2.Units-Dimension.pdf - Brains Education

BRAINS EDUCATION, Plot no-79, Bapuji Nagar, Bhubaneswar, www.brainsedu.in UNITS & DIMENSION 31

UNITS & DIMENSION

IIT-JEE Syllabus

1. Unit & Dimensions

2. Dimensional analysis

Least count

4. Significant figure

5. Methods of measurement

6. Methods of measurement and Error analysis for physical quantities

Total No. of questions in Units & Dimension are:

Solved examples…....…………………………..…07

Exercise # 1 …….……………………………….…15

Exercise # 2 …….……………………………….…20

Exercise # 3 …….……………………………….…21

Exercise # 4 ……………………………………..…07

Exercise # 5 ……………………………………..…12

Total No. of questions………………..82

*** Students are advised to solve the questions of exercises in the same sequence or as

directed by the faculty members.


Index : Preparing your own list of Important/Difficult Questions

Instruction to fill

(A) Write down the Question Number you are unable to solve in column A below, by Pen.

(B) After discussing the Questions written in column A with faculties, strike off them in the

manner so that you can see at the time of Revision also, to solve these questions again.

(C) Write down the Question Number you feel are important or good in the column B.

EXERCISE NO.

COLUMN :A COLUMN :B

Questions i am unable to solve in first attempt

Good/Important questions

1

2

3

4

5

Advantages

1. It is advised to the students that they should prepare a question bank for the revision as it is

very difficult to solve all the questions at the time of revision.

2. Using above index you can prepare and maintain the questions for your revision.


KEY CONCEPT

1. Physical quantity

The quantities by means of which we describe the

laws of physics are called physical quantities.

1.1 Fundamental Quantities

Physical quantities which are independent of

each other and cannot be further resolved into

any other physical quantity are known as

fundamental quantities. There are seven

fundamental quantities. In SI system of units.

1.2 Derived Quantities :

Physical quantities which depend upon

fundamental quantities or which can be

derived from fundamental quantities are

known as derived quantities.

2. Unit

Measurement of any physical quantity involves

comparison with a certain basic, arbitrarily chosen,

internationally accepted reference standard called

unit.

2.1 Fundamental unit :

The unit for the fundamental or the base

quantities are called fundamental or base

units.

2.2 Derived unit :

The unit of the derived quantities are called

derived unit.

3. Principle system of units

3.1 (i) C.G.S. system [centimetre (cm) ; gram (g)

and second (s)]

(ii) F.P.S system [foot ; pound ; second]

(iii) M.K.S. system [meter ; kilogram ; second]

(iv) S.I. (system of international)

3.2 In SI system their are seven base unit :

Fundamental Units Symbol

quantity

(i) Length metre m

(i) Mass kilogram kg

(iii) Time second s

(iv) Electric current ampere A

(v) Thermodynamic kelvin K

temperature

(vi) Luminous candela cd

intensity

(vii) Amount of mole mol.

substance

3.3 supplementary units :

Besides the seven fundamental units two

supplementary units are defined.

(i) the unit for the plane angle is radian with the

symbol rad

(ii) the unit for the solid angle is steradian with

the symbol sr.

4. Dimensions

4.1 Dimensions of a physical quantity are the

powers to which the fundamental quantities

must be raised to represent the given physical

quantity.

llustration :

Force (Quantity) = mass × acceleration

= mass × time

velocity = mass ×

2)time(

length

= mass × length × (time)–2

So dimensions of force : 1 in mass

1 in length

–2 in time

and Dimensional formula : [MLT–2]


4.2 Dimension of seven base quantities :

Fundamental quantity Dimension

(i) Length [L]

(ii) Mass [M]

(iii) Time [T]

(iv) Electric current [A]

(v) Thermodynamic temperature [K]

(vi) Luminous intensity [cd]

(vii) Amount of substance [mol]

5. Dimensional formula

It is an expression which shows how and which of the fundamental units are required to represent the unit of

physical quantity.

Different quantities with units. symbol and dimensional formula,

Quantity Symbol Formula S.I. Unit D.F.

Displacement s — Metre or m M0LT0

Area A l × b (Metre)2 or m2 M0L2T0

Volume V l × b × h (Metre)3 or m3 M0L3T0

Velocity v v = t

s

m/s M0LT–1

Momentum p p = mv kgm/s MLT–1

Acceleration a a = t

v

m/s2 M0LT–2

Force F F = ma newton or N MLT–2

Impulse – F × t N.sec MLT–1

Work W F. d N.m ML2T–2

Energy KE or U K.E. =

2

1mv2

P.E. = mgh

joule or J ML2T–2

Power P P = t

W watt or W ML2T–3

Density d d = mass/volume kg/m3 ML–3T0

Pressure P P = F/A pascal or Pa ML–1T–2

Torque = r × F N.m. ML2T–2


Quantity Symbol Formula S.I. Unit D.F.

Angular displacement = radius

arc radian or rad M0L0T0

Angular velocity = t

rad/sec M0L0T–1

Angular acceleration = t

rad/sec2 M0L0T–2

Moment of Inertia I I = mr2 kg-m2 ML2T0

Angular momentum J or L J = mvr s

m.kg 2

ML2T–1

Frequency or f f = T

1 hertz or Hz M0L0T–1

Stress — F/A N/m2 ML–1T–2

Strain —

;

A

A;

V

V —- M0L0T0

Young's modulus

Y Y =

/

A/F

N/m2 ML–1T–2

Surface tension T

F or

A

W

m

N;

2m

J ML0T–2

Force constant (spring) k F = kx N/m ML0T–2

Coefficient of viscosity F =

dx

dvA kg/ms(poise in C.G.S) ML–1T–1

Gravitational constant G

F = 2

21

r

mGm

G = 21

2

mm

Fr

2

2

kg

mN M–1L3T–2

Gravitational potential Vg Vg = m

PE

kg

J M0L2T–2

Temperature — kelvin or K M0L0T0 K+1

Heat Q Q = m × S × t joule or calorie ML2T–2

Specific heat S Q = m × S × t kelvin.kg

jouleor Jkg

–1K

–1

M0L2T–2 K–1

Latent heat L Q = mL kg

Joule or Jkg

–1

M0L2T–2

Coefficient of thermal

conductivity K Q =

d

t)(KA 21

Ksecm

Joule or Jm

–1J

–1K

–MLT–3 K

–1


Quantity Symbol Formula S.I. Unit D.F. 1

Mechanical equivalent

of heat J W = JH — M0L0T0

Charge Q or q I = t

Q coulomb or C M0L0TA

Current I — ampere or A M0L0T0A

Electric permittivity 0 F = 04

1

.

2

21

r

qq

m.N

.)coul( 2

or 2

2

mN

C

M–1L–3A2T4

Electric Potential V V = q

W joule/coul ML2T–3A–1

Intensity of electric field E E = q

F N/coul. MLT–3A–1

Capacitance C Q = CV farad M–1L–2T4A2

Dielectric constant

or relative permittivity r r =

0

— M0L0T0

Resistance R V = IR Ohm ML2T–3A–2

Conductance S S = R

1 mho M–1L–2T–3A2

Specific resistance

or resistivity =

RA ohm × meter ML3T–3A–2

Conductivity or

specific conductance =

1 Mho/meter M–1L–3T3A2

Magnetic induction B F = qvBsin

or F = BIL tesla or weber/m2 MT–2A–1

Magnetic flux e = dt

d weber ML2T–2A–1

Magnetic intensity H B = H A/m M0L–1T0A

Magnetic permeability

of free space or medium 0 B =

4

0

2r

sindlI 2amp

N MLT–2A–2

Coefficient of self or

Mutual inductance L e = L .

dt

Id henery ML2T–2A–2

Electric dipole moment p p = q × 2l C.m. M0LTA

Magnetic dipole moment M M = NIA amp.m2 M0L2AT0


6. Application of dimensional analysis

(a) Checking the dimensional consistency of

equations.

(b) Deducing relation among the physical

quantities.

(c) To find the unit of a given physical

quantity in a given system of units

7. Absolute error

a= |Measured value – True value|

= |a – a0|

8. Fractional error & percentage error

(i) fractional error = 0a

a,

(ii) percentage error = 100a

a

0

9. Combination of errors

if

(i) x = a + b x = a + b

(ii) x = a – b x = a + b

(iii) x = a / b x

x=

b

b

a

a

(iv)x = an x

x= n

a

a

(v) x = p

mn

c

ba

x

x=

c

c|p|

b

b|m|

a

a|n|

10. Common rules for counting signification

Rule 1. All non zero digits are significant.

Ex.x = 1234 has four significant figures.

Rule 2. All zero occurring between two non zero

digits are significant

Ex.x = 1007 has four significant figure.

Again x = 1.0809 has five significant

figure

Rule 3. In a number less than one all zeros to the

right of decimal point and to the left of a non zero digit are NOT significant.

Ex. x = 0.0084 has only two significant digits

Rule 4. All zeros on the right of the last non zero

digit in the decimal part are significant

Ex.x = 0.00800 has three significant

figure 8, 0, 0. The zeros before 8 are not

significant. Again 1.00 has three

significant figures

11. Arithmatical operations with significant

figures

(a) Addition and subtraction: –

In addition or subtraction, the number of

decimal places in the result should equal the

smallest number of decimal places of terms in

the operation.

(b) Multiplication and Division: –

In multiplication and division, the number of

significant figures in the product or in the

quotients is the same as the smallest number

of significant figures in any of the factors.

Ex. 64125.92823.10

9500

As 9500 has minimum number of significant

figures (i.e. 2), therefore, the quotient can

have only two significant digits. On rounding

off. we obtain the quotient = 930

12. Multiplication factors & SI prefixes

Multiplication factors Prefix Symbol

1018 Exa. E

1015 Peta P

1012 Tera T

109 Giga G

106 Mega M

103 Kilo k

102 Hecto h

10 Deca da

10–1 Deci d

10–2 Centi c


10–3 Milli m

10–6 Micro µ

10–9 Nano n

10–12 Pico p

10–15 Femto f

10–18 Atto a

13. Some of the non SI units in common use

are

(a) For length/distance

(i) Astronomical unit 1 AU = 1.496 × 1011 m

(ii) Light year, 1 y = 9.64 × 1015 m

(iii) Parallactic second 1 pc = 3.084 × 1016 m

= 3.26 y

(iv) Micron or 1 µm = 10–6 m

micrometer

(v) Nanometer 1 nm = 10–9 m

(vi) Angstrom unit 1 Å = 10–10 m

(vii) X - ray unit 1 xu = 10–13 m

(viii) Fermi 1 f = 10–15 m

(ix) Yard 1 yd = 0.9144 m

(x) Foot 1 ft = 0.3048 m

(xi) Inch 1 in = 0.0254 m

(xii) Mile 1 Mile = 1609.344 m

= 1.61 km

(xiii) Nautical mile 1 n mile = 1852 m

(b) For mass

(i) Pound 1 lb = 0.4536 kg

(ii) Slug 1 slug = 14.59 kg

(iii) Quintal 1 q = 100 kg

(iv) Metric tone 1 t = 1000 kg

(v) Atomic mass unit 1 amu = 1u

= 1.66 × 10–27 kg

(c) For time

(i) Minute 1 min = 60 s

(ii) Hour 1 h = 60 × 60 s

(iii)Day 1 day = 24 h

= 24 × 60 × 60 s

(iv)Year 1 yr = 365.25 days

= 3.156 × 107 s

(v) Shake 1 shake = 10–8 s

(d) For other quantities

(i) Barn (for area) 1 barn = 10–28 m2

(ii) Litre (for volume) 1 = 103 cc = 10–3 m3

where cc represents

cubic centimeter

i.e. cm3.

(iii) Gallon (for volume) In U.S.A.,

1 gallon = 3.7854

(iv) Pascal (for pressure) 1 Pa = 1 Nm–2

Pressure exerted by 1 atm = 1.01 × 105 Pa

earth’s atmosphere

(v) Electron volt 1 eV = 1.6 × 10–19 J

(for energy/work)

(vi) Erg (for energy/ work) 1 erg = 10–7 J

(vii) Kilowatt hour 1 kWh = 3.6 × 106 J

(for energy)

(viii) Horse Power 1hp = 746 W

(for power)

(ix) Dioptre 1 D = 1 m–1

(for power of a lens)

(x) Degree (for angle) 1º = rad180

π


SOLVED EXAMPLES

Ex.1 Mass of the proton is 1.6 × 10–27 kg.

Calculate the number of protons in a piece of

metal whose net mass is one gram. Express

your answer in order of magnitude.

(A) 1023 (B) 1025

(C) 1024 (D) 1022

Sol 1 gm = 10–3 kg

number of protons in one gram metal

= 27

3

106.1

10

= 6.25 × 1023

Thus the number of protons in one gram

metal is of the order of 1024.

Hence correct answer is (C).

Ex.2 Solve with regards to significant figures.

638.1

54.39996.0

Sol. 638.1

54.39996.0 = 2.1603

The answer should be up to 3 significant

figures. Therefore the correct answer is 2.16.

Ex.3 If velocity, force and time are taken to be

fundamental quantities find the dimensions

formula for a mass.

(A) KV–1 FT–1 (B) K V–1 FT

(C) K V F–1 T–1 (D) K V–1 F–1 T

Sol. Let the mass is represented by M then

M = f (V, F, T)

Assuming that the function is product of

power functions of V, F and T

M = KVx Fy Tz

Where k is a dimension less constant of

proportionality. The above equation

dimensionally becomes.

[M] = [LT–1]x [MLT–2]y [T]z

i.e. [M] = [My] [Lx + y T – x – 2y + z ]

So equation becomes

[M] = [My Lx + y T– x – 2y + z]

For dimentionally correct expression,

y = 1, x + y = 0 and – x – 2y + z = 0

x = –1, y = 1 and z = 1.

therefore M = KV–1 FT.

Hence correct answer is (B).

Ex.4 Column 1 gives three physical quantities.

Select the appropriate units for these from

choices given in column. Some of the

physical quantities may have more than one

choice.

I II

Capacitance Ohm × second

Inductance Coul2 joule–1

Magnetic- Coulomb (volt)–1

inductance Newton(amp.-m)–1

Volt-sec (Ampere)–1

Sol. (I) q = CV

i.e. [C] =

V

q

so [C] = [M–1 L–2 T4 A2]

U = 2

1LI2

i.e. [L] =

2I

U so [L] = [M1 L2 T–2 A–2]

F = Bil sin

i.e. [B] =

i

F so [B] = [MT–2 A–1]

(II) Now the dimensions from given units are

ohm × sec [R] [T] [ML2 T–3 A–2] [T]

= [ML2 T–2 A–2]

Coul2 – joule–1

w

q2

]TL[

]TA[22

22

M

= [M–1 L–2 T4 A2]

Coul (volt)–1

V

q

]TML[

]TA[22

22

= [M–1 L–2 T4 A2]

newton (amp-m)–1

i

F

]AL[

MLT 2

= [M T–2 A–1 ]

volt sec (amp)–1

Aq

TW

]ATA[

]T][TML[ 22

= [ML2 T–2 A–2 ]


Comparing dimensions of II with I we find

that, capacitance has units [coulomb2 . joule–1

and coulomb (volt)–1] inductance has units

[ohm.sec and volt.sec (ampere)–1] and

magnetic induction has units [newton

(ampere.m)–1]

Ex.5 A certain physical quantity is calculated from

the formula 3

(a2 – b2) h, where h, a and b

are all length. The quantity being calculated

is-

(A) Velocity (B) Length

(C) Area (D) Volume

Sol. Given quantity is = (a2 – b2) h

dimension of h = [L]

dimensions of a2 – b2 = [L2 – L2] = L2

Therefore the dimensions of the given

quantity are [L3]. Thus the quantity being

measured is volume

Ans.(D)

Ex.6 When a current of 2.5 ± 0.5 ampere flows

through a wire, it develops a potential

difference of 20 ± 1 volt. Find the resistance

of the wire.

(A) 6.0 ± 3 (B) 7.0 ± 2

(C) 8.0 ± 2 (D) 9.0 ± 3

Sol. R = I

V =

5.05.2

120

= 8 ± R

the error in the measurement is

R

R =

V

V +

I

I

= 20

1+

5.2

5.0

= 0.05 + 0.2 = 0.25

R = 0.25 R = 0.25 × 8 = 2

Thus the resistance of the wire with the error is

= 8 ± 2 ohm.

Hence correct answer is (C).

Ex.7 In an experiment the values of two resistances

were measured to be as given below. R1 = 5.0

± 0.2 ohm and R2 = 10.0 ± 0.1 ohms. Find

their combined resistance in series

Sol. In series R = R1 + R2

R ± R = (R1 + R2) ± (R1 + R2)

R = [(5 + 10) ± 0.3] = [15 ± 0.3]

or R = [15 ± 2%] because the error in percentage

is = 15

1003.0 = 2%


EXERCISE # 1

Questions

based on Units, System of units

Q.1 Which is the correct unit for measuring

nuclear radii ?

(A) micron (B) millimetre

(C) angstrom (D) Fermi

Q.2 Which of the following is not a unit of time ?

(A) microsecond (B) leap year

(C) lunar month (D) light year

Dimension, finding dimensional formula

Questions

based on

Q.3 A quantity X is defined as the ratio of the

angular to linear momentum of an object.

Then the dimensions of X are -

(A) M0L1T0 (B) M1L1T1

(C) M1L2T–2 (D) M0L–1T0

Q.4 The dimensional formula of angular velocity

r

v is-

(A) M0 L0 T–1

(B) M L T–1

(C) M0 L0 T1 (D) M L0 T–2

Questions

based on Principle of homogeneity of dimension

Q.5 There are two different quantities A and B

having different dimensions. Then which of

the following operation is dimensionally

correct ?

(A) A + B (B) A – B (C) A/B (D) eA/B

Q. 6 A wave is represented by

y = a sin (At – Bx + C)

where A, B, C are constants and t is in seconds

& x is in metre. The Dimensions of A, B, C

are-

(A) T–1, L, M0L0T0

(B) T–1, L–1, M0L0T0

(C) T, L, M

(D) T–1, L–1, M–1

Q.7 A dimensionless quantity -

(A) never has a unit (B) always has a unit

(C) may have a unit (D) does not exist

Q.8 If v =

p, then the dimensions of are (p

is pressure, is density and v is speed of

sound has their usual dimension) -

(A) M0L0T0 (B) M0L0T–1

(C) M1L0T0 (D) M0L1T0

Application of dimensional analysis : Deriving new relation

Questions

based on

Q.9 If energy (E), velocity (V) and force (F), be

taken as fundamental quantities, then what

are the dimensions of mass -

(A) EV2 (B) EV–2

(C) FV–1 (D) FV–2

Application of dimensional analysis : Checking the validity of equation

Questions

based on

Q.10 The formula S = ut – 3

1at2 where S is the

distance travelled, u is the initial velocity, a is

the acceleration and t is the time is -

(A) dimensionally correct only

(B) dimensionally incorrect only

(C) dimensionally and numerically correct

(D) dimensionally and numerically wrong

Questions

based on Significant digits, Rounding, Error

Q.11 The volume of one sphere is 1.76 c.c. The

volume of 25 such spheres (according to the

idea of significant figures) is -

(A) 44.00 cc

(B) 44.0 c.c

(C) 44 c.c

(D) 0.44 × 102 c.c.


Q.12 The percentage error in the measurement of

mass and speed are 2% and 3% respectively.

How much will be the maximum error in the

estimate of Kinetic energy obtained by

measuring mass and speed -

(A) 11% (B) 8% (C) 5% (D) 4%

True or false type questions

Q.13

00µ

1

has the dimensions of velocity and is

numerically equal to velocity of light.

Q.14 In the dimensional analysis of the equation,

(velocity)x = (pressure difference)3/2 ×

(density)–3/2. The value of x comes out to be 3.

Fill in the blanks type questions

Q.15 The dimensions of pressure gradient are

..........


EXERCISE # 2

(Only single correct answer type

questions) Part-A

Q.1 Which of the following pairs of physical

quantities has different dimensions -

(A) stress, pressure

(B) Young’s modulus, energy density

(C) density, relative density

(D) energy, torque

Q.2 Which of the following is a dimensional

constant ?

(A) refractive index

(B) dielectric constant

(C) relative density

(D) gravitational constant

Q.3 The velocity v of a particle is given in terms

of time t by the equation

v = at + b/t + c

The dimensions of a, b & c are -

(A) L2, T, LT2 (B) LT2, LT, L

(C) LT–2, L, LT–1 (D) L, LT, T2

Q.4 Express L (length) in terms of G, h and c -

(A) G–1/2. h1/2 c1/2 (B) G1/2 h1/2 c–3/2

(C) G1/2 h1/2 c–5/2 (D) G h c

Q.5 If Force = (x/density) + C is dimensionally

correct, the dimension of x are -

(A) MLT–2 (B) MLT–3

(C) ML2T–3 (D) M2L–2T–2

Q.6 The equation of the stationary wave is

y = 2A sin

ct2cos

x2

Which of the following statements is wrong ?

(A) the unit of ct is same as that of

(B) the unit of x is same as that of

(C) the unit of 2 c/ is same as that of

2x/t

(D) the unit of c/ is same as that of x/

Q.7 If the velocity of light c, acceleration due to

gravity g, and the atmospheric pressure P are

taken as the fundamental units, then the unit

of mass will be -

(A) 1kg (B) 81 kg

(C) 9 × 1018 kg (D) 81 × 1034 kg

Q.8 The time dependence of a physical quantity P

is found to be of the form P = 2t

0eP where

t is time and is some constant. Then the

constant will -

(A) be dimensionless

(B) have dimensions of T–2

(C) have dimensions of P

(D) have dimensions of P multiplied by T–2

Q.9 If the speed of light (c), acceleration due to

gravity (g) and pressure (p) are taken as

fundamental units, the dimensions of

gravitational constant (G) are -

(A) c0gp–3 (B) c2g3p–2

(C) c0g2p–1 (D) c2g2p–2

Q.10 If P is radiation pressure, c represents speed of

light and Q is radiation energy striking a unit

area per second, then non zero integers x, y

and z such that PxQyCz is dimensionless are-

(A) x = 1, y = 1, z = –1 (B) x = 1, y = –1, z = 1

(C) x = –1, y = 1, z = 1 (D) x = 1, y = 1, z = 1

Q.11 Turpentine oil is flowing through a tube of

length l, radius r, the pressure difference

between the two ends of the tube is p. The

viscosity of the oil is given by

= v4

)xr(p 22

where v is velocity at distance x from the axis

of tube then the dimension of are -

(A) M0L0T0 (B) MLT–1

(C) ML2T–2 (D) ML–1T–1


Q.12 The dimensional formula of a physical

quantity x is [M–1L3T–2]. The error in

measuring the quantities M, L, and T are 2%,

3% and 4%. The maximum percentage of

error that occurs in measuring, the quantity x

is -

(A) 9 (B) 10

(C) 14 (D) 19

Q.13 The heat dissipated in a resistance can be

obtained by the measurement of resistance,

current and time. If the maximum error in the

measurement of these quantities is 1%, 2%,

and 1% respectively. The maximum error in

the determination of the dissipated heat is -

(A) 4% (B) 6% (C) 4/3% (D) 2%

Q.14 The period of oscillation of a non linear

oscillator depends on the mass ‘m’, with

dimensions of M, a restoring force constant

‘K’, with dimensions of ML–2

T–2

and the

amplitude A, with dimensions of L.

Dimensional analysis shows that the period of

oscillation should be proportional to -

(A) k

mA (B) A

2m/k

(C) k

mA 1 (D) A

2K

3/m

One or more than one correct

answer type questions Part-B

Q.15 Which of the following combination have the

dimension of time ? L, C, R represents as usual.

(A) RC (B) LC

(C) R/L (D) C/L

Q.16 The dimension of Boltzmann’s constant are

same as that of -

(A) pressure

(B) stefan's constant

(C) plank’s constant

(D) none of these

Q.17 Choose the correct statements -

(A) A dimensionally correct equation may be

correct

(B) A dimensionally incorrect equation must

be incorrect

(C) A dimensionally correct equation may be

incorrect

(D) A dimensionally incorrect equation may

be correct

Assertion-Reason type questions Part-C

The following questions consists of two

statements each, printed as Assertion and

Reason. While answering these questions you

are to choose any one of the following four

responses.

(A) If both Assertion and Reason are true and

the Reason is correct explanation of the

Assertion.

(B) If both Assertion and Reason are true but

Reason is not correct explanation of the

Assertion.

(C) If Assertion is true but the Reason is false.

(D) If Assertion is false but Reason is true.

Q.18 Assertion : light year is a unit of time.

Reason : light year is the distance travelled

by light in vacuum in one year.

Q.19 Assertion : The equation y = x + t cannot be

true, where x, y are distance and t is time.

Reason : Quantities with different

dimensions cannot be added.

Column matching Part-D

Q.20 Column-I Column-II

(A) Radiation energy (P) Joule/m2

(B) Surface tension (Q) ML2T

–2

(C) Torque (R) ML–1

T–1

(D) Coefficient of viscosity (S) MT–2


EXERCISE # 3

Subjective Type Questions Part-A

Q.1 If a composite physical quantity in terms of

moment of inertia I, force F, velocity v, work

W and length L is defined as ,

Q = (IFv2 /WL3),

find the dimensions of Q and identify it.

Q.2 Check whether the following equations are

dimensionally correct.

(a) 22 xa

dx=

a

1sin

–1

a

x, where x and a

stand for distances.

(b) =2

1

I

mg,

where I = moment of inertia & is length

Q.3 The speed of a particle as a function of time is

represented by V = A1 cos A2t. What are the

dimensions and S.I. units of constants A1 and

A2 ?

Q.4 Taking force, length and time to be the

fundamental quantities find the dimension of-

(A) density (B) pressure

(C) momentum (D) energy

Q.5 The SI and C.G.S. units of energy are joule

and erg respectively. How many ergs are

equal to one joule ?

Q.6 Young’s modulus of steel is 19 x 1010 N/m2.

Express it in dyne/ cm2. Here dyne is the

C.G.S. unit of force

Q.7 The heat produced in a wire carrying an

electric current depends on the current, the

resistance and the time, Assuming that the

dependence is of the product of powers type,

guess an equation between these quantities

using, dimensional analysis. The dimensional

formula of resistance is ML2–2T–3 and heat is

a form of energy.

Q.8 The frequency of vibration of a string

depends on the length L between the nodes,

the tension F in the string and its mass per

unit length M. Guess the expression for its

frequency from dimensional analysis.

Q.9 The kinetic energy K of a rotating body

depends on its moment of inertia and its

angular speed . Assuming the relation to be

K = ab where is a dimensionless

constant, find a and b. M of the sphere about

its diameter is 2/5 Mr2.

Q.10 The refractive index (µ) of water in an

experiment is recorded as 1.29, 1.33, 1.34,

1.31, 1.33 and 1.36 respectively. Determine

(i) mean value of refractive index

(ii) mean absolute error

(iii) relative error and the percentage error

Q.11 The position of a particle at any time is given

by, s(t) = a

v0 (1 – e–at

), where a > 0 and v0 are

constants. What are the dimensions of a and v0 ?

Q.12 Test if following equation are dimensionally

correct, where symbols have their usual

meaning –

(a) h = rg

cosS2

(b) v =

p

(c) v =

8

tpr 4

(d) n = 2

1

I

mg

where h = height, S = Surface tension,

= density, p = pressure, V = volume,

= coefficient of viscosity, = frequency

and I = moment of inertia

Q.13 The distance covered by a particle in time t is

given by x = a + bt + ct2 + dt3, find the

dimensions of a, b, c & d.


Q.14 The volume of liquid flowing per second Q

through a tube depends upon (i) coefficient of

viscosity of fluid , (ii) radius of the tube r,

(iii) the pressure gradient (P/). Deduce by

method of dimension the formula for the

volume of liquid flowing per second.

Q.15 Using dimensional analysis show that de

Broglie wavelength , associated with a

particle of mass m moving with velocity v is

given by mv

h

where h is Planck’s constant.

Q.16 A calorie is a unit of heat energy and it equals

about 4.2 J where 1 J = 1 kg m2s–2. Suppose,

we employ a system of units in which unit of

mass equals kg, the unit of length equals

m, and unit of time is sec. Show that a

calorie has a magnitude of 4.2 –1–22; in

terms of the new units.

Q.17 Assuming that the largest mass that can be

moved by a flowing river depends on velocity

of flow, density of river water and on

acceleration due to gravity, show that the

mass varies as the sixth power of velocity of

flow.

Q.18 The density of a sphere is measured by

measuring the mass and diameter. If it is

known that the maximum percentage errors in

the measurement are 2% and 3% then what is

the maximum percentage error in the

measurement of density.

Passage based objective questions Part-B

Passage (Q. 19 to 21)

The surface of sea-bed is identical to that of land

in all respects i.e. there are valleys, mountain’s,

volcanoes, plains etc, only difference being there

is water in place of air on land. There are often

explosions is active volcanoes on sea-bed and

suppose a small bubble is formed due to one such

explosion and it oscillates with time period T such

that T PadbEc where P is pressure, d is density

of water and E is total energy of explosion

Q.19 The value of a is -

(A) 1/2 (B) –3/4

(C) –5/6 (D) –2

Q.20 The time period T is directly proportional to -

(A) P (B) E

(C) d (D) none of these

Q.21 The time period T is directly proportional to -

(A) E1/3

(B) density (d)

(C) P (D) none of these


EXERCISE # 4

Old IIT-JEE Objective type questions

Q.1 A quantity X is given by 0L t

V

where 0 is

permittivity of free space, L is length, V is a

potential difference and t is a time interval.

The dimensional formula for x is same as that

of- [IIT-2001]

(A) resistance (B) charge

(C) voltage (D) current

Q.2 A cube has a side 1.2 × 10–2m. Its volume

will be recorded as - [IIT-2003]

(A) 1.728 × 10–6 m3 (B) 1.72 × 10–6m3

(C) 1.7 × 10–6 m3 (D) .72 × 10–6 m3

Q.3 A wire is of mass (0.3 ± .003) gm. The radius

is (0.5 ± 0.005) cm and length is (6 ± .06) cm.

The maximum percentage error in density is -

[IIT-2004]

(A) 3% (B) 4% (C) 8% (D) 16%

Q.4 P =

exp

BK

z

Temperature

P Pressure

KB Boltzmann constant

z Distance

Dimension of is - [IIT-2004]

(A) M0 L0 T0 (B) M–1L1T2

(C) M0L2T0 (D) ML–1T–2

Q.5 Which of the following physical quantities do

not have the same dimensions ? [IIT-2005]

(A) Pressure, Young’s modulus, stress

(B) Electromotive force, voltage, potential

(C) Heat, Work, Energy

(D) Electric dipole, electric field, flux

Passage for Q.No.6 & 7

A dense collection of equal number of

electrons and positive ions is called neutral

plasma. Certain solids containing fixed

positive ions surrounded by free electrons can

be treated as neutral plasma. Let 'N' be the

number density of free electrons, each of

mass 'm'. When the electrons are subjected to

an electric field, they are displaced relatively

away from the heavy positive ions. If the

electric field becomes zero, the electrons

begin to oscillate about the positive ions with

a natural angular frequency 'p', which is

called the plasma frequency. To sustain the

oscillations, a time varying electric field

needs to be applied that has an angular

frequency , where a part of the energy is

absorbed and a part of it is reflected. As

approaches p, all the free electrons are set to

resonance together and all the energy is

reflected. This is the explanation of high

reflectivity of metals. [IIT-2011]

Q.6 Taking the electronic charge as 'e' and the

permittivity as '0', use dimensional analysis

to determine the correct expression for p.

(A) 0mε

Ne (B)

Ne

mε0

(C) 0

2

mε

Ne (D)

2

0

Ne

mε

Q.7 Estimate the wavelength at which plasma

reflection will occur for metal having the

density of electrons N 4 × 1027

m–3

. Take

0 = 10–11

and m 10–30

, where these

quantities are in proper SI unit-

(A) 800 nm (B) 600 nm

(C) 300 nm (D) 200 nm


EXERCISE # 5(ARCHIVES)

Old IIT-JEE questions

Q.1 Which of the following pairs have same

dimensions ? [IIT-1986]

(A) Torque and work

(B) Angular momentum and work

(C) Energy and Young's modulus

(D) Light year and wavelength

Q.2 Column I gives three physical quantities.

Select the appropriate units for these from

choices given in Column II. Some of the

physical quantities may have more than one

choice [IIT-1990]

(a) Capacitance (i)

(b) Inductance (ii)

(c) Magnetic Induction (iii)

Column I

(iv)

Column II

(v)

Ohm-second

Coulomb2 Joule

–1

Coulomb (volt)–1

Newton (ampere

meter)–1

Volt second

(ampere)–1

Q.3 A highly rigid cubical block A of small mass

M and side L is fixed rigidly on the another

cubical block of same dimensions and of low

modulus of rigidity such that the lower face

of A completely covers the upper face of B.

The lower face of B is rigidly held on a

horizontal surface. A small force F is applied

perpendicular to one of the side faces of A.

After the force is withdrawn, block A

executes small oscillations, the time period of

which is given by - [IIT-92]

(A) 2 )LM( (B) 2 )L/M(

(C) 2 )/ML( (D) 2 )L/M(

Q.4 Match the physical quantities given in column

I with dimensions expressed in column II in

tabular form : [IIT-1993]

(a) Angular Momentum (i) ML2T

–2

(b) Latent Heat (ii) ML2Q

–2

(c) Torque (iii) ML2T

–1

(d) Capacitance (iv) ML3T

–1Q

–2

(e) Inductance (v) M–1

L–2

T2Q

2

(f) Resistivity (vi) L2T

–2

Column I Column II

Q.5 The frequency of oscillation of an object of

mass m suspended by means of spring of

force constant K is given by f = CmxKy,

where C is a dimensionless constant. The

value of x and y are - [IIT-1994]

(A) x = 2

1, y =

2

1 (B) x = –

2

1,y =

2

1

(C) x = 2

1, y = –

2

1 (D) x = –

2

1,y = –

2

1

Q.6 In the formula X = 3YZ2, X and Z have

dimensions of capacitance and magnetic

induction respectively. What are dimensions of

Y in MKSQ system ? [IIT-1995]

(A) [M–3L–1T3Q4] (B) [M–3L–2T4Q4]

(C) [M–2L–2T4Q4] (D) [M–3L–2T4Q1]

Q.7 The pairs of physical quantities that have the

same dimensions are - [IIT-1995]

(A) Reynolds number and coefficient of

friction

(B) Latent heat and gravitational potential

(C) Curie and frequency of a light wave

(D) Planck’s constant and torque

Q.8 The dimensions of electrical conductivity is

........................... [IIT-1997]


Q.9 The equation of state of a real gas is given by

2V

aP (V – b) = RT

where P, V and T are pressure, volume and

temperature respectively and R is the

universal gas constant. The dimensions of the

constant a in the above equation is ..............

[IIT-1997]

Q.10 The SI unit of the inductance, the henry can

by written as - [IIT-1998]

(A) weber/ampere

(B)volt-second/ampere

(C) joule/(ampere)2

(D) ohm-second

Q.11 Let 0 denote the dimension formula of the

permittivity of the vacuum and µ0 that of

permeability of the vacuum, then –

[IIT-1998]

(A) [0] = M–1L–3T2I

(B) [0] = M–1L–3T4I2

(C) [µ0] = MLT–2I–2

(D) [µ0] = ML2T–1I

Q.12 The dimension of

2

10E

2 (0 : permittivity

of free space, E electric field) is– [IIT-2000]

(A) MLT–1 (B) ML2T–2

(C) ML–1T–2 (D) ML2T–1


ANSWER KEY

EXERCISE # 1

Q.No. 1 2 3 4 5 6 7 8 9 10 11 12

Ans. D D A A C B C A B A B B

13. True 14. True 15. [ML–2

T–2

]

EXERCISE # 2

PART-A

Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Ans. C D C B D D D B C B D D B C

PART-B

Q.No. 15 16 17 18 19

Ans. A,D D A,B,C D A

PART-C

Q.No. 18 19

Ans. D A

PART-D

20. A Q ; B P, S ; C Q ; D R

EXERCISE # 3

PART-A

1. [Q] = MT–2. The quantity may be surface tension, force constant or surface energy. The physical quantity will

not be unique.

2. (a) Incorrect (b) Correct

3. [LT– 1] and [T– 1] ; m/s and rad/s. 4. (a) FL–4T2, (b) FL–2, (c) FT, (d) FL

5. 10+7 erg 6. 19 × 1011 dyne/cm2. 7. H = KI2Rt


8. = L

K

m

F 9. KE = kI2 10. (i) 1.33, (ii) 0.02, (iii) 1.5 %

11. [T–1], [LT–1]

12. All are dimensionally correct 13. [a] = L, [b] = LT–1, [c] = LT–2, [d] = LT–3

14. Q

4Pr 18. 11 %

PART-B

Q.No. 19 20 21

Ans. C C A

EXERCISE # 4

Qus. 1 2 3 4 5 6 7

Ans. D C B C A C B

EXERCISE # 5

1. (A,D) 2. (a) (ii) & (iii), (b) (i) & (v), (c) (iv) 3. (D)

4. (a) (iii), (b) (vi), (c) (i), (d) (v), (e) (ii), (f) (iv)

5. (B) 6. (B) 7. (A) 8. [M– 1 L– 3 T 3 A2]

9. [M L5 T– 2] 10.(A,B,C,D) 11. (B,C) 12.(C)

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