CLASS 11 PHYSICS • UNITS AND MEASUREMENTS

Dimensions and Dimensional Analysis

Master dimensions, dimensional formulas, homogeneity, derivations and JEE/NEET level dimensional analysis techniques.

CBSENEETJEE MainJEE AdvancedIBIGCSEA-Level

Searching for a Physics Tutor?

For CBSE, NEET, JEE Main, JEE Advanced, IB, IGCSE and A-Level Physics, contact Kumar Sir for one-to-one online Physics classes.

Phone: +91-9958461445Email: kumarsirphysics@gmail.com Website: https://kumarphysicsclasses.com WhatsApp Kumar Sir

1. Dimensions

Dimensions show how a physical quantity depends on fundamental quantities. They do not give the numerical value; they tell the physical nature of a quantity.

Mass

Length

Time

Current

Temperature

Amount

mol

Luminous Intensity

Use

Formula checking, relation derivation and unit conversion.

2. Dimensional Formula

A dimensional formula represents a physical quantity in terms of powers of fundamental dimensions. It is written inside square brackets.

Quantity	Formula	Dimensional Formula
Velocity	displacement / time	[LT^-1]
Acceleration	velocity / time	[LT^-2]
Force	mass × acceleration	[MLT^-2]
Energy	force × displacement	[ML²T^-2]
Power	energy / time	[ML²T^-3]
Pressure	force / area	[ML^-1T^-2]
Density	mass / volume	[ML^-3]

Rules: multiply dimensions when quantities multiply, divide powers when quantities divide, and raise every dimension to the given power.

3. Linear Motion vs Angular Motion

Angular motion has analogues of linear quantities. This comparison is heavily useful in rotational mechanics and JEE questions.

Linear Motion	Angular Motion	Explanation
Displacement s	Angular displacement θ	s measures length moved; θ measures angle swept.
Velocity v = ds/dt	Angular velocity ω = dθ/dt	Both are rates of change.
Acceleration a = dv/dt	Angular acceleration α = dω/dt	Both measure change of velocity with time.
Mass m	Moment of inertia I	I is rotational analogue of inertia.
Force F	Torque τ	Torque causes angular acceleration.
Linear momentum p = mv	Angular momentum L = Iω	Momentum analogue in rotation.
Work = F × s	Work = τ × θ	Force-displacement pair becomes torque-angle pair.
Kinetic energy = 1/2 mv²	Rotational kinetic energy = 1/2 Iω²	Same mathematical structure.
Newton's second law: F = ma	Rotational analogue: τ = Iα	Cause equals inertia times acceleration.

4. Important Dimensional Formulae

Complete Class 11 and Class 12 formula reference for quick revision.

Chapter Area	Quantity	Dimensional Formula
Mechanics	Velocity	[LT^-1]
Mechanics	Acceleration	[LT^-2]
Mechanics	Momentum	[MLT^-1]
Mechanics	Force	[MLT^-2]
Mechanics	Impulse	[MLT^-1]
Mechanics	Work/Energy	[ML²T^-2]
Mechanics	Power	[ML²T^-3]
Mechanics	Pressure	[ML^-1T^-2]
Mechanics	G	[M^-1L³T^-2]
Mechanics	Angular momentum	[ML²T^-1]
Mechanics	Torque	[ML²T^-2]
Mechanics	Moment of inertia	[ML²]
Properties of Matter	Density	[ML^-3]
Properties of Matter	Surface tension	[MT^-2]
Properties of Matter	Stress	[ML^-1T^-2]
Properties of Matter	Strain	Dimensionless
Properties of Matter	Young modulus	[ML^-1T^-2]
Properties of Matter	Viscosity coefficient	[ML^-1T^-1]
Heat	Heat	[ML²T^-2]
Heat	Specific heat	[L²T^-2K^-1]
Heat	Thermal conductivity	[MLT^-3K^-1]
Thermodynamics	Gas constant R	[ML²T^-2K^-1mol^-1]
Waves	Frequency	[T^-1]
Waves	Wavelength	[L]
Waves	Wave velocity	[LT^-1]
Electrostatics	Charge	[IT]
Electrostatics	Electric field	[MLT^-3I^-1]
Electrostatics	Potential	[ML²T^-3I^-1]
Electrostatics	Capacitance	[M^-1L^-2T⁴I²]
Electrostatics	Permittivity ε₀	[M^-1L^-3T⁴I²]
Current Electricity	Resistance	[ML²T^-3I^-2]
Current Electricity	Conductance	[M^-1L^-2T³I²]
Current Electricity	Resistivity	[ML³T^-3I^-2]
Current Electricity	Conductivity	[M^-1L^-3T³I²]
Magnetism	Magnetic field	[MT^-2I^-1]
Magnetism	Magnetic flux	[ML²T^-2I^-1]
Magnetism	Permeability μ₀	[MLT^-2I^-2]
EMI	Inductance	[ML²T^-2I^-2]
AC	Reactance	[ML²T^-3I^-2]
Optics	Refractive index	Dimensionless
Modern Physics	Planck constant h	[ML²T^-1]
Modern Physics	Speed of light c	[LT^-1]
Modern Physics	Bohr radius	[L]

5. Dimensionless Quantities

A dimensionless quantity has no dimensions because it is a ratio of similar quantities or a pure number. Angle θ in radian is arc / radius, so it is dimensionless.

Common Dimensionless Quantities

Angle θ
Radian and solid angle
Strain
Refractive index
Relative density
Coefficient of friction

Functions

sin θ, cos θ and tan θ require θ to be dimensionless. e^x, ln x and log x also require x to be dimensionless.

Why?

Mathematical functions are defined for pure numbers. A dimensional argument like sin(5 m) has no physical meaning.

6. Principle of Homogeneity

According to the principle of dimensional homogeneity, every term in a physically correct equation must have the same dimensions.

Verification 1

Question: Check s = ut + 1/2 at².

Assumption: All terms must have dimension of length.

Dimensional Formula: [s]=[L], [ut]=[LT^-1][T], [at²]=[LT^-2][T²]

Substitution: [ut]=[L], [at²]=[L]

Calculation: All terms are [L].

Final Answer: Dimensionally correct.

Exam Tip: Constants like 1/2 are dimensionless.

Verification 2

Question: Check v² = u² + 2as.

Assumption: Every term must be [L²T^-2].

Dimensional Formula: [v²] = [L²T^-2], [as]=[LT^-2][L]

Substitution: [as]=[L²T^-2]

Calculation: Both sides match.

Final Answer: Dimensionally correct.

Exam Tip: Check each added term separately.

7. Checking Correctness of Formula

Method: write dimensions of LHS, write dimensions of every term on RHS, compare powers of M, L and T, then decide correctness.

NEET-level Example

Question: Check T = 2π√(l/g).

Assumption: T should be time.

Dimensional Formula: [l/g] = [L]/[LT^-2]

Substitution: [T²], square root = [T]

Calculation: Correct

Final Answer: Correct

Exam Tip: Remember constants have no dimensions.

JEE-level Example

Question: Check F = mv²/r.

Assumption: Force should be [MLT^-2].

Dimensional Formula: [m][v²]/[r]

Substitution: [M][L²T^-2]/[L] = [MLT^-2]

Calculation: Correct

Final Answer: Correct

Exam Tip: Centripetal force passes dimensional check.

Incorrect Formula

Question: Check s = ut + at.

Assumption: s should be [L].

Dimensional Formula: [ut]=[L], [at]=[LT^-2][T]

Substitution: [at]=[LT^-1]

Calculation: Terms differ.

Final Answer: Incorrect

Exam Tip: Added terms must have same dimensions.

Formula Check

Question: Check pressure = energy / volume.

Assumption: Pressure dimension [ML^-1T^-2].

Dimensional Formula: Energy/volume = [ML²T^-2]/[L³]

Substitution: [ML^-1T^-2]

Calculation: Correct

Final Answer: Correct

Exam Tip: Many formulae can be checked by basic units.

8. Derivation of Relations

Dimensional derivation assumes a relation with unknown powers and compares dimensions to find powers.

Simple Pendulum

Question: For a simple pendulum, assume T = kL^ag^b. Find a and b.

Assumption: T depends on length L and acceleration g.

Dimensional Formula: [T] = [L]^a[LT^-2]^b = [L^a+bT^-2b]

Substitution: Compare powers: L: a+b=0, T: -2b=1

Calculation: b = -1/2, a = 1/2

Final Answer: T = k√(L/g)

Exam Tip: Use dimensions to get dependence, not constant 2π.

Wave Velocity

Question: For a stretched string, assume v = kT^aμ^b. Find relation.

Assumption: T is tension, μ is mass per unit length.

Dimensional Formula: [LT^-1] = [MLT^-2]^a[ML^-1]^b

Substitution: M: a+b=0, L: a-b=1, T: -2a=-1

Calculation: a=1/2, b=-1/2

Final Answer: v = k√(T/μ)

Exam Tip: Tension symbol T is not time here.

Surface Tension

Question: Find dimensions of surface tension from work = surface tension × area.

Assumption: Surface tension S = work / area.

Dimensional Formula: [S] = [ML²T^-2] / [L²]

Substitution: [MT^-2]

Calculation: S = [MT^-2]

Final Answer: [MT^-2]

Exam Tip: Surface tension also equals force/length.

Energy Density

Question: Find dimension of energy density.

Assumption: Energy density = energy / volume.

Dimensional Formula: [ML²T^-2] / [L³]

Substitution: [ML^-1T^-2]

Calculation: Same as pressure.

Final Answer: [ML^-1T^-2]

Exam Tip: Energy density and pressure share dimensions.

Gravitational Problem

Question: Find dimension of G from F = Gm₁m₂/r².

Assumption: G = Fr² / m².

Dimensional Formula: [G] = [MLT^-2][L²] / [M²]

Substitution: [M^-1L³T^-2]

Calculation: Dimension of G obtained.

Final Answer: [M^-1L³T^-2]

Exam Tip: Rearrange formula first.

Electric Problem

Question: Find dimension of electric field from F = qE.

Assumption: E = F/q, q = It.

Dimensional Formula: [E] = [MLT^-2] / [IT]

Substitution: [MLT^-3I^-1]

Calculation: Dimension of E obtained.

Final Answer: [MLT^-3I^-1]

Exam Tip: Charge dimension is [IT].

9. Limitations of Dimensional Analysis

No Numerical Constants

It cannot determine 2, 1/2, π or other pure constants.

No + or − Signs

It cannot tell whether terms add or subtract.

No Functions

It cannot derive sin θ, cos θ, e^x, ln x or log x forms.

No Scalar Vector Distinction

Work and torque share dimensions but are different physical quantities.

Incomplete Dependencies

If variables are missing, dimensional derivation becomes incomplete.

Dimensionless Groups

It cannot decide unknown functions of dimensionless ratios.

10. Errors in Dimensional Analysis

Confusing Tension and Time

In string waves, T may mean tension, not time dimension.

Ignoring Constants

Constants are dimensionless, but missing constants can make numerical answer wrong.

Wrong Charge Dimension

Charge has dimension [IT], not [I/T].

Adding Unlike Terms

Terms in an equation must have identical dimensions.

Forgetting Powers

Every dimension must be raised to the same power as the quantity.

Using Units as Dimensions

Metre is a unit; L is the dimension.

11. JEE Frequently Asked Questions

JEE Exam-style Question

Question: Find dimensions of Planck constant h.

Assumption: h has dimension of energy × time.

Dimensional Formula: [h] = [ML²T^-2][T]

Substitution: [ML²T^-1]

Calculation: [ML²T^-1]

Final Answer: [ML²T^-1]

Exam Tip: Use E = hν.

JEE Exam-style Question

Question: Is sin(vt) meaningful if v is velocity and t is time?

Assumption: Argument of sin must be dimensionless.

Dimensional Formula: [vt]=[L]

Substitution: [L] is not dimensionless

Calculation: Not meaningful unless multiplied by wave number.

Final Answer: Not dimensionally valid.

Exam Tip: Trig arguments must be pure numbers.

JEE Exam-style Question

Question: Check x = A sin(ωt). Find dimension of A.

Assumption: sin argument is dimensionless and x is length.

Dimensional Formula: [A] = [x]

Substitution: [A]=[L]

Calculation: Amplitude is length.

Final Answer: [L]

Exam Tip: Coefficient carries dimension of dependent variable.

JEE Exam-style Question

Question: Find dimensions of coefficient of viscosity η from F = ηAv/y.

Assumption: η = Fy/(Av).

Dimensional Formula: [η]=[MLT^-2][L]/([L²][LT^-1])

Substitution: [ML^-1T^-1]

Calculation: Done.

Final Answer: [ML^-1T^-1]

Exam Tip: Rearrange carefully.

12. NEET Question Bank: 60 MCQs

NEET MCQ 1

Question: Dimensional formula of force is

(a) [MLT^-2](b) [ML²T^-2](c) [LT^-1](d) [M^-1L³T^-2]

Correct Answer: [MLT^-2]

Explanation: Force = mass × acceleration.

Exam Tip: Know common formulae.

NEET MCQ 2

Question: Dimensional formula of energy is

(a) [ML²T^-2](b) [MLT^-2](c) [ML²T^-3](d) [ML^-1T^-2]

Correct Answer: [ML²T^-2]

Explanation: Energy = force × displacement.

Exam Tip: Energy and work have same dimensions.

NEET MCQ 3

Question: Which is dimensionless?

(a) Refractive index(b) Pressure(c) Force(d) Power

Correct Answer: Refractive index

Explanation: It is speed in vacuum / speed in medium.

Exam Tip: Ratios of same quantities are dimensionless.

NEET MCQ 4

Question: Principle of homogeneity states that

(a) all terms have same dimensions(b) all units are same numbers(c) constants have dimensions(d) vectors are scalars

Correct Answer: all terms have same dimensions

Explanation: Physical equation must be dimensionally homogeneous.

Exam Tip: Check every term.

NEET MCQ 5

Question: Dimension of G is

(a) [M^-1L³T^-2](b) [MLT^-2](c) [ML²T^-1](d) [IT]

Correct Answer: [M^-1L³T^-2]

Explanation: Use F = Gm₁m₂/r².

Exam Tip: Memorize G.

NEET MCQ 6

Question: Dimension of h is

(a) [ML²T^-1](b) [MLT^-2](c) [M^-1L³T^-2](d) [LT^-1]

Correct Answer: [ML²T^-1]

Explanation: Use E = hν.

Exam Tip: Planck constant = action.

NEET MCQ 7

Question: Angle in radians is

(a) dimensionless(b) [L](c) [T](d) [LT^-1]

Correct Answer: dimensionless

Explanation: Angle = arc/radius.

Exam Tip: Ratio of lengths.

NEET MCQ 8

Question: Dimension of pressure is

(a) [ML^-1T^-2](b) [MLT^-2](c) [ML²T^-2](d) [M]

Correct Answer: [ML^-1T^-2]

Explanation: Pressure = force / area.

Exam Tip: Pressure = energy density.

NEET MCQ 9

Question: Dimensional analysis cannot determine

(a) numerical constants(b) dimensions(c) homogeneity(d) units

Correct Answer: numerical constants

Explanation: Pure numbers cannot be found by dimensions.

Exam Tip: Know limitations.

NEET MCQ 10

Question: Charge has dimension

(a) [IT](b) [I/T](c) [MT](d) [MLT]

Correct Answer: [IT]

Explanation: Charge = current × time.

Exam Tip: Useful in electricity.

NEET MCQ 11

Question: Dimensional formula of force is

(a) [MLT^-2](b) [ML²T^-2](c) [LT^-1](d) [M^-1L³T^-2]

Correct Answer: [MLT^-2]

Explanation: Force = mass × acceleration.

Exam Tip: Know common formulae.

NEET MCQ 12

Question: Dimensional formula of energy is

(a) [ML²T^-2](b) [MLT^-2](c) [ML²T^-3](d) [ML^-1T^-2]

Correct Answer: [ML²T^-2]

Explanation: Energy = force × displacement.

Exam Tip: Energy and work have same dimensions.

NEET MCQ 13

Question: Which is dimensionless?

(a) Refractive index(b) Pressure(c) Force(d) Power

Correct Answer: Refractive index

Explanation: It is speed in vacuum / speed in medium.

Exam Tip: Ratios of same quantities are dimensionless.

NEET MCQ 14

Question: Principle of homogeneity states that

(a) all terms have same dimensions(b) all units are same numbers(c) constants have dimensions(d) vectors are scalars

Correct Answer: all terms have same dimensions

Explanation: Physical equation must be dimensionally homogeneous.

Exam Tip: Check every term.

NEET MCQ 15

Question: Dimension of G is

(a) [M^-1L³T^-2](b) [MLT^-2](c) [ML²T^-1](d) [IT]

Correct Answer: [M^-1L³T^-2]

Explanation: Use F = Gm₁m₂/r².

Exam Tip: Memorize G.

NEET MCQ 16

Question: Dimension of h is

(a) [ML²T^-1](b) [MLT^-2](c) [M^-1L³T^-2](d) [LT^-1]

Correct Answer: [ML²T^-1]

Explanation: Use E = hν.

Exam Tip: Planck constant = action.

NEET MCQ 17

Question: Angle in radians is

(a) dimensionless(b) [L](c) [T](d) [LT^-1]

Correct Answer: dimensionless

Explanation: Angle = arc/radius.

Exam Tip: Ratio of lengths.

NEET MCQ 18

Question: Dimension of pressure is

(a) [ML^-1T^-2](b) [MLT^-2](c) [ML²T^-2](d) [M]

Correct Answer: [ML^-1T^-2]

Explanation: Pressure = force / area.

Exam Tip: Pressure = energy density.

NEET MCQ 19

Question: Dimensional analysis cannot determine

(a) numerical constants(b) dimensions(c) homogeneity(d) units

Correct Answer: numerical constants

Explanation: Pure numbers cannot be found by dimensions.

Exam Tip: Know limitations.

NEET MCQ 20

Question: Charge has dimension

(a) [IT](b) [I/T](c) [MT](d) [MLT]

Correct Answer: [IT]

Explanation: Charge = current × time.

Exam Tip: Useful in electricity.

NEET MCQ 21

Question: Dimensional formula of force is

(a) [MLT^-2](b) [ML²T^-2](c) [LT^-1](d) [M^-1L³T^-2]

Correct Answer: [MLT^-2]

Explanation: Force = mass × acceleration.

Exam Tip: Know common formulae.

NEET MCQ 22

Question: Dimensional formula of energy is

(a) [ML²T^-2](b) [MLT^-2](c) [ML²T^-3](d) [ML^-1T^-2]

Correct Answer: [ML²T^-2]

Explanation: Energy = force × displacement.

Exam Tip: Energy and work have same dimensions.

NEET MCQ 23

Question: Which is dimensionless?

(a) Refractive index(b) Pressure(c) Force(d) Power

Correct Answer: Refractive index

Explanation: It is speed in vacuum / speed in medium.

Exam Tip: Ratios of same quantities are dimensionless.

NEET MCQ 24

Question: Principle of homogeneity states that

(a) all terms have same dimensions(b) all units are same numbers(c) constants have dimensions(d) vectors are scalars

Correct Answer: all terms have same dimensions

Explanation: Physical equation must be dimensionally homogeneous.

Exam Tip: Check every term.

NEET MCQ 25

Question: Dimension of G is

(a) [M^-1L³T^-2](b) [MLT^-2](c) [ML²T^-1](d) [IT]

Correct Answer: [M^-1L³T^-2]

Explanation: Use F = Gm₁m₂/r².

Exam Tip: Memorize G.

NEET MCQ 26

Question: Dimension of h is

(a) [ML²T^-1](b) [MLT^-2](c) [M^-1L³T^-2](d) [LT^-1]

Correct Answer: [ML²T^-1]

Explanation: Use E = hν.

Exam Tip: Planck constant = action.

NEET MCQ 27

Question: Angle in radians is

(a) dimensionless(b) [L](c) [T](d) [LT^-1]

Correct Answer: dimensionless

Explanation: Angle = arc/radius.

Exam Tip: Ratio of lengths.

NEET MCQ 28

Question: Dimension of pressure is

(a) [ML^-1T^-2](b) [MLT^-2](c) [ML²T^-2](d) [M]

Correct Answer: [ML^-1T^-2]

Explanation: Pressure = force / area.

Exam Tip: Pressure = energy density.

NEET MCQ 29

Question: Dimensional analysis cannot determine

(a) numerical constants(b) dimensions(c) homogeneity(d) units

Correct Answer: numerical constants

Explanation: Pure numbers cannot be found by dimensions.

Exam Tip: Know limitations.

NEET MCQ 30

Question: Charge has dimension

(a) [IT](b) [I/T](c) [MT](d) [MLT]

Correct Answer: [IT]

Explanation: Charge = current × time.

Exam Tip: Useful in electricity.

NEET MCQ 31

Question: Dimensional formula of force is

(a) [MLT^-2](b) [ML²T^-2](c) [LT^-1](d) [M^-1L³T^-2]

Correct Answer: [MLT^-2]

Explanation: Force = mass × acceleration.

Exam Tip: Know common formulae.

NEET MCQ 32

Question: Dimensional formula of energy is

(a) [ML²T^-2](b) [MLT^-2](c) [ML²T^-3](d) [ML^-1T^-2]

Correct Answer: [ML²T^-2]

Explanation: Energy = force × displacement.

Exam Tip: Energy and work have same dimensions.

NEET MCQ 33

Question: Which is dimensionless?

(a) Refractive index(b) Pressure(c) Force(d) Power

Correct Answer: Refractive index

Explanation: It is speed in vacuum / speed in medium.

Exam Tip: Ratios of same quantities are dimensionless.

NEET MCQ 34

Question: Principle of homogeneity states that

(a) all terms have same dimensions(b) all units are same numbers(c) constants have dimensions(d) vectors are scalars

Correct Answer: all terms have same dimensions

Explanation: Physical equation must be dimensionally homogeneous.

Exam Tip: Check every term.

NEET MCQ 35

Question: Dimension of G is

(a) [M^-1L³T^-2](b) [MLT^-2](c) [ML²T^-1](d) [IT]

Correct Answer: [M^-1L³T^-2]

Explanation: Use F = Gm₁m₂/r².

Exam Tip: Memorize G.

NEET MCQ 36

Question: Dimension of h is

(a) [ML²T^-1](b) [MLT^-2](c) [M^-1L³T^-2](d) [LT^-1]

Correct Answer: [ML²T^-1]

Explanation: Use E = hν.

Exam Tip: Planck constant = action.

NEET MCQ 37

Question: Angle in radians is

(a) dimensionless(b) [L](c) [T](d) [LT^-1]

Correct Answer: dimensionless

Explanation: Angle = arc/radius.

Exam Tip: Ratio of lengths.

NEET MCQ 38

Question: Dimension of pressure is

(a) [ML^-1T^-2](b) [MLT^-2](c) [ML²T^-2](d) [M]

Correct Answer: [ML^-1T^-2]

Explanation: Pressure = force / area.

Exam Tip: Pressure = energy density.

NEET MCQ 39

Question: Dimensional analysis cannot determine

(a) numerical constants(b) dimensions(c) homogeneity(d) units

Correct Answer: numerical constants

Explanation: Pure numbers cannot be found by dimensions.

Exam Tip: Know limitations.

NEET MCQ 40

Question: Charge has dimension

(a) [IT](b) [I/T](c) [MT](d) [MLT]

Correct Answer: [IT]

Explanation: Charge = current × time.

Exam Tip: Useful in electricity.

NEET MCQ 41

Question: Dimensional formula of force is

(a) [MLT^-2](b) [ML²T^-2](c) [LT^-1](d) [M^-1L³T^-2]

Correct Answer: [MLT^-2]

Explanation: Force = mass × acceleration.

Exam Tip: Know common formulae.

NEET MCQ 42

Question: Dimensional formula of energy is

(a) [ML²T^-2](b) [MLT^-2](c) [ML²T^-3](d) [ML^-1T^-2]

Correct Answer: [ML²T^-2]

Explanation: Energy = force × displacement.

Exam Tip: Energy and work have same dimensions.

NEET MCQ 43

Question: Which is dimensionless?

(a) Refractive index(b) Pressure(c) Force(d) Power

Correct Answer: Refractive index

Explanation: It is speed in vacuum / speed in medium.

Exam Tip: Ratios of same quantities are dimensionless.

NEET MCQ 44

Question: Principle of homogeneity states that

(a) all terms have same dimensions(b) all units are same numbers(c) constants have dimensions(d) vectors are scalars

Correct Answer: all terms have same dimensions

Explanation: Physical equation must be dimensionally homogeneous.

Exam Tip: Check every term.

NEET MCQ 45

Question: Dimension of G is

(a) [M^-1L³T^-2](b) [MLT^-2](c) [ML²T^-1](d) [IT]

Correct Answer: [M^-1L³T^-2]

Explanation: Use F = Gm₁m₂/r².

Exam Tip: Memorize G.

NEET MCQ 46

Question: Dimension of h is

(a) [ML²T^-1](b) [MLT^-2](c) [M^-1L³T^-2](d) [LT^-1]

Correct Answer: [ML²T^-1]

Explanation: Use E = hν.

Exam Tip: Planck constant = action.

NEET MCQ 47

Question: Angle in radians is

(a) dimensionless(b) [L](c) [T](d) [LT^-1]

Correct Answer: dimensionless

Explanation: Angle = arc/radius.

Exam Tip: Ratio of lengths.

NEET MCQ 48

Question: Dimension of pressure is

(a) [ML^-1T^-2](b) [MLT^-2](c) [ML²T^-2](d) [M]

Correct Answer: [ML^-1T^-2]

Explanation: Pressure = force / area.

Exam Tip: Pressure = energy density.

NEET MCQ 49

Question: Dimensional analysis cannot determine

(a) numerical constants(b) dimensions(c) homogeneity(d) units

Correct Answer: numerical constants

Explanation: Pure numbers cannot be found by dimensions.

Exam Tip: Know limitations.

NEET MCQ 50

Question: Charge has dimension

(a) [IT](b) [I/T](c) [MT](d) [MLT]

Correct Answer: [IT]

Explanation: Charge = current × time.

Exam Tip: Useful in electricity.

NEET MCQ 51

Question: Dimensional formula of force is

(a) [MLT^-2](b) [ML²T^-2](c) [LT^-1](d) [M^-1L³T^-2]

Correct Answer: [MLT^-2]

Explanation: Force = mass × acceleration.

Exam Tip: Know common formulae.

NEET MCQ 52

Question: Dimensional formula of energy is

(a) [ML²T^-2](b) [MLT^-2](c) [ML²T^-3](d) [ML^-1T^-2]

Correct Answer: [ML²T^-2]

Explanation: Energy = force × displacement.

Exam Tip: Energy and work have same dimensions.

NEET MCQ 53

Question: Which is dimensionless?

(a) Refractive index(b) Pressure(c) Force(d) Power

Correct Answer: Refractive index

Explanation: It is speed in vacuum / speed in medium.

Exam Tip: Ratios of same quantities are dimensionless.

NEET MCQ 54

Question: Principle of homogeneity states that

(a) all terms have same dimensions(b) all units are same numbers(c) constants have dimensions(d) vectors are scalars

Correct Answer: all terms have same dimensions

Explanation: Physical equation must be dimensionally homogeneous.

Exam Tip: Check every term.

NEET MCQ 55

Question: Dimension of G is

(a) [M^-1L³T^-2](b) [MLT^-2](c) [ML²T^-1](d) [IT]

Correct Answer: [M^-1L³T^-2]

Explanation: Use F = Gm₁m₂/r².

Exam Tip: Memorize G.

NEET MCQ 56

Question: Dimension of h is

(a) [ML²T^-1](b) [MLT^-2](c) [M^-1L³T^-2](d) [LT^-1]

Correct Answer: [ML²T^-1]

Explanation: Use E = hν.

Exam Tip: Planck constant = action.

NEET MCQ 57

Question: Angle in radians is

(a) dimensionless(b) [L](c) [T](d) [LT^-1]

Correct Answer: dimensionless

Explanation: Angle = arc/radius.

Exam Tip: Ratio of lengths.

NEET MCQ 58

Question: Dimension of pressure is

(a) [ML^-1T^-2](b) [MLT^-2](c) [ML²T^-2](d) [M]

Correct Answer: [ML^-1T^-2]

Explanation: Pressure = force / area.

Exam Tip: Pressure = energy density.

NEET MCQ 59

Question: Dimensional analysis cannot determine

(a) numerical constants(b) dimensions(c) homogeneity(d) units

Correct Answer: numerical constants

Explanation: Pure numbers cannot be found by dimensions.

Exam Tip: Know limitations.

NEET MCQ 60

Question: Charge has dimension

(a) [IT](b) [I/T](c) [MT](d) [MLT]

Correct Answer: [IT]

Explanation: Charge = current × time.

Exam Tip: Useful in electricity.

13. JEE Main Question Bank

JEE Main Practice

Question: For a simple pendulum, assume T = kL^ag^b. Find a and b.

Assumption: T depends on length L and acceleration g.

Dimensional Formula: [T] = [L]^a[LT^-2]^b = [L^a+bT^-2b]

Substitution: Compare powers: L: a+b=0, T: -2b=1

Calculation: b = -1/2, a = 1/2

Final Answer: T = k√(L/g)

Exam Tip: Use dimensions to get dependence, not constant 2π.

JEE Main Practice

Question: For a stretched string, assume v = kT^aμ^b. Find relation.

Assumption: T is tension, μ is mass per unit length.

Dimensional Formula: [LT^-1] = [MLT^-2]^a[ML^-1]^b

Substitution: M: a+b=0, L: a-b=1, T: -2a=-1

Calculation: a=1/2, b=-1/2

Final Answer: v = k√(T/μ)

Exam Tip: Tension symbol T is not time here.

JEE Main Practice

Question: Find dimensions of surface tension from work = surface tension × area.

Assumption: Surface tension S = work / area.

Dimensional Formula: [S] = [ML²T^-2] / [L²]

Substitution: [MT^-2]

Calculation: S = [MT^-2]

Final Answer: [MT^-2]

Exam Tip: Surface tension also equals force/length.

JEE Main Practice

Question: Find dimension of energy density.

Assumption: Energy density = energy / volume.

Dimensional Formula: [ML²T^-2] / [L³]

Substitution: [ML^-1T^-2]

Calculation: Same as pressure.

Final Answer: [ML^-1T^-2]

Exam Tip: Energy density and pressure share dimensions.

JEE Main Practice

Question: Check T = 2π√(l/g).

Assumption: T should be time.

Dimensional Formula: [l/g] = [L]/[LT^-2]

Substitution: [T²], square root = [T]

Calculation: Correct

Final Answer: Correct

Exam Tip: Remember constants have no dimensions.

JEE Main Practice

Question: Check F = mv²/r.

Assumption: Force should be [MLT^-2].

Dimensional Formula: [m][v²]/[r]

Substitution: [M][L²T^-2]/[L] = [MLT^-2]

Calculation: Correct

Final Answer: Correct

Exam Tip: Centripetal force passes dimensional check.

14. JEE Advanced Question Bank

JEE Advanced Practice

Question: Two quantities have same dimensions. Must they represent same physical quantity?

Assumption: Same dimensions do not guarantee same physical meaning.

Dimensional Formula: Work and torque both have [ML²T^-2].

Substitution: Dimensions match.

Calculation: Physical nature differs: work scalar, torque vector.

Final Answer: No.

Exam Tip: Dimensional equality is necessary, not sufficient.

JEE Advanced Practice

Question: Why cannot dimensional analysis derive s = ut + 1/2 at² completely?

Assumption: It cannot decide constants or addition of terms.

Dimensional Formula: All terms have [L].

Substitution: Homogeneity only checks dimensions.

Calculation: 1/2 cannot be obtained.

Final Answer: It cannot derive full equation.

Exam Tip: Remember limitations.

JEE Advanced Practice

Question: Can e^kx be valid if x is length? Find dimension of k.

Assumption: kx must be dimensionless.

Dimensional Formula: [k][x] = 1

Substitution: [k][L]=1

Calculation: [k]=[L^-1]

Final Answer: [L^-1]

Exam Tip: Exponential argument must be dimensionless.

JEE Advanced Practice

Question: If y = A sin(kx - ωt), find dimensions of k and ω.

Assumption: Arguments of sin are dimensionless.

Dimensional Formula: [kx]=1, [ωt]=1

Substitution: [k]=[L^-1], [ω]=[T^-1]

Calculation: Done.

Final Answer: k: [L^-1], ω: [T^-1]

Exam Tip: Wave phase is dimensionless.

15. CBSE School Questions

1 Mark

Question: Define dimensional formula.

Answer: Expression showing powers of fundamental dimensions in a physical quantity.

2 Mark

Question: State two limitations of dimensional analysis.

Answer: It cannot find numerical constants and cannot decide plus or minus signs.

3 Mark

Question: Verify v² = u² + 2as dimensionally.

Answer: Each term has [L²T^-2], so equation is dimensionally correct.

5 Mark

Question: Derive time period of simple pendulum using dimensions.

Answer: Assume T = kL^ag^b. Comparing dimensions gives a=1/2 and b=-1/2, so T = k√(L/g).

16. IB Physics Questions

IB 1

Question: Why must equations be dimensionally homogeneous?

Solution: Otherwise quantities of different physical nature would be added or equated.

IB 2

Question: What is a dimensionless ratio?

Solution: Ratio of two quantities with same dimensions.

IB 3

Question: Why is log x valid only for dimensionless x?

Solution: Logarithm is defined for pure numbers.

17. IGCSE Questions

IGCSE 1

Question: State dimension of speed.

Solution: [LT^-1]

IGCSE 2

Question: State one use of dimensional analysis.

Solution: Checking correctness of equations.

IGCSE 3

Question: Give one dimensionless quantity.

Solution: Refractive index.

18. A-Level Questions

A-Level 1

Question: Find dimension of magnetic flux.

Solution: [ML²T^-2I^-1]

A-Level 2

Question: Find dimension of capacitance.

Solution: [M^-1L^-2T⁴I²]

A-Level 3

Question: Why can dimensional analysis not determine π?

Solution: π is dimensionless numerical constant.

19. Assertion Reason Questions

Options: (a) Both A and R are true and R explains A. (b) Both are true but R does not explain A. (c) A true, R false. (d) A false, R true.

Assertion Reason 1

Assertion: Dimensional homogeneity is necessary for a correct physical equation.

Reason: Unlike dimensional terms cannot be added.

Answer: (a)

Explanation: Correct principle.

Assertion Reason 2

Assertion: All dimensionally correct equations are physically correct.

Reason: Dimensional analysis cannot test constants and signs.

Answer: (d)

Explanation: Assertion false, reason true.

Assertion Reason 3

Assertion: Angle in radian is dimensionless.

Reason: Angle is arc length divided by radius.

Answer: (a)

Explanation: Ratio of lengths.

Assertion Reason 4

Assertion: sin θ can have dimensional θ.

Reason: Trigonometric arguments must be dimensionless.

Answer: (d)

Explanation: Assertion false, reason true.

Assertion Reason 5

Assertion: Work and torque have same dimensions.

Reason: Same dimensions need not mean same physical quantity.

Answer: (b)

Explanation: Both true but reason does not explain equality directly.

Assertion Reason 6

Assertion: Dimensional analysis can determine 2π in pendulum formula.

Reason: Numerical constants are dimensionless.

Answer: (d)

Explanation: It cannot find constants.

Assertion Reason 7

Assertion: Charge has dimension [IT].

Reason: Current is charge per unit time.

Answer: (a)

Explanation: q = It.

Assertion Reason 8

Assertion: Refractive index is dimensionless.

Reason: It is ratio of two speeds.

Answer: (a)

Explanation: Correct.

Assertion Reason 9

Assertion: Pressure and energy density have same dimensions.

Reason: Both reduce to [ML^-1T^-2].

Answer: (a)

Explanation: Correct.

Assertion Reason 10

Assertion: e^x can have dimensional x.

Reason: Exponential argument must be pure number.

Answer: (d)

Explanation: Assertion false, reason true.

Assertion Reason 11

Assertion: Dimensional homogeneity is necessary for a correct physical equation.

Reason: Unlike dimensional terms cannot be added.

Answer: (a)

Explanation: Correct principle.

Assertion Reason 12

Assertion: All dimensionally correct equations are physically correct.

Reason: Dimensional analysis cannot test constants and signs.

Answer: (d)

Explanation: Assertion false, reason true.

Assertion Reason 13

Assertion: Angle in radian is dimensionless.

Reason: Angle is arc length divided by radius.

Answer: (a)

Explanation: Ratio of lengths.

Assertion Reason 14

Assertion: sin θ can have dimensional θ.

Reason: Trigonometric arguments must be dimensionless.

Answer: (d)

Explanation: Assertion false, reason true.

Assertion Reason 15

Assertion: Work and torque have same dimensions.

Reason: Same dimensions need not mean same physical quantity.

Answer: (b)

Explanation: Both true but reason does not explain equality directly.

Assertion Reason 16

Assertion: Dimensional analysis can determine 2π in pendulum formula.

Reason: Numerical constants are dimensionless.

Answer: (d)

Explanation: It cannot find constants.

Assertion Reason 17

Assertion: Charge has dimension [IT].

Reason: Current is charge per unit time.

Answer: (a)

Explanation: q = It.

Assertion Reason 18

Assertion: Refractive index is dimensionless.

Reason: It is ratio of two speeds.

Answer: (a)

Explanation: Correct.

Assertion Reason 19

Assertion: Pressure and energy density have same dimensions.

Reason: Both reduce to [ML^-1T^-2].

Answer: (a)

Explanation: Correct.

Assertion Reason 20

Assertion: e^x can have dimensional x.

Reason: Exponential argument must be pure number.

Answer: (d)

Explanation: Assertion false, reason true.

Assertion Reason 21

Assertion: Dimensional homogeneity is necessary for a correct physical equation.

Reason: Unlike dimensional terms cannot be added.

Answer: (a)

Explanation: Correct principle.

Assertion Reason 22

Assertion: All dimensionally correct equations are physically correct.

Reason: Dimensional analysis cannot test constants and signs.

Answer: (d)

Explanation: Assertion false, reason true.

Assertion Reason 23

Assertion: Angle in radian is dimensionless.

Reason: Angle is arc length divided by radius.

Answer: (a)

Explanation: Ratio of lengths.

Assertion Reason 24

Assertion: sin θ can have dimensional θ.

Reason: Trigonometric arguments must be dimensionless.

Answer: (d)

Explanation: Assertion false, reason true.

Assertion Reason 25

Assertion: Work and torque have same dimensions.

Reason: Same dimensions need not mean same physical quantity.

Answer: (b)

Explanation: Both true but reason does not explain equality directly.

Assertion Reason 26

Assertion: Dimensional analysis can determine 2π in pendulum formula.

Reason: Numerical constants are dimensionless.

Answer: (d)

Explanation: It cannot find constants.

Assertion Reason 27

Assertion: Charge has dimension [IT].

Reason: Current is charge per unit time.

Answer: (a)

Explanation: q = It.

Assertion Reason 28

Assertion: Refractive index is dimensionless.

Reason: It is ratio of two speeds.

Answer: (a)

Explanation: Correct.

Assertion Reason 29

Assertion: Pressure and energy density have same dimensions.

Reason: Both reduce to [ML^-1T^-2].

Answer: (a)

Explanation: Correct.

Assertion Reason 30

Assertion: e^x can have dimensional x.

Reason: Exponential argument must be pure number.

Answer: (d)

Explanation: Assertion false, reason true.

20. Case Study Questions

Case Study: Simple Pendulum

A student wants to find how time period depends on length and g using dimensions.

Questions: Which variables are assumed?; What relation is obtained?; Can 2π be found?; Why?

Answers: L and g; T = k√(L/g); no; dimensions cannot find numerical constants.

Explanation: The case applies dimensional formulas, homogeneity and limitations in exam context.

Case Study: Wave Motion

Velocity of wave on a string depends on tension and mass per unit length.

Questions: Write assumed relation; Final relation; Dimension of tension; Dimension of μ

Answers: v = kT^aμ^b; v = k√(T/μ); [MLT^-2]; [ML^-1].

Explanation: The case applies dimensional formulas, homogeneity and limitations in exam context.

Case Study: Rotational Motion

Linear and angular quantities have analogous roles in rotational dynamics.

Questions: Analogue of force?; Analogue of mass?; Analogue of F=ma?; Work relation?

Answers: Torque; moment of inertia; τ = Iα; work = τθ.

Explanation: The case applies dimensional formulas, homogeneity and limitations in exam context.

Case Study: Dimensional Homogeneity

A student checks s = ut + at and finds the second RHS term has dimension of velocity.

Questions: Is equation correct?; Why?; Correct term?; Rule used?

Answers: No; unlike terms added; 1/2 at²; homogeneity.

Explanation: The case applies dimensional formulas, homogeneity and limitations in exam context.

Case Study: Dimensionless Quantities

A function e^kx appears in a physics formula where x is distance.

Questions: What must be dimension of kx?; Dimension of k?; Why?; Is e^5m meaningful?

Answers: Dimensionless; [L^-1]; exponential argument is pure number; no.

Explanation: The case applies dimensional formulas, homogeneity and limitations in exam context.

21. Quick Revision Notes

One Page Formula Sheet

Velocity: [LT^-1]
Force: [MLT^-2]
Energy: [ML²T^-2]
Power: [ML²T^-3]

Dimensionless List

Angle θ
Strain
Refractive index
Coefficient of friction
sin θ, cos θ, e^x arguments

Exam Tricks

Every added term must match dimensions
Constants cannot be found dimensionally
Same dimensions do not prove same quantity
Always make trig/log/exponential arguments dimensionless

Most important JEE/NEET concepts: homogeneity, dimensions of G/h/ε₀/μ₀, dimensionless arguments, pendulum derivation, wave velocity derivation and limitations.

Searching for a Physics Tutor?

For CBSE, NEET, JEE Main, JEE Advanced, IB, IGCSE and A-Level Physics, contact Kumar Sir for one-to-one online Physics classes.

Phone: +91-9958461445Email: kumarsirphysics@gmail.com Website: https://kumarphysicsclasses.com WhatsApp Kumar Sir