Units and Dimensions

Only One Alternative is Correct

1. E, m, L, G denote energy, mass, angular momentum constant & gravita on constant

𝐄𝐋𝟐
respec vely. The dimensions of will be that of :
𝐦𝟓 𝐆 𝟐
a) angle b) length c) mass d) me

2. The dimensional formula for which of the following pair is not the same?
a) impulse and momentum
b) torque and work
c) stress and pressure
d) momentum and angular momentum

3. If the speed of light (c), accelera on due to gravity (g) and pressure (p) are taken as
fundamental units, the dimensional of gravita onal constants (G) are :
a) [c2 g3 p2] b) [c0 g2 p–1] c)[c2 g2 p–2] d) [c0 g p–3]

4. Which of the following combina ons of three dimensionally different physical

quan es P, Q, R can never be a meaningful quan ty?
a) PQ – R b) PQ/R c) (P – Q)/R d) (PR – Q2)/QR

5. In a view unit system, 1 unit of me is equal to 10 second, 1 unit of mass is 5kg and
1 unit of length is 20 m. In the new system of units, 1 unit of energy is equal to :
𝟏
a) 20 joule b) joule c) 4 joule d) 16 joule
𝟐𝟎
 𝐚 𝐚 𝐭𝟐
6. The dimensions of in the equa on, P = where P is the pressure, x is the
𝐛 𝐛𝐱
distance and t is the me, are :
a) [M2 LT–3] b) [MT–2] c) [LT–3] d) [ML3T–1]

7. The me dependence of a physical quan ty p is given by p = p0e(-at2)where 𝛂 is

constant and t is me. The constant 𝛂 :
a) is dimensionless b) has dimensions [T–2]
c) has dimensions [T2] d) has dimensions of p

8. If area (A), velocity and density (𝛒) are base units, then the dimensional formula of
force can be represented as :
a) [Av𝛒] b) [Av2𝛒] c) [Av𝛒2] d) [A2v𝛒]

𝐑𝟐 𝐏𝟐
9. Two forces P and Q act at a point and have resultant R. If Q is replaced by
𝐐

ac ng in the direc on opposite to that of Q, the resultant :

a) remains same b) becomes half
c) becomes twice d) none of these

10. If instead of mass, length and me as fundamental quan es, we choose velocity,
accelera on and force as fundamental quan es express their dimensions by v, a
and F respec vely, then the dimensions of Young’s modulus will be expressed as :
a) [Fa2v–4] b) [F2v–1a] c) [Fa2v–1] d) [Fav–2]
11. Which of the following statement is correct about conversions of units =, for
example 1 m = 100 cm?
a) Conversion of units have different dimensions on each side of the equal sign but
not the same units
b) Conversion of units have same dimensions on each side of the equal sign but not
the same units
c) If a larger unit is used then numerical value of physical quan ty is large
d) Due to conversion of units physical quan ty to be measured will change

12. If the speed v of a par cle of mass m as func on of me t is given by

𝐤
v = 𝛚Asin 𝐭 , where A has dimension of length
𝐦

a) The argument of trigonometric func on must be a dimensionless quan ty

b) Dimensional formula of 𝛚 is [LT–1]
c) Dimensional formula of k is [MLT–2]

𝐊
d) Dimensional formula of is [T]
𝐦

13. If P and Q have different non-zero dimensions, which of the following opera ons is
possible?
𝐏
a) P + Q b) PQ c) P – Q d) 1 –
𝐐

14. In the formula X = 3YZ2, X and Z have dimensions of capacitance and magne c
induc on respec vely. What are the dimensions of Y in MKS system?
a) [M–3L–1T3Q4] b) [M–3L–2T4Q4]
c) [M–2L–2T4Q4] d) [M–3L–2T4Q1]
15. A cube has side of length 1.2 × 10–2m. Calculate its volume :
a) 1.7 × 10–6 m3 b) 1.73 × 10–6 m3
c) 1.70 × 10–6 m3 d) 1.732 × 10–6 m3

𝛂 𝛂𝐳
16. Pressure depends on a distance as, P = exp , where 𝛂,𝛃 are constant, z is
𝛃 𝐤𝛉

distance, k is Boltzmann’s constant and 𝛉 is temperature. The dimensions of 𝛃 are :

a) [M0L0T0] b) [M–1–1T–1] c) [M0L2T0] d) [M–1L–1T2]

17. A wire of length l = 6 ± 0.06 cm and radius r = 0.5 ± 0.005 cm and mass m = 0.3 ±
0.003 g. Maximum percentage error in density is :
a) 4% b) 2% c) 1% d) 6.8%

18. Which of the following sets have different dimensions?

a) Pressure, Young’s modulus, stress
b) Emf, poten al difference, electric poten al
c) Heat, work done, energy
d) Dipole moment, electric flux, electric field

19. Which of the pair have same dimensions?

a) Force and strain b) Force and stress
c) Angular velocity and frequency d) Energy and strain

20. The physical quan es not having sane dimensions are :

a) torque and work b) momentum and Planck’s constant
c) stress and Young’s modulus d) speed and (𝛍0 𝛆0) –1/2
21. The dimensions of coefficient of viscosity is :
a) [ML–1T–1] b) [MLT–2] c) [ML0T–2] d) [MLT–1]

22. A par cle is moving eastwards with a velocity of 5 m/s. In 10 sec, the velocity
changes to 5 m/s northwards. The average accelera on in this me is :
𝟏
a) zero b) ms–2towards north-west
√𝟐
𝟏 𝟏
c) ms–2towards north-east d) ms–2towards north
√𝟐 √𝟐

23. Out of the following the only pair that does not have iden cal dimensions is :
a) angular momentum and Planck’s constant
b) moment of iner a and moment of a force
c) work and torque
d) impulse and momentum

24. Which of the following units denotes the dimension ML2 / Q2 , where Q denotes
the electric charge?
a) weber (Wb) b) Wb/m2 c) henry (H) d) H/m2

25. The dimension of magne c field in M, L, T and C (column) is given as :

a) [MLT–1C–1] b) [MT2C–2] c) [MT–1C–1] d) [MT–2C–1]

More than One Alternative are Correct

26. Which of the following dimensions are correctly matched? (𝛉 = temperature)

a) Angular momentum-[M1L2T–1] b) Torque-[M1L2T–2]
c) Stefan’s constant-[M1T–3 𝛉–4] d) Planck’s constant-[M1L2T–2]
 𝐚
27. The gas equa on for n moles of a real gas is 𝐏 + (V – b) = nRT where P is the
𝐯𝟐

pressure, V is the volume, T is the absolute temperature, R is the molar gas constant
and a, b are arbitrary constants. Which of the following have the same dimensions
as those of PV?
a) nRT b) a/V c) Pb d) ab / V2

28. The dimensions of the quan es in one (or more) of the following pairs are the
same. Iden fy the pair(s) :
a) Torque and work b) Angular momentum and work
c) Energy and Young’s modulus d) Light-year and wavelength

29. The dimensions of length are expressed as Gx cy hz, where G,c and h are the
universal gravita onal constant, speed of light and Planck’s constant respec vely,
then :
a) x = (1/2), y = (1/2) b) x = (1/2), z = (1/2)
c) y = (–3/2), z = (1/2) d) y = (1/2), z = (3/2)

30. The pairs of physical quan ty that have the same dimension in (are):
a) Reynolds number and coefficient of fric on
b) Curie and frequency of a light wave
c) Latent heat and gravita onal poten al
d) Planck’s constant and torque
 Matching Type Problems
31. Match the column :

Column-Ⅰ(Quan ty) Column-Ⅱ(Unit)

(a) Energy density (Energy per (p) dyne / cm2

unit volume)
(b) Force constant of a spring (q) Kg-m/s

(c) Pressure (r) erg/cm2

(s) pascal

32. Suppose that two students are trying to make a news measurement system so that
they can use it like a code measurement system and others do not understand it.
Instead of taken 1 kg, 1 m and 1 sec, as basic unit they took unit of mass as 𝛂 kg,
the unit of length as 𝛃 m and the unit of mes as 𝛄 second. They called power in
new system as ACME then match the two columns.

Column-Ⅰ Column-Ⅱ

(a) 1N in new system (p) 𝛂–1 𝛃–2 𝛄2

(b) 1J in new system (q) 𝛂–1 𝛃–1 𝛄1

(c) 1 pascal (SI unit of pressure) (r) 𝛂–1 𝛃 𝛄2

in new system
(d) 𝛂 ACME in wa (s) 𝛂–2 𝛃2 𝛄–3
33. Match the physical quan es given in columnⅠwith dimensions expressed in
terms of mass (M), length (L), me (T), and charge (Q) given in columnⅡ

Column-Ⅰ Column-Ⅱ

(a) Angular momentum (p) [ML2T–2]

(b) Torque (q) [ML2T–1]

(c) Inductance (r) [M–1L–2T2Q2]

(d) Latent heat (s) [ML2Q–2]

(e) Capacitance (t) [ML3T-1Q–2]

(f) Resis vity (u) [L3T–2]

Comprehension Based Problems

34. A student forgot Newton’s formula for speed of sound but he knows there were
speed (v), pressure (p) and density (d) in the formula. He then start using
dimensional analysis method to find the actual rela on
v = kpx dy

Where k is a dimensionless constant. On the basis of above passage answer the

following ques ons:
i) The value of x is :
𝟏 𝟏
a) 1 b) c) – d) 2
𝟐 𝟐
ii) The value of y is :
𝟏 𝟏
a) 1 b) c) – d) 2
𝟐 𝟐
iii) If the density will increase the speed of sound will :
a) increase b) decrease c) unchanged d) none of these
 Answer Key

Only One Alterna ve is Correct

Q 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

A a d b c a b b b a a a a b b a

Q 16 17 18 19 20 21 22 23 24 25

A c a d c b a b b c c

More than One Alterna ve are Correct

26 27 28 29 30

(a,b,c) (a,b,c,d) (a,c) (a,b) (a,d)

Matching Type Problems

1. (a)- p, s; (b)-r ; (c)- p, s
2. (a)-q ; (b)-p (c)-r ; (d) s
3. (a)-q ; (b)-p ; (c)-s (d) u ; (e)-r : (f)-t

Comprehension Based Problems

1. (b) 2. (c) 3. (b)