force is of the form m a v b r c. Find the values a, b and c. Ans. Dimensionally, [Force] -22- MLT 2 m a (LT 1 ) b L c Equating dimensions of M, L and T a 1, b + c 1, 2 b Substituting b in b + c 2 + c 1 c Substituting the value of a, b, c in (1 ) (Mass) a (Velocity) b (Length) c (1) F M 1 v 2 r 1 1. What are the four fundamental forces? F mv 2 r very Short ANSWER Questions Ans. (i) (ii) Gravitational force Electromagnetic force (iii) Weak nuclear force (iv) Strong nuclear force 2. What is Parallax? Ans. When same object is viewed from two locations, the positions of object appear to be different. This apparent change in position, depending on the position of observation is called Parallax. 3. What is Least Count Error? Ans. Using an instrument, measurement below least count cannot be taken. This uncertainity of the order of least count is called least count error (L C Error). 4. What are the advantages of dimensional analysis? Ans. (i) (ii) It is used to check correctness of an equation. It is used to deduce new equations. (iii) It is used to convert units from one system to another. 5. Name the strongest force in nature. What is its range? Ans. Strong Nuclear force of range What is the relation between light year and Parsec? Ans. 1 parsec 3.26 light year. 7. Name some physical quantities that have same dimension. Ans. Work, Energy and Torque.

23 Units and Measurements If a physical quantity is represented by X M a L b T c and the percentage errors in the measurements of M, L and T are a,b and g respectively. What will be the percentage error in X? Ans. % error in measurement of X aa + bb + cg 9. Given relative error in the measurement of length is What is the percentage error? Ans. Percentage error relative error % 10. Name the technique used in locating (a) an under water obstacle. (b) position of an aeroplane in space. Ans. (a) SONAR Sound Navigation and Ranging. (b) RADAR Radio Detection and Ranging. 11. Give examples of dimensional constants and dimensionless constants. Ans. Dimensional Constant : Gravitational Constant, Planck s Constant. Dimensionless Contant : π, θ 12. Deduce dimensional formulae for Boltzmann s constant. Ans. [K] [Heat] [Temperature] ML 2 T 2 K ML2 T 2 K Does magnitude of a quantity change with change in the system of units? Ans. No, magnitude of a quantity does not change. 14. Can a body have zero mass and zero weight? Ans. No, mass of a body can never be zero but it weight may be zero. 15. Are inertial and gravitational mass of a body different from one another? Ans. No, they are equivalent. 16. How many fermi are there in one metre? Ans. 1 fermi m 1 1 m fermi 17. Is light year a unit of time? Ans. No, it is a unit of distance. 18. Define a parsec. Ans. It is the distance at which an arc of length 140 subtends an angle of one second of an arc. 1 parsec m 19. The mass of an electron is kg. How many electrons would make 1 kg? Ans. Number of electrons in 1 kg 20. Define atomic mass unit. 1 mass of electron 1kg

24 -24- Ans. One a.m. u is defined as 1 th mass of one 12 6 C 12 atom i.e., 1 a. m. u kg 21. Which is the most accurate clock? Ans. Atomic clocks are so far the most accurate clocks. 22. Can there be a physical quantity which has no units and no dimensions. Ans. Yes, for example strain has no units and no dimensions. 23. Name a scalar and a vector quantity which have same dimensions. Ans. Speed is a scalar and velocity is a vector both have the same dimensions. 24. What are the dimensions of angular displacement? Ans. Angular displacement length of arc radius i.e., dimensionless. 25. Are all constants dimensionless. Comment. Ans. No, all constants are not dimensionless. For example, universal gravitational constant has dimension. 26. Find the dimensional formula of latent heat. Ans. Latent heat [Quantity of heat] [Mass] ML2 T 2 M L2 T Can a quantity have units, but still be dimensionless. Ans. Yes, for example angle is dimensionless but it has units. 28. Write the dimensions of potential energy. Ans. Potential energy mgh [P. E] MLT 2. L ML 2 T What are the dimentions of rate of flow. Ans. Rate of flow represents volume flowing per second. Its dimensional formula is L 3 T Define significant figures. Ans. The digits whose values are accurately known in a particular measurment are called its significant figures. 31. The length of a table as measured by two students is given as 2.5 m and 2.54 m. Which of the two measurements is more accurate and why? Ans. The second measurement i.e., 2.54 m is more accurate. 32. What do you understand by absolute error. Ans. The difference of the standard value and the observed value of a physical quantity is called absolute error.

25 Units and Measurements -25- Short ANSWER Questions 1. State the principle of homogenity. Test the dimensional homogenity of equations. (i) S ut at2 (ii) S n u + a 2 (2n - 1) Ans. (i) Dimensions of LHS [S] M 0 L 1 T 0 Dimensions of RHS [ut] [at 2 ] LT -1. T LT -2.T 2 L The equation is dimensionally homogenous. a (ii) S n u + (2n - 1) 2 LHS [S n ] L RHS [u] LT -1 LHS RHS. This is dimensionally incorrect. 2. In Van der wall s gas equation [ P + a 2 ](v - b) RT. Determine the dimensions of a and b. v Ans. Since dimensionally similar quantities can only be added. [P] [ a v 2 ] [a] [Pv 2 ] ML -1 T -2.(L 3 ) 2 ML 5 T -2 Also [b] [v] L 3 3. Magnitude of force experienced by an object moving with speed v is given by F Kv 2. Find dimensions of K. Ans. F Kv 2 [K] [F] [v 2 ] MLT-2 ML [LT -1 ] A book with printing error contains four different formulae for the displacement. Choose the correct formulae. (a) y a sin 2π T. t (b) y a sin vt (c) y a T sin ( t a ) (d) y a T (sin 2π T. t + cos 2π T. t) Ans. The argument of sine and and cosine function is dimensionless quantities. 2π (a) y a sin T. t i.e., [ 2πt T ] T0, is dimensionless and is correct and [a] L [y] (b) y a sin vt Here [vt] LT -1.T L, so this equation is incorrect. (c) y a T sin ( t a ) [ t a ] L-1 T, it is incorrect. (d) y a T (sin 2π T. t + cos 2π T. t)

26 -26- Though 2πt T is dimensionless [ a T ] LT 1 [y] and the equation is incorrect. 5. Find the dimensions of a b in equation P b - l2, where P is power, l is length and t is time. at Ans. By principle of homogenity, [b] [l 2 ] L 2 [P] [ b - l2 at ] [a] [b - l2 ] [P][t] L 2 ML 2 T -3.T M-1 T 2 [a b] M -1 L 2 T 2 6. What is the basic difference between inertial mass, gravitational mass and weight of a body? Ans. Inertial mass of a body is measure of inertia of the body. It is equal to force required to produce unit acceleration in the body. Gravitational mass of a body is the force experienced by the body in a gravitational field of unit intensity. Weight of a body is the force with which the body is attracted towards the center of earth. 7. What is the technique used for measuring large time intervals? Ans. We use radioative dating technique for large time intervals. Large time intervals are measured by studying the ratio of number of radioactive atom decayed to the number of surviving atoms in the specimen. 8. Derive SI unit of Joule (J) in terms of fundamental units. Ans. Joule is the unit of work, work force distance mass acceleration distance Mass. distance. distance time 2 unit is kg m 2 s 2 9. The mass of a box measured by a grocer s balance is 2.3kg. Two gold pieces 20.15g and 20.17g are added to the box. (i) What is the total mass of the box? (ii) What is the difference in masses of the pieces to correct significant figures. Ans. Mass of box 2.3 kg (i) Mass of gold pieces g kg Total mass (ii) The difference in masses is 0.02g. ( ) 10. If the error in measurement of mass of a body be 3% and in the measurement of velocity be 2%.What will be the possible error in calculation of kinetic energy? Ans. K E 1 2 mv2 AK K 100 Am m Av v 100 3% + 2 2% 7% 11. A calorie is a unit of heat or energy and it equals 4.2 J where 1J 1kgm 2 s 2. Suppose we employ a system of units in which unit of mass is kg, unit of length is βm, and unit of time is gs. What will be the magnitude of calorie in terms of this new system?

27 Units and Measurements -27- Ans. n 2 n 1 [ m 1 m 2 ] a. [ L 1 L 2 ] b. [ T 1 T 2 ] c 4.2 [ kg αkg ]a. [ m βm ]b.[ s γs ]c and n α 1 β 2 γ 2. Here a 1, b 2 c The length of a rod as measured in an experiment was found to be 2.48m, 2.46m, 2.49m, 2.50m and 2.48m. Find the average length, absolute error and the percentage error. Express the result with error limit Ans. m, Average length 2.48 m 5 then, m 1 - m m 2 - m m 3 - m m 4 - m Mean absolute error, m 5 - m Percentage error m absolute error average Correct length with error limit 2.48 ± % 13. When the planet Jupiter is at a distance of million km from the Earth, its angular diameter is measured to be of arc. Calculate the diameter of Jupiter. Ans. Given θ 35.72, D km we have rad θ rad d Dθ km 14. A laser light beamed at the moon takes 2.56s and to return after reflection at the moon s surface. What will be the radius of lunar orbit? Ans. Given t 2.56s t time taken by the laser beam to go to moon distance between earth and moon, d c m A physical quantity is measured as Q (2.1 ± 0.5) units. Calculate the percentage error in t 2 t 2 (i) Q 2 (ii) 2Q

28 Ans. (i) Let P Q 2 given Q 2.1, Q p 2 Q p Q p % 48% p (ii) Let R 2Q R R R R Q Q % 24% 16. Pressure is defined as momentum per unit volume. Is it true? Ans. Pressure [Pressure] but [momentum] [volume] Force Area MLT 2 ML 1 T 2 L 2 MLT 1 L 3 ML 2 T 1 Hence dimensionally it is not true. 17. The rotational kinetic energy of a body is given by E ½ Iω 2, where ω is angular veloity of the body. Use this equation to get dimensional formula of I. Ans. Here E ½ Iω 2 or I 2E ω 2 [I] 2 [Energy] ML2 T 2 [ω] 2 [T 1 ] 2 ML2 18. Write the dimensions of each of the following in terms of mass, length, time; reynold number. Ans. Reynold number, N R [N R ] ρdv η ML 3 L LT 1 M ML 1 T o L o T o Find the value of х in the relation y Tх cosθ. I. Where y is young's modulus, T is time period, I is torque and L is legth. Ans. y T х [T х ] Equating dimensions of T, х 0 T х cosθ. I L 3 yl 3 cosθ. I L 3 ML 1 L 3 ML 2 T 2 M o L o T o 20. Give the dimensional formula of thermal conductivity. Ans. The coefficient of thermal conductivity is, k Qd A(θ 1 - θ 2 )t WhereQ, d, A, (θ 1 θ 2 ) and t stand for energy, distance, area, temperature difference and time.

29 Units and Measurements If х a + bt + ct 2 where х is in metres and t in seconds. Find the units of b. Ans. Here, х a + bt + ct 2 b [х] [t] LT Find the sum of cm and cm. Ans. Let х cm unit is ms 1 y cm cm х + y ( ) cm 23. The resistance R is the ratio of potential difference V and current I. What is the % error in R if V is (200 ± 3) volt and I (20 ± 0.4)A. Ans. R R 100 ± [ V V + I I ] ± [ ] % 24. What are the dimensions of a and b in the relation. F at + bх, where F is force and х is distance. Ans. Here [a] [F] [t] [b] [F] [х] MLT 2 T MLT 2 L MLT 3 MT 2 numerical problems 1. How many newtons are contained in 2 kg? Mass of one newton is kg. [Hint : ] 2. Convert a velocity 30 ms 1 in kilometre per hour. [Hint : 108 Kmh 1 ] 3. The dimensions of a block are m m 1.87 m. Express its volume in appropriate significant figures. [Hint : 29.8 m 3 ] 4. Two forces F 1 and F 2 acting simultaneously, on a particle are measured as F 1 (26 ± 0.2)N, F 2 (13 ± 0.3)N. What will be the resultant if they act (i) in same direction (ii) in opposite direction? [Hint : (13 ± 0.5)N]

30 5. A field has measurements of (17.85 ± 0.7) m and (21.52 ± 0.5) m. Calculate the area of the field. [Hint : ( ± ) m 2 ] The length of two rods are recorded as l 1 (25.2 ± 0.1) cm and l 2 (16.8 ± 0.1) cm. Find the sum of the lengths of the two rods with the limits of error. [Hint : (42 ± 0.2) cm] 7. A potential difference of v 150 ± 2 volt when applied across a resistance R gives a current of 15 ± 0.5 ampere. Calculate percentage error in R given by R V I. [Hint : 4%] 8. In a simple pendulum experiment, the length of the pendulum is m. The time period is 1.91s. Write the value of acceleration due to gravity to correct significant figures and round it off. [Hint : 9.8 ms 2 ] 9. The mass of 25 cm 3 of a certain substance is 87.2 g. Calculate its density with due regard to significant figures. [Hint : 3.48 g cm 3 ] a 10. A physical quantity X is given by the relation X 2 b. If the percentage error of measurement in a, c b and c are 3% and 4% respectively, then calculate the percentage error in X. [Hint : 9%] 11. Check the correctness of the following relation by the method of dimensions t ρr3 T period of oscillation ρ density, r radius and T force of surface tension., where t is time [Hint : [L. H. S] [R. H. S], hence relation is not correct.] 12. Velocity of sound depends upon the coefficient of elasticity E of the medium and the density ρ of the medium. Obtain the expression for ν by the method of dimensional analysis. [Hint : b ½, a ½, ν K E p ] 13. Force of visocity F acting on a spherical body moving through a fluid depends upon its velocity v, radius r and coefficent of viscosity η of the fluid. Obtain an expression for F. [Hint : F kηrν] 14. Find the unit of length, mass and time if the unit of force, velocity and energy respectively are 100 N, 10 ms 1 and 500J. [Hint : M 5 kg] 15. Following is the equation of Bernoulli's : P + hρg + ½ ρv 2 k Here p is pressure, h is height, ρ is density, g is acceleration due to gravity, ν is velocity and k is a constant. Show that equation is dimensionally correct. [Hint : All three quantities have same dimensions, so the equation is correct.] 16. Check the dimensional correctness of the equation v(t) v (0) + at. 17. Check the accuracy of the relation t 2π l g for a simple pendulum.

31 Units and Measurements Obtain dimensional formula of gravitational constant. [Hint : [G] M 1 L 3 T 2 ] 19. Check the dimensional accuracy of the relation λ h mv 20. How many significant figures are present in the mass difference of a proton and a neutron? m p kg, m n kg [Hint : four] 21. The length, breadth and thickness of a rectangular sheet of metal are m, m and 2.01 cm respectively. Given the area and volume of the sheet to correct significant figures. [Hint : Area - 4, Volume - 3] 22. A physical quantity P is related to four observables a, b, c and d as P a3 b 2. The error in the c d measurement of a, b, c and d are 1%, 3%, 4% and 2% respectively. What is the percentage error in the quantity P. [Hint : 13] 23. The side of a square is measured to an accuracy of 0.1 cm. If the side is of length 25.2 cm. Find the percentage error in the measurement of area. [Hint : 0.79] 24. Using dimensions, check the correctness of equation for kinetic energy. 25. Radius of a circle is 4.12 cm. Calculate the area considering the idea of significant figures. [Hint : 53.3 cm 2 ] hots questions 1. The specific heat capacity of a particular solid is given by C at 3 where T is the absolute temperature close to 0 kelvin and a is a constant characteristic of the solid. What are the units of 'a' in terms of base units of S. I? 2. One of the equation which occurs in high vaccum technique Q Kr3 (P 1 P 2 ) M where K is a l RT dimentionless constant, P 1 and P 2 are the pressure at each end of a tube of radius r and length l, m is the molar mass, R is the molar gas constant and T is the temperature on absolute scale. What are the units of Q in terms of base units of S. I? F 3. The drag coefficient C D of a car moving with a speed ν through air of density ρ is given by C D ρv 2 A, Where F is the drag force exerted on the car and A is the maximum cross-sectional area of the car perpendicular to the direction of travel. Show that C D is dimensionless. 4. The energy of a photon of light of frequency f is given by hf, where h is planck's constant. Determine the units of h in terms of base units of S.I.

32 A liquid having a small depth but a large volume is forced by applied pressure to escape through an orifice with a velocity v. Obtain an expression for the velocity of liquid in terms of applied pressure and density? 6. Kepler discovered that the orbital periods T of the planets about the sun are related to their distances r from the sun. From Newton's law, the following relationship may be derived. T 2 4π2 GM. r3. Here M is the mass of the sun. Obtain units of G in terms of the base units of S. I. 7. The velocity ν of a particle is given in terms of time t by the equation v at + dimensions of a, b and c? 8. Obtain the dimensional formulae of w and k from the equation y a sin (wt kх). b. What are the t + c 9. In the equation y A sin (wt - kх), obtain the dimensional formula of w and k. Given х is distance and t is time. 10. A plane lamina has an irregular surface. Suggest some indirect method to find its surface area. 11. A long jumper can cover a distance of 5m. The same is measured with measuring tape marked in feet. (a) (b) (c) What is a unit? Is there any variation for the quantity measured. If we use another unit, say foot? Give the relation between the unit and the numerical value of a quantity. hints / answers 1. specific heat capacity Given a Units of 'a' are m 2 s 2 K 4 Q m T ML2 T 2 MK 1 L 2 T 2 K 1 C T 3 L2 T 2 K 1 K 3 2. [K] M o t o T o [P 1 - P 2 ] ML 1 T 2 [r] [M] [R] JK 1 mol 1 Kg mol 1 M 1 mol 1 ML 2 T 2 K 1 mol 1 L Q 3 ML 1 T 2 L Units of Q is Kgs 1 M mol 2 ML 2 T 2 K 1 mol 1 K MT 1 3. [F] MLT 2 [ρ] ML 3 [v] LT 1 [A] L 2 [C D ] MLT 2 ML 3 L 2 T 2 L 2 Mo L o T o

33 Units and Measurements 4. E hf [E] ML 2 T 2 f T 1 h ML 2 T 2 Unit is T 1 ML T Kg m s 5. Let ν KP a ρ b Taking dimensions, LT 1 (ML 1 T 2 ) a (ML 3 ) b i.e., a + b 0 2a 1 a 3b 1 a ½ b ½ ν k P ρ 4π 6. Given T 2 2 GM. r3 G i.e., [G] Unit of G is Kg 1 m 3 s 2. 4π 2 T 2 M. r3 L 3 T 2 M M 1 L 3 T 2 [ν] 7. [ν] [at] a [t] [t] [c] T [ b T ] [ν] [b] LT 1 T L LT 2 T LT 2 8. Since (wt kх) is an angle, it should be dimensionless. Therefore both wt and kх should be dimensionless. [wt] constant [kх] constant 1 [w] t T 1 [k] L 1 9. The given equation is y A sin(wt - kх) Here, wt angle [angle] [w] T 1 [t] [angle] Also kх angle [k] L 1 [х] 10. The given lamina is placed on a plane, uniformly thick quality paper and the paper pattern that fits the plane lamina precisely is cut out. The weight of paper pattern is taken w 1. From the same quality paper a large square piece is cut down and its weight is taken w 2. Let A 2 be the area of the square piece. If A 1 is the area of the lamina then, w 1 A 1 A 1 w 2 A 2 w 1 w 2 A 2

34 11. (a) The standard measure used for the comparison of a physical quatity is called a unit (b) (c) No change. Quantity, Q η μ η numerical value, μ unit SCERT PREVIOUS YEAR QUESTIONS 1. Fill in the blanks: (a) 1 micron m (b) Hertz is the unit of 2. (a) m A student writes the equation for the relativistic variation of mass with velocity as m 0 1- x 2 / c, 2 where m 0 is the rest mass and c is the speed of light. What is the dimensional formula for x? (b) If the percentage error in the measurement of radius R of a sphere is 2%, then what is the percentage error in its volume? 3. Mechanical power is represented by P Fν + Aν 3 ρ, where F is the force, v is the velocity, A is the area and ρ is the density. (a) (b) (c) The dimensional formula of power is. Check the dimensional validity of the above equation. Which of the following equations can t be obtained by the dimensional method? (i) T K (l/g) (ii) E kmν 2 (iii) P hρg (iv) N N 0 e -λ t 4. Give examples for the following: (a) (b) (c) A dimensionless, unitless physical quantity. A dimensionless physical quantity but having unit in SI sytem. Two physical quantities which have the same dimensions. 5. A company manufacturing PVC pipes claims in an advertisement that the volume of water flowing out through pipe in a given volume V KA 2 ut where A is the area, u is the speed of the flow, t is the time and K is a dimensionless constant. (a) (b) Name the principle that can be used to check the dimensional correctness of this equation. Check the equation and state whether the claim can be correct. 6. Pick out quantities with same dimensional formula and write their SI unit. (a) Work, Energy (b) Moment, Momentum (c) Impulse, Momentum 7. All measurements are associated with errors. (a) (b) What are errors in measurements? In an experiment for determining g, a boy obtained following values. Find the absolute error in each measurement.

35 Units and Measurements Trial Obtained value of g Fill in the blanks. Quantity C.G.S SI Dimensions a Force N b dyne/cm 2 M 1 L 1 T 2 c Relative velocity M 0 L 1 T 1 d Light year 9. A particular vernier calipers have a mainscale with 0.5 mm in one division and has 20 divisions on its vernier scale. A screw gauge of 100 divisions on its circular scale has a pitch of 0.1 mm (a) (b) (c) Find the count of vernier calipers. Which instrument will give more precise value in a measurement and why? What is the need for measurement? 10. (a) Express 4 light years distance in SI unit. (b) Do A o and AU stand for same length. 11. (a) Name a scalar or vector quantity which have same dimensions. l (b) Calculate percentage error in the determination of g 4π 2 2, when l and t are measured with t ± 2% and ± 3% errors respectively. 12. The unit of length convenient on the atomic scale is known as an amstrong and is denoted by A o. 1A o m. The size of the hydrogen atom is 0.5A o. What is the total atomic volume in m 3 of a mole of hydrogen atoms? 13. The dimensional formula of force is MLT 2. (a) (b) (c) What does it mean? What is the difference between dimensional formula and dimensional equation? Give the dimensional formula of relative density. 14. The diameter of sphere is 2.78 m. Calculate its volume in due regard to significant figures. hints / answers 1. (a) 10-6 m (b) Frequency 2. (a) [x] [c] LT -1 (b) % error in volume 3 R R % 6% 3. (a) ML 2 T -3 (b) [p] [Fν] [ Aν 3 ρ]

36 -36- ML 2 T -3 [ML -2 (LT) -1 ] [ L 2 (LT -1 ) 3. ML -3 ] [ML 2 T -3 ] [ML 2 T -3 ], so the equation is correct. (c) N N 0 e -λt 4. (a) Strain or relative density or refractive index (b) Angle (c) (i) impulse - momentum (ii) work - energy 5. (a) Principle of homogenity (b) [V] [KA 2 ut] L 3 [ L 4 LT 1 T] [L 5 ], equation is wrong. 6. (a) [work] [energy] ML 2 T 2 S I unit is Kgm 2 s 2 or J (b) [moment ] [momentum] (c) [impulse] [momentum] [MLT 1 ] Unit is Kgms 1 7. (a) The uncertainity in a measurement is called error. (b) True value of ḡ Absolute error in measurements, g 1 g 1 - ḡ g 2 g 2 - ḡ g 3 g 3 - ḡ (a) dyne, MLT 2 (b) Pressure/stress, Nm 2 or Pascal. (c) Cms 1, ms 1 (d) Cm, m, L mm 9. (a) Least count of vernier calipers mm mm (b) Least count of screw gauge mm 100 The screw gauge has more resolution. Hence more precised. 10. (a) 1 light year m 4 light year m (b) No. 1 AU 1 astronomical unit m 1 A o m 11. (a) Speed is a scalar quantity. Velocity is a vector quantity. Both have LT 1 dimension. l (b) Given, g 4π 2 t 2 g g 100 ± ( l + 2 t l t ) 100 ± ( ) 100 ± 8% 12. Here r 0.5 A o m V 1 Volume of each hydrogen atom 4 3 πr ( ) m 3

37 Units and Measurements (a) It tells how and which of the fundamental units are required to represent the unit of a physical quantity. (b) (c) Dimensional quantity is the equation obtained by equating the physical quantity with its dimensional formula. Being a mere number, it has no dimensional formula. 14. Radius of sphere, r Volume v 11.3 m m πr (1.39) m 3 entrance EXAM CORNER 1. If the time period of osillation of a pendulum is measured as 2.5 using a stop watch with the least count ½ s, then the permissible error in the measurement is (a) 10% (b) 30% (c) 15% (d) 20% 2. The displacement of a particle moving along X-axis with respect to time t is х at + bt 2 - ct 3. The dimesions of c is (a) [T 3 ] (b) [LT 2 ] (c) [LT 3 ] (d) [LT 3 ] 3. If F denotes force and t time, then in the equation F at + bt 2, the dimensions of a and b respectively are (a) [LT -4 ] and [LT -1 ] (b) [LT -1 ] and [LT -4 ] (c) [MLT -4 ] and [MLT -1 ] (d) [MLT -1 ] and [MLT -4 ] 4. The mass and volume of a body are found to be (5 ± 0.05) kg and (1 ± 0.05) m 3 respectively. Then the maximum possible percentage error in its density is (a) 6% (b) 3% (c) 10% (d) 5% 5. Dimensional formula of Stefan's constant is (a) [MT -3 K -4 ] (b) [MLT -2 K -4 ] (c) [ML 2 T -2 ] (d) [ML -2 L 0 ] 6. The percentage errors in the measurment of length and time period of a simple pendulum are 1% and 2% respectively. Then the maximum error in the measurement of acceleration due to gravity. (a) 8% (b) 3% (c) 4% (d) 5% 7. If C is the capacitance and V is the potential the dimensional formula for CV 2 is (a) [ML 2 T -1 ] (b) [ML -2 T -3 ] (c) [ML 2 T -2 ] (d) [ML -2 T -2 ] 8. The values of two resistors are R 1 (6 ± 0.3) kω and R 2 (10 ± 0.2) kω. The percentage error in the equivalent resistance when they are connected in parallel is. (a) 5.125% (b) 2% (c) 3.125% (d) % 9. The physical quantity angular momentum has the same dimensions as that of (a) work (b) force (c) momentum (d) planck's constant

38 A physical quantity P is related to four measurable quantities a, b, c and d as follows P a3 b 2 c d. The percentage errors of measurment in a, b, c and d are 1%, 3%, 4% and 2%. The percentage error in P is (a) 10% (b) 13% (c) 5% (d) 15% 11. Which of the following sets of quantities have same dimensional formula? (a) (b) (c) (d) Frequency, Angular frequency and Angular momentum. Surface tension, Stress and Spring constant. Acceleration, Momentum and Retardation. Work, Energy and Torque. 12. A physical quantity A is related to four observables a, b, c and d as follows: A a3 b 2. The percentage c d errors of measurement in a, b, c and d are 1%, 3%, 2% and 2% respectively. What is the percentage error in A? (a) 12% (b) 27% (c) 5% (d) 14% 13. The dimensional formula of magnetic flux is: (a) [M 1 L 0 T 2 A 1 ] (b) [M 1 L 2 T - 1 A 1 ] (c) [M 1 L 2 T 1 A 2 ] (d) [M 1 L 2 T 0 A 1 ] 14. In an experiement to measure the height of a bridge by dropping stone into water underneath, the error in measurement of time is 0.1s at the end of 2s, then the error in estimation of height of bridge will be: (a) 0.49 m (b) 0.98 m (c) 1.37 m (d) 1.96 m 15. The dimensions of kinectic energy is (a) [M 2 L 2 T 1 ] (b) [M 1 L 2 T 1 ] (c) [ML 2 T - 2 ] (d) [M 1 L 2 T 1 ] 16. Dimensions ML - 1 T - 1 are related to (a) torque (b) work (c) energy (d) coefficient of viscosity 17. If the velocity of light C, gravitational constant G and planck's constant h are chosen as fundamental units, the dimensions of length L in the new system is: (a) [L 1 C 1 G 1 ] (b) [h ½ C ½ G -½ ] (c) [h 1 C 3 G 1 ] (d) [h ½ C 3/2 G ½ ] 18. A body is moving at a speed of near 0.3 ms -1, measure its speed with an accuracy about 1%, using a sampling distance 3mm, the measuring clock should have a least count of the order of (a) 0.1 s (b) 0.01 s (c) s (d) s 19. If M Mass, L Length, T Time and I Electric Current, then the dimensional formula for electrical resistance is given by (a) [M 1 L 2 T -3 I -2 ] (b) [M 1 L 2 T -3 I 2 ] (c) [M 1 L 2 T 3 I -2 ] (d) [M 1 L 2 T 3 I 2 ] 20. Identify the pair which has different dimensions. (a) Planck's constant and linear momentum. (b) Impulse and linear momentum.

39 Units and Measurements -39- (c) Angular momentum and frequency. (d) Pressure and young's modulus. 21. The dimensional formula [M 0 L 2 T -2 ] stands for (a) Torque (b) Angular Momentum (c) latent heat (d) Co-efficient of thermal conductivity 22. The SI unit of the coefficient of viscosity is (a) NM 2 (b) NS (c) NSM -2 (d) NM 2 S 23. The dimensions of the quantity, where h is planck's constant, V is the frequency and C is the velocity of light is (a) MT -1 (b) MLT -1 (c) MLT -2 (d) ML 2 T Light year is used as a measure of (a) small time (b) work (c) large distance (d) mass 25. The velocity of a particle is given as v a + bt + ct 2. If the velocity is measured in ms -1, then units of 'a' and 'c' are (a) ms -1 and ms -3 (b) ms -2 and ms -1 (c) m 2 s -1 and ms 2 (d) ms +2 and ms In which of the following systems of units, a weber is the unit of magnetic flux? (a) CGS (b) MKS (c) SI (d) None of these 27. With the usual notations, the following equation s t u + ½ a (2t - 1) is (a) (b) (c) (d) only numerically correct only dimensionally correct Both numerically and dimensionally correct Neither numerically nor dimensionally correct 28. If the velocity of light c, gravitational constant G and Planck s constant h are chosen as fundamental units, the dimensions of length L in the new system are (a) [hcg -1 ] (b) [h ½ c ½ G -½ ] (c) [hc -3 G 1 ] (d) [h ½ c - 2/3 G ½ ] 29. Using mass (M), length (L), time (T) and current (A) as fundamental quantities, the dimensions of magnetic permeability are (a) [M - 1 LT - 2 A] (b) [MLT - 2 A - 1 ] (c) [MLT -2 A - 2 ] (d) [MLT - 1 A - 1 ] e The dimensions of 4π ε 0 hc, Where e, ε, h and c are the electronic charge, electric permittivity, Planck s 0 constant and velocity of light in vacuum respectively, are (a) [M 0 L 0 T 0 ] (b) [ML 0 T 0 ] (c) [M 90 LT 0 ] (d) [M 0 L 0 T] 31. x 3yz 2, find dimension of y in (MKSA) system, if x and z are dimension of capacity and magnetic field respectively (a) [M -3 L -2 T -4 A -1 ] (b) [ML -2 ] (c) [M -3 L -2 T 4 A 4 ] (d) [M -3 L -2 T 8 A 4 ]

40 If the acceleration due to gravity is 1Gms - 2 and units of length and time are changed in kilometre and hours respectively, the numerical value of acceleration is (a) (b) (c) (d) If E energy, G gravitational constant, I impulse and M mass, then dimensions of GIM2 same as that of E 2 (a) time (b) mass (c) length (d) force are 34. If the edge of a cube is a (1.20 ± 0.01) 10-2 m then its volume will be recorded as (a) (1.728 ± 0.01) 10-6 m 3 (b) (1.728 ± 0.03) 10-6 m 3 (c) (1.73 ± 0.25) 10-6 m 3 (d) (1.73 ± 0.025) 10-6 m The speed (v) of ripples on the surface of water depends on surface tension (σ) density (ρ) and wavelength (λ). The square of speed (v) is proportional to (a) σ ρ λ (b) ρ σ λ (c) λ σ ρ (d) ρλσ 36. Dimensions of resistance in an electrical circuit, in terms of dimension of mass M, of length L, of time T and of current I, would be (a) [ML 2 T 3 I 1 ] (b) [ML 2 T 2 ] (c) [ML 2 T 1 I 1 ] (d) [ML 2 T 3 I 2 ] 37. In the relation y a cos (wt - kx), the dimensional formula for k is (a) [M 0 L 1 T 1 ] (b) [M 0 LT 1 ] (c) [M 0 L 1 T 0 ] (d) [M 0 LT] α 38. In the relation p β e α z p is pressure, z is distance, k is Boltzmann constant and θ is the kθ temperature. The dimensional formula of β will be (a) [M 0 L 2 T 0 ] (b) [ML 2 T] (c) [ML 0 T 1 ] (d) [M 0 L 2 T 1 ] 39. A physical quantity A is calculated from the relation A a2 b 3. The percentage errors of measurement c d in a, b, c and d are 1%, 3%, 2% and 2% respectively. What is the maximum percentage error possible in the quantity a? (a) 12% (b) 7% (c) 5% (d) 14% 40. If the length of rod A is 3225 ± 0.01 cm and that of B is cm, then the rod B is longer than rod A by (a) (0.94 ± 0.00) cm (b) (0.94 ± 0.01) cm (c) (0.94 ± 0.02) cm (d) (0.094 ± 0.005) cm 41. If L cm, B 2.1 cm, then value of L + B is (a) cm (b) 4.43 cm (c) 4.4 cm (d) 4 cm 42. If error in measurement of radius of sphere is 1%, what will be the error in measurement of volume? 1 (a) 1% (b) % (c) 3% (d) 10% The dimensions of αb 4 (σ Stefan s constant and b Wein s constant) are (a) [M 0 L 0 T 0 ] (b) [ML 4 T 3 ] (c) [ML 2 T] (d) [ML 6 T 3 ]

41 Units and Measurements 44. Units of CV are of ρε 0 (a) charge (b) current (c) time (d) frequency The dimensions of a b in the equation p a t2 bх (a) [M 2 LT -3 ] (b) [MT - 2 ] (c) [ML 3 T - 2 ] (d) [LT - 3 ] 46. The dimensions of the quantity hc are where p is pressure, х is distance and t is time are (a) [ML 2 T - 1 ] (b) [MLT - 1 ] (c) [ML 3 T - 2 ] (d) [ML 3 T - 1 ] 47. The relative density of a metal may be found by hanging a block of the metal from a spring balance and noting that in air the balance is (5.00 ± 0.05)N while in water it reads (4.00 ± 0.05) N. Then density would be quoted as (a) 5.00 ± 0.05 (b) 5.00 ± 11% (c) 500 ± 0.10 (d) 5.00 ± 6% 48. In an experiment, refractive index of glass is observed to be 1.45, 1.56, 1.54, 1.44, 1.55, The mean absolute error in the expel is (a) ± 0.04 (b) 0.02 (c) (d) ± The coefficient of viscosity (η) of a liquid method of flow through a capillary tube is by the formula it where η π R 4 P 8 l Q R radius of the capillary tube, p pressure difference between its ends l length of the tube, and Q volume of liquid flowing per sec. Which quantity must be measured most accurately? (a) R (b) l (c) P (d) Q 50. In the experiment of determination of the speed of sound using a resonance column (a) prongs of the tuning fork are kept in a vertical plane. (b) (c) (d) prongs of the tuning fork are kept in a horizontal plane. in one of the two resonances observed, the length of the resonating air column is close to the wavelength of sound in air. in one of the two resonance observed, the length of the resonating air column is close to half of the wavelength of sound in air. 51. Which of the following units denotes the dimensions [ML 2 / Q 2 ], where Q denotes the electric charge? (a) Wb/m 2 (b) henry (H) (c) H/m 2 (d) weber (Wb) 52. Which of the following sets share different dimensions? (a) (b) (c) (d) Pressure, Young's modulus, stress Emf, potential difference, electric potential Heat, work done, energy Dipole moment, electric flux, electric field

42 53. Out of the following pairs, which one does not have identical dimensions? -42- (a) Angular momentum and Planck's constant (b) Impulse and momentum (c) Moment of inertia and moment of a force (d) Work and torque 54. Which one of the following represent the correct dimensions of the coefficient of viscosity? (a) [ML - 1 T - 2 ] (b) [MLT - 1 ] (c) [ML - 1 T - 1 ] (d) [ML - 2 T - 2 ] 55. Dimensions of 1 µ 0 ε 0, where symbols have the usual meaning, are (a) [L - 1 T] (b) [L 2 T 2 ] (c) [L 2 T - 2 ] (d) [LT - 1 ] 56. The physical quantities not having same dimensions are (a) torque and work (b) momentum and Planck's constant (c) stress and Young's modulus (d) speed and (µ 0 ε 0 ) ½ 57. The physical process of measurement is a (a) Comparison process (b) Contrast process (c) Conversion process (d) Computation process 58. Light year is the unit of (a) intensity of light (b) time (c) distance (d) velocity of light 59. If gram cm 2 sec 1 х joule sec., then number х is equal to (a) (b) (c) (d) The value of g is 9.8 m/s 2. Its value in km/min 2 is (a) 15.3 (b) 25.3 (c) 35.3 (d) The dimensional formula of prepogation constant k is (a) M o LT 1 (b) M o L 1 T o (c) M 1 L 1 T o (d) M 1 L 1 T o 62. Of the following pairs the one which does not have the same dimension is (a) Specific heat and latent heat (b) Tension and surface tension (c) Moment of inertia and moment of momentum (d) Momentum and impulse 63. If the velocity of light C, the constant of gravitation G and planck's constant h be chosen as fundamental units, the dimension of mass in the new system will be (a) h ½ C ½ G ½ (b) h ½ C ½ G ½ (c) h ½ C ½ G ½ (d) h ½ C ½ G ½ 64. The unit of time in that system in which the unit of length is 'metre', unit of mass 'kg' and unit of force 'kg wt' is 1 (a) 9.8 sec (b) (9.8)2 sec (c) 9.8 sec (d) 9.8 sec 65. A cube has side cm. The number of significant figures in surface area is (a) 1 (b) 3 (c) 2 (d) The order of magnitude of 5000 is (a) 0 (b) 1 (c) 2 (d) 3

43 Units and Measurements If velocity v, acceleration a and force F are taken as fundamental quantities, the dimensions of youngs modulus is (a) Fa 2 v 2 (b) Fa 2 v 3 (c) Fa 2 v 4 (d) Fa 2 v 5 68 Which of the following is not measured in units of energy? (a) Couple angle turned (b) Moment of inertia (angular velocity) 2 (c) Force distance (d) Impulse time 69. Dimensions of mometum are: (a) 0, 1, 2 (b) 1, 2, 1 (c) 1, 1, 1 (d) 2, 1, Which of the following relations is dimensionally correct. (a) V st (b) V at (c) m Fa (d) W νr 71. In the gas equation (p + a V 2 )(V b) Rθ, where θ is the absolute temperature, P is the pressure and V is the volume. The dimensional formula of the constant a is (a) ML 2 T 1 (b) ML 5 T 2 (c) L 3 (d) ML 1 T A number has significant figures (a) 3 (b) 4 (c) 5 (d) None of these 73. The result of is: (a) (b) 46.5 (c) 46.4 (d) The least count of a stop watch is 1.5s. The time of 20 oscillations of a pendulum is 25s. What is the maximum percentage error in this measurement. (a) 1.6% (b) 16% (c) 0.08% (d) 0.8% 75. Unit of intesity of illumination is (a) lumen (b) candela (c) lux (d) photon 76. One second is defined to be equal to: (a) periods of krypton clock. (c) periods of cesium clock. (b) periods of krypton clock. (d) periods of cesium clock. 77. One nanometer is equal to (a) 10 9 mm (b) 10 6 cm (c) 10 7 cm (d)10 9 cm 78. Which one of the following is not a unit of time. (a) lunar month (b) leap year (c) parallactic second (d) solar day 79. Distance z travelled by a particle is defined by, z α + βt + γt 2. Dimensions of γ are (a) LT - 1 (b) L - 1 T (c) LT - 2 (d) LT Nm 2 /Kg 2 is a unit of (a) surface tension (b) permittivity (c) gravitational constant (d) torque

44 is equal to -44- (a) 2.88 (b) (c) 2.9 (d) None of these 82. The radius of ball is (5.4 ± 0.2) cm. The percentage error in the volume of the ball is, (a) 11% (b) 4% (c) 7% (d) 9% 83. The significant figures in are (a) 6 (b) 5 (c) 4 (d) The Hubble constant has the dimension of (a) time (b) time - 1 (c) length (d) mass 85. The dimensions of resistance capacitance are same as that of (a) current (b) energy (c) frequency (d) time 86. The number of significant figures in is (a) 2 (b) 3 (c) 4 (d) Length cannot be measured by (a) fermi (b) micron (c) debye (d) light year 88. S I unit of power is (a) Joule (b) erg (c) Newton (d) Watt L 89. Dimensions of RCV are (a) A - 1 (b) A - 2 (c) A (d) A The voltage v (100 ± 5) V and current I (0 ± 0.2) A, the percentage error in R is (a) 5.2% (b) 2.5% (c) 7% (d) 25% 91. Which of the following unit is not that of mutual inductance? (a) Henry (b) Weber (c) Ohm second (d) Volt second ampere The number of significant figures in is (a) 3 (b) 6 (c) 5 (d) The SI unit of electron mobility is (a) m 2 s - 1 v - 1 (b) ms v - 1 (c) ms - 1 v (d) m 2 s - 1 v The SI unit of entropy (a) Joule/Kelvin (b) Nm (c) Calorie/second (d) Joule/Calorie 95. The joule second is the unit of (a) energy (b) linear momentum (c) angular momentum (d) Power 96. The unit 1 Nm - 1 is equivalent to (a) 1 erg cm - 1 (b) 1 erg cm - 2 (c) 1 Jm - 1 (d) 1Jm The dimension of coefficient of viscosity are (a) ML - 1 T - 1 (b) MLT - 2 (c) ML 0 T - 2 (d) MLT - 1

45 Units and Measurements 98. Parsec is the unit of -45- (a) time (b) distance (c) frequency (d) angular acceleration 99. SI unit of velocity is (a) m/s (b) m sec - 2 (c) m hr - 2 (d) m/hr 1. a b c d 26. a b c d 51. a b c d 76. a b c d 2. a b c d 27. a b c d 52. a b c d 77. a b c d 3. a b c d 28. a b c d 53. a b c d 78. a b c d 4. a b c d 29. a b c d 54. a b c d 79. a b c d 5. a b c d 30. a b c d 55. a b c d 80. a b c d 6. a b c d 7. a b c d 31. a b c d 32. a b c d 56. a b c d 57. a b c d 81. a b c d 82. a b c d 8. a b c d 33. a b c d 58. a b c d 83. a b c d 9. a b c d 34. a b c d 59. a b c d 84. a b c d 10. a b c d 35. a b c d 60. a b c d 85. a b c d 11. a b c d 12. a b c d 13. a b c d 14. a b c d 15. a b c d 36. a b c d 37. a b c d 38. a b c d 39. a b c d 40. a b c d 61. a b c d 62. a b c d 63. a b c d 64. a b c d 65. a b c d 86. a b c d 87. a b c d 88. a b c d 89. a b c d 90. a b c d 16. a b c d 17. a b c d 18. a b c d 19. a b c d 20. a b c d 41. a b c d 42. a b c d 43. a b c d 44. a b c d 45. a b c d 66. a b c d 67. a b c d 68. a b c d 69. a b c d 70. a b c d 91. a b c d 92. a b c d 93. a b c d 94. a b c d 95. a b c d 21. a b c d 22. a b c d 23. a b c d 24. a b c d 25. a b c d 46. a b c d 47. a b c d 48. a b c d 49. a b c d 50. a b c d 71. a b c d 72. a b c d 73. a b c d 74. a b c d 75. a b c d 96. a b c d 97. a b c d 98. a b c d 99. a b c d