JEE Physics Units and Dimensions MCQs Set 09

Physics Objective Questions and Answers: Units and Dimensions

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Multiple Choice Questions

Question 1: If the gravitational constant G, Planck's constant h and the velocity of light c are chosen as fundamental units, the dimensions of the radius of gyration are:
(a) \( h^{1/2}c^{-3/2}G^{-1/2} \)
(b) \( h^{-1/2}c^{3/2}G^{1/2} \)
(c) \( h^{1/2}c^{3/2}G^{1/2} \)
(d) \( h^{1/2}c^{-3/2}G^{1/2} \)
Show Answer & Explanation

Answer: (d) \( h^{1/2}c^{-3/2}G^{1/2} \)

Explanation:
1. Let L = Gᵃcᵇhᶜ and match the powers.
2. M: -a + c = 0; L: 3a + b + 2c = 1; T: -2a - b - c = 0.
3. Solving: a = c = 1/2, b = -3/2.

Teacher's Note:
1. Solve three linear equations for the exponents.
2. The result is the Planck length.

Question 2: The dimensions of \( \frac{1}{\sqrt{\varepsilon_0\mu_0}} \) are that of:
(a) velocity
(b) distance
(c) time
(d) capacitance
Show Answer & Explanation

Answer: (a) velocity

Explanation:
1. c = 1/√(μ₀ε₀) is the speed of light in vacuum.
2. So its dimensions are LT⁻¹.

Teacher's Note:
1. Maxwell's relation gives the speed of light.
2. Recall c² = 1/(μ₀ε₀).

Question 3: Which of the following are dimensionless?
(i) acceleration due to gravity
(ii) strain
(iii) Mach number
(iv) refractive index
Which of the following is/are correct?
(a) ii, iii, iv
(b) i, ii, iv
(c) i, iii, iv
(d) i, ii, iii
Show Answer & Explanation

Answer: (a) ii, iii, iv

Explanation:
1. g has dimensions LT⁻².
2. Strain, Mach number and refractive index are ratios of like quantities.

Teacher's Note:
1. Ratios of like quantities have no dimensions.
2. g is an acceleration.

Question 4: Which of the following are dimensionless constants?
(i) gravitational constant
(ii) Reynolds number
(iii) Mach number
(iv) permeability
Which of the following is/are correct?
(a) ii, iii
(b) i, iv
(c) i, ii
(d) iii, iv
Show Answer & Explanation

Answer: (a) ii, iii

Explanation:
1. G and μ₀ have dimensions.
2. Reynolds number and Mach number are pure numbers.

Teacher's Note:
1. Dimensionless constants are pure numbers.
2. Check each for ratios of like quantities.

Question 5: Select the correct statement(s):
(i) a dimensionally correct equation may be correct
(ii) a dimensionally correct equation may be incorrect
(iii) a dimensionally incorrect equation may be correct
(iv) a dimensionally incorrect equation may be incorrect
Which of the following is/are correct?
(a) ii, iii, iv
(b) i, iii, iv
(c) i, ii, iv
(d) i, ii, iii
Show Answer & Explanation

Answer: (c) i, ii, iv

Explanation:
1. Dimensional correctness is necessary but not sufficient.
2. So a dimensionally correct equation may or may not be correct, while a dimensionally incorrect equation is always incorrect.

Teacher's Note:
1. Dimensional analysis can reject an equation but cannot prove it.
2. A wrong-dimension equation can never hold.

Question 6: Which of the following pairs have the same dimensions?
(i) torque and work
(ii) angular momentum and work
(iii) energy and Young's modulus
(iv) light year and wavelength
Which of the following is/are correct?
(a) iii, iv
(b) i, iv
(c) ii, iii
(d) i, ii
Show Answer & Explanation

Answer: (b) i, iv

Explanation:
1. Torque and work both have ML²T⁻².
2. Light year and wavelength are lengths.
3. Angular momentum is ML²T⁻¹ and Young's modulus is ML⁻¹T⁻².

Teacher's Note:
1. Compare the powers of M, L and T.
2. Torque and work share dimensions despite different physical meanings.

Question 7: The pair(s) of physical quantities that have the same dimensions is/are:
(i) Reynolds number and coefficient of friction
(ii) latent heat and gravitational potential
(iii) Curie and frequency of a light wave
(iv) Planck's constant and torque
Which of the following is/are correct?
(a) i, iii, iv
(b) ii, iii, iv
(c) i, ii, iii
(d) i, ii, iv
Show Answer & Explanation

Answer: (c) i, ii, iii

Explanation:
1. Reynolds number and coefficient of friction are dimensionless.
2. Latent heat and gravitational potential are L²T⁻².
3. Curie (activity) and frequency are T⁻¹.
4. Planck's constant is ML²T⁻¹ and torque is ML²T⁻².

Teacher's Note:
1. Compare the dimensions pair by pair.
2. Planck constant and torque differ by a factor of T.

Question 8: Regarding the expression \( \int\frac{dx}{\sqrt{a^2-x^2}}=\frac1a\sin^{-1}\frac ax \), where x and a stand for distances:
(i) it is mathematically correct
(ii) it is mathematically incorrect
(iii) it is dimensionally correct
(iv) it is dimensionally incorrect
Which of the following is/are correct?
(a) ii, iv
(b) ii, iii
(c) i, iii
(d) i, iv
Show Answer & Explanation

Answer: (a) ii, iv

Explanation:
1. The integral is sin⁻¹(x/a), not (1/a) sin⁻¹(a/x).
2. The left side is dimensionless but the right side has dimensions L⁻¹, so it is dimensionally incorrect too.

Teacher's Note:
1. Check the standard integral.
2. Compare dimensions of both sides.

Question 9: Which of the following pairs have the same dimension?
(i) \( \text{electric flux and }q/\varepsilon_0 \)
(ii) \( \text{electric flux and }\mu_0 i \)
(iii) \( h/e\text{ and electric flux} \)
(iv) \( h/e\text{ and magnetic flux} \)
Which of the following is/are correct?
(a) iii, iv
(b) ii, iii
(c) i, ii
(d) i, iv
Show Answer & Explanation

Answer: (d) i, iv

Explanation:
1. Gauss's law gives flux = q/ε₀.
2. h/e = ML²T⁻²A⁻¹, the same as magnetic flux, not electric flux.

Teacher's Note:
1. Use Gauss's law for the first pair.
2. h/e is the unit of magnetic flux.

Question 10: A book with many printing errors contains different formulas for the displacement y of a particle executing periodic motion, where a is the maximum displacement, v the speed and T the time period. Which of the following formulas are wrong on dimensional grounds?
(i) \( y=a\sin\frac{2\pi t}{T} \)
(ii) \( y=a\sin vt \)
(iii) \( y=\frac at\sin\frac ta \)
(iv) \( y=(a\sqrt2)\left(\sin\frac{2\pi t}{T}+\cos\frac{2\pi t}{T}\right) \)
Which of the following is/are correct?
(a) i, ii
(b) i, iv
(c) iii, iv
(d) ii, iii
Show Answer & Explanation

Answer: (d) ii, iii

Explanation:
1. The argument of sine must be dimensionless.
2. In (ii), vt is a length. In (iii), t/a is a time divided by a length.
3. Formulas (i) and (iv) have dimensionless arguments.

Teacher's Note:
1. Check the argument of every trigonometric function.
2. Also, y must have the dimensions of length.

Question 11: If the dimensions of length are expressed as \( G^xc^yh^z \), where G, c and h are the gravitational constant, speed of light and Planck's constant, then:
(i) \( x=\tfrac12,\ y=\tfrac12 \)
(ii) \( x=\tfrac12,\ z=\tfrac12 \)
(iii) \( y=\tfrac32,\ z=\tfrac12 \)
(iv) \( y=-\tfrac32,\ z=\tfrac12 \)
Which of the following is/are correct?
(a) ii, iii
(b) ii, iv
(c) i, iii
(d) i, iv
Show Answer & Explanation

Answer: (b) ii, iv

Explanation:
1. Matching M, L, T gives x = 1/2, y = -3/2, z = 1/2.
2. So (ii) and (iv) are correct.

Teacher's Note:
1. Solve for the exponents of the Planck length.
2. Check each option against x = z = 1/2 and y = -3/2.

Question 12: Assertion (A): L/R and CR both have the same dimensions.
Reason (R): L/R and CR both have the dimensions of time.
(a) Both Assertion and Reason are true and Reason is the correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
(c) Assertion is true but Reason is false.
(d) Assertion is false but Reason is true.
Show Answer & Explanation

Answer: (a) Both Assertion and Reason are true and Reason is the correct explanation of Assertion.

Explanation:
1. Both are time constants of circuits.
2. So the assertion holds and the reason explains it.

Teacher's Note:
1. L/R and RC are time constants.
2. Equal dimensions follow from both being time.

Question 13: Assertion (A): Angle and strain are dimensionless.
Reason (R): Angle and strain have no unit.
(a) Both Assertion and Reason are true and Reason is the correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
(c) Assertion is true but Reason is false.
(d) Assertion is false but Reason is true.
Show Answer & Explanation

Answer: (c) Assertion is true but Reason is false.

Explanation:
1. Both are ratios of like quantities, so they are dimensionless.
2. But an angle is measured in radians (or degrees), so it does have a unit. The reason is false.

Teacher's Note:
1. Dimensionless does not mean unitless.
2. Angles are measured in radian or degree.

Question 14: Assertion (A): The units of the Rydberg constant R are m⁻¹.
Reason (R): \( \frac1\lambda=R\left(\frac1{n_1^2}-\frac1{n_2^2}\right) \) follows from Bohr's formula.
(a) Both Assertion and Reason are true and Reason is the correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
(c) Assertion is true but Reason is false.
(d) Assertion is false but Reason is true.
Show Answer & Explanation

Answer: (a) Both Assertion and Reason are true and Reason is the correct explanation of Assertion.

Explanation:
1. The bracket is dimensionless, so R has the dimensions of 1/λ, i.e. m⁻¹.
2. The formula therefore explains the units.

Teacher's Note:
1. Equate dimensions on both sides.
2. The bracket has no dimensions.

Question 15: Match the quantities in Column I with their dimensions in Column II.
A. Angular momentum
B. Torque
C. Surface tension
D. Coefficient of viscosity
(p) \( ML^{-1}T^{-1} \)
(q) \( MT^{-2} \)
(r) \( ML^2T^{-1} \)
(s) \( ML^2T^{-2} \)
(a) A-p, B-q, C-r, D-s
(b) A-r, B-s, C-p, D-q
(c) A-s, B-r, C-q, D-p
(d) A-r, B-s, C-q, D-p
Show Answer & Explanation

Answer: (d) A-r, B-s, C-q, D-p

Explanation:
1. Angular momentum ML²T⁻¹, torque ML²T⁻², surface tension MT⁻², viscosity ML⁻¹T⁻¹.

Teacher's Note:
1. Work out the dimensions or units of each item in Column I.
2. Then match with the identical entry in Column II.

Question 16: Match the quantities in Column I with their dimensions in Column II.
A. Energy
B. Moment of inertia
C. Angular acceleration
D. Angular momentum
(p) \( ML^2T^{-1} \)
(q) \( ML^2T^{0} \)
(r) \( ML^2T^{-2} \)
(s) \( M^0L^0T^{-2} \)
(a) A-q, B-r, C-s, D-p
(b) A-r, B-q, C-s, D-p
(c) A-r, B-s, C-q, D-p
(d) A-p, B-q, C-r, D-s
Show Answer & Explanation

Answer: (b) A-r, B-q, C-s, D-p

Explanation:
1. Energy ML²T⁻², moment of inertia ML², angular acceleration T⁻², angular momentum ML²T⁻¹.

Teacher's Note:
1. Work out the dimensions or units of each item in Column I.
2. Then match with the identical entry in Column II.

Question 17: Match the quantities in Column I with their dimensions in Column II.
A. Spring constant
B. Pascal
C. Hertz
D. Joule
(p) \( ML^2T^{-2} \)
(q) \( T^{-1} \)
(r) \( MT^{-2} \)
(s) \( ML^{-1}T^{-2} \)
(a) A-r, B-s, C-q, D-p
(b) A-s, B-r, C-q, D-p
(c) A-r, B-q, C-s, D-p
(d) A-p, B-s, C-r, D-q
Show Answer & Explanation

Answer: (a) A-r, B-s, C-q, D-p

Explanation:
1. Spring constant MT⁻², pascal ML⁻¹T⁻², hertz T⁻¹, joule ML²T⁻².

Teacher's Note:
1. Work out the dimensions or units of each item in Column I.
2. Then match with the identical entry in Column II.

Question 18: Match the quantities in Column I with their units in Column II.
A. Capacitance
B. Magnetic induction
C. Inductance
D. Resistance
(p) volt (ampere)⁻¹
(q) volt-second (ampere)⁻¹
(r) newton (ampere)⁻¹ (metre)⁻¹
(s) coulomb² (joule)⁻¹
(a) A-s, B-r, C-q, D-p
(b) A-s, B-q, C-r, D-p
(c) A-p, B-r, C-q, D-s
(d) A-q, B-s, C-p, D-r
Show Answer & Explanation

Answer: (a) A-s, B-r, C-q, D-p

Explanation:
1. C = Q²/(2U), B = F/(IL), L = V s/A and R = V/A.

Teacher's Note:
1. Work out the dimensions or units of each item in Column I.
2. Then match with the identical entry in Column II.

Question 19: Match the quantities in Column I with their units in Column II.
A. Distance between the earth and stars
B. Inter-atomic distance in a solid
C. Size of the nucleus
D. Wavelength of an infrared laser
(p) micron
(q) angstrom
(r) light year
(s) fermi
(a) A-p, B-q, C-r, D-s
(b) A-q, B-r, C-s, D-p
(c) A-r, B-q, C-s, D-p
(d) A-r, B-s, C-q, D-p
Show Answer & Explanation

Answer: (c) A-r, B-q, C-s, D-p

Explanation:
1. Stars are light years away; atoms are angstroms apart; nuclei are fermis; infrared wavelengths are microns.

Teacher's Note:
1. Work out the dimensions or units of each item in Column I.
2. Then match with the identical entry in Column II.

Question 20: Match List I with List II (dimensions).
A. Boltzmann constant
B. Coefficient of viscosity
C. Planck constant
D. Thermal conductivity
(p) \( [ML^2T^{-1}] \)
(q) \( [ML^{-1}T^{-1}] \)
(r) \( [MLT^{-3}K^{-1}] \)
(s) \( [ML^2T^{-2}K^{-1}] \)
(a) A-q, B-s, C-r, D-p
(b) A-p, B-q, C-s, D-r
(c) A-s, B-r, C-p, D-q
(d) A-s, B-q, C-p, D-r
Show Answer & Explanation

Answer: (d) A-s, B-q, C-p, D-r

Explanation:
1. k = E/T, η = ML⁻¹T⁻¹, h = E/ν, κ = P L/(A ΔT).

Teacher's Note:
1. Work out the dimensions or units of each item in Column I.
2. Then match with the identical entry in Column II.

Question 21: Out of the following pairs, which one does NOT have identical dimensions?
(a) impulse and momentum
(b) angular momentum and Planck's constant
(c) moment of inertia and moment of a force
(d) work and torque
Show Answer & Explanation

Answer: (c) moment of inertia and moment of a force

Explanation:
1. Moment of inertia is ML²; moment of a force is ML²T⁻².
2. The other pairs are identical.

Teacher's Note:
1. Torque equals moment of a force.
2. Compare the powers of T.

Question 22: In the relation \( P=\frac\alpha\beta e^{-\frac{\alpha z}{k\theta}} \), P is pressure, z is distance, k is the Boltzmann constant and θ is the temperature. The dimensional formula of β will be:
(a) \( [M^0L^2T^0] \)
(b) \( [M^1L^0T^{-1}] \)
(c) \( [M^0L^2T^{-1}] \)
(d) \( [M^1L^1T^1] \)
Show Answer & Explanation

Answer: (a) \( [M^0L^2T^0] \)

Explanation:
1. αz/(kθ) is dimensionless, so [α] = [kθ]/L = MLT⁻².
2. β = α/P = MLT⁻²/(ML⁻¹T⁻²) = L².

Teacher's Note:
1. The exponent must be dimensionless.
2. Find α first, then β.

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