Which pair of physical quantities has the same dimensional formula?

Surface tension and spring constant

Units & Dimensions

31 NEET-UG practice questions on Units & Dimensions, part of Physics. The first 25 are below; the first 5 with answers and explanations (tap Show answer after you have chosen), the rest in Learn with a free account.

Question 1 · Hard
Suppose force F, acceleration A and time T are chosen as the base quantities of a system of units. In this system, what are the dimensions of angular momentum?
1. A[F A² T³]
2. B[F A T³]
3. C[F A⁻¹ T³]
4. D[F A T²]
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Answer: B · [F A T³]
Mass = F/A and length = AT² (from acceleration × time²). Angular momentum = mass × length²/time = (F/A)(A²T⁴)/T = FAT³. Check: angular momentum = length × linear momentum = (AT²)(FT) = FAT³.
- A: This takes mass as F instead of F/A.
- C: This writes length with a wrong power of A.
- D: This gives the dimensions of energy, F × length.
Remember this
First write mass and length in the new base quantities, then substitute into the usual formula.
Question 2 · Medium
Match List-I with List-II. List-I (quantity) A. Young's modulus B. Coefficient of viscosity C. Torque D. Planck's constant List-II (dimensional formula) I. [ML²T⁻¹] II. [ML⁻¹T⁻¹] III. [ML⁻¹T⁻²] IV. [ML²T⁻²]
1. AA-II, B-III, C-I, D-IV
2. BA-II, B-III, C-IV, D-I
3. CA-III, B-II, C-I, D-IV
4. DA-III, B-II, C-IV, D-I
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Answer: D · A-III, B-II, C-IV, D-I
Young's modulus is stress, [ML⁻¹T⁻²]. Viscosity from F = ηA(dv/dx) is [ML⁻¹T⁻¹]. Torque has the dimensions of energy, [ML²T⁻²], and Planck's constant, E/ν, is [ML²T⁻¹].
- A: This makes both of the swaps above.
- B: This swaps Young's modulus and viscosity, which differ by one power of T.
- C: This swaps torque and Planck's constant.
Remember this
Planck's constant has the dimensions of angular momentum; torque has the dimensions of energy.
Question 3 · Medium
A screw gauge has a pitch of 0.5 mm and 50 circular divisions. With the jaws closed and nothing between them, it already reads 0.04 mm. For a wire, the pitch scale shows 1.5 mm and circular division 28 is on the reference line. What is the true diameter of the wire?
1. A1.78 mm
2. B1.60 mm
3. C1.82 mm
4. D1.74 mm
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Answer: D · 1.74 mm
Least count = 0.5/50 = 0.01 mm, so the observed reading = 1.5 + 28 × 0.01 = 1.78 mm. The gauge reads 0.04 mm too much, so the true value = 1.78 − 0.04 = 1.74 mm.
- A: This ignores the zero error.
- B: This takes the least count as half its true value.
- C: This adds the zero error instead of subtracting it.
Remember this
True reading = observed reading − zero error, keeping the sign of the zero error.
Question 4 · Hard
An empty beaker has a mass of (50.0 ± 0.1) g. With a liquid in it, the mass is (90.0 ± 0.1) g. The volume of the liquid is (20.0 ± 0.1) cm³. How should the density of the liquid be reported?
1. A(2.00 ± 0.3) g/cm³
2. B(2.00 ± 0.02) g/cm³
3. C(2.00 ± 0.015) g/cm³
4. D(2.00 ± 0.01) g/cm³
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Answer: B · (2.00 ± 0.02) g/cm³
Mass of liquid = 40.0 g, and in a difference the absolute errors add: ±0.2 g, which is 0.5%. The volume error is 0.1/20 = 0.5%, so density has a 1% error. Density = 40.0/20.0 = 2.00 g/cm³, and 1% of it is ±0.02.
- A: This adds absolute errors in the quotient instead of relative errors.
- C: This uses only ±0.1 g for the mass found by subtraction.
- D: This subtracts the two mass errors, so only the volume error remains.
Remember this
Add absolute errors for a sum or difference; add relative errors for a product or quotient.
Question 5 · Medium
Which pair of physical quantities has the same dimensional formula?
1. APressure and force
2. BSurface tension and spring constant
3. CTorque and angular momentum
4. DImpulse and kinetic energy
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Answer: B · Surface tension and spring constant
Surface tension is force per unit length and the spring constant is force per unit extension, so both are [MT⁻²].
- A: Pressure is force per unit area, so it differs from force by [L⁻²].
- C: Torque is [ML²T⁻²] but angular momentum is [ML²T⁻¹].
- D: Impulse is [MLT⁻¹] but kinetic energy is [ML²T⁻²].
Remember this
Force per unit length gives [MT⁻²], whatever the physical setting.
Question 6 · Easy
Twenty divisions on the sliding scale of a measuring instrument span the same length as nineteen main scale divisions of 1 mm each. Find the least count of the instrument.
1. A0.1 mm
2. B0.5 mm
3. C0.95 mm
4. D0.05 mm
Answer and explanation in LearnFree account
Question 7 · Easy
A student records the mass of a metal pellet as 0.0050200 kg. How many significant figures does this reading have?
1. A6
2. B5
3. C8
4. D3
Answer and explanation in LearnFree account
Question 8 · Medium
Arrange the following quantities in increasing order of the power of length L in their dimensional formulae. A. Pressure B. Impulse C. Angular momentum D. Surface tension
1. AD, A, C, B
2. BA, B, D, C
3. CA, D, B, C
4. DD, A, B, C
Answer and explanation in LearnFree account
Question 9 · Easy
From Newton's law of gravitation, what is the dimensional formula of the gravitational constant G?
1. A[M⁻¹L³T⁻²]
2. B[M⁻²L³T⁻²]
3. C[ML³T⁻²]
4. D[M⁻¹L²T⁻²]
Answer and explanation in LearnFree account
Question 10 · Medium
To find g with a simple pendulum, a student measures the length as (80.0 ± 0.2) cm and the time for 40 oscillations as (64.0 ± 0.4) s. Using g = 4π²L/T², what is the percentage error in g?
1. A1.0%
2. B0.875%
3. C1.25%
4. D1.5%
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Question 11 · Hard
Small drops of a liquid oscillate in shape when disturbed. Assume the frequency f of these oscillations depends only on the surface tension S, the density ρ and the radius r of the drop. Dimensional analysis then gives f proportional to
1. A√(S/(ρr))
2. B√(Sρ/r³)
3. C√(ρr³/S)
4. D√(S/(ρr³))
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Question 12 · Medium
Match List-I with List-II. List-I (Physical quantity) A. Coefficient of viscosity B. Surface tension C. Universal gravitational constant D. Universal gas constant List-II (SI unit) I. N m² kg⁻² II. J mol⁻¹ K⁻¹ III. Pa s IV. N m⁻¹ Choose the correct answer from the options given below.
1. AA-I, B-IV, C-III, D-II
2. BA-III, B-IV, C-II, D-I
3. CA-III, B-IV, C-I, D-II
4. DA-IV, B-III, C-I, D-II
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Question 13 · Medium
The kinetic energy of a body is found from K = p²/2m. If the momentum p is measured with an error of 2% and the mass m with an error of 1%, what is the maximum percentage error in K?
1. A3%
2. B4%
3. C5%
4. D1%
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Question 14 · Medium
Three rods of lengths 12.4 cm, 3.26 cm and 0.587 cm are laid end to end. Reported with the correct number of significant figures, what is their total length?
1. A16.25 cm
2. B16.2 cm
3. C16.247 cm
4. D16 cm
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Question 15 · Easy
Planck's constant, h, appears in the relation E = hν. Which quantity has the same dimensional formula as h?
1. APower
2. BEnergy
3. CAngular momentum
4. DLinear momentum
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Question 16 · Medium
In a new system of units, the unit of mass is 5 kg, the unit of length is 10 m and the unit of time is 2 s. What is the numerical value of an energy of 100 J in this system?
1. A0.8
2. B8
3. C0.05
4. D1.25
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Question 17 · Easy
Pressure is force per unit area. Which of the following has the same dimensional formula as pressure?
1. AEnergy per unit volume
2. BForce per unit length
3. CPower per unit area
4. DMomentum per unit area
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Question 18 · Hard
A student finds the density of a small metal sphere. Its mass is measured as 50.0 g with an error of ± 0.5 g, and its diameter as 2.00 cm using a vernier calliper of least count 0.01 cm. Taking the error in the diameter as one least count, what is the maximum percentage error in the density?
1. A1.5%
2. B1.75%
3. C2.5%
4. D0.5%
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Question 19 · Medium
In an experiment the potential difference across a resistor is measured as (100 ± 5) V and the current through it as (10 ± 0.2) A. How should the resistance R = V/I be reported?
1. A(10 ± 5.2) Ω
2. B(10 ± 0.7) Ω
3. C(10 ± 0.07) Ω
4. D(10 ± 0.3) Ω
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Question 20 · Medium
Assertion A: An equation that is dimensionally consistent must be physically correct. Reason R: An equation that is dimensionally inconsistent must be wrong. Choose the correct answer.
1. ABoth A and R are correct and R is the correct explanation of A
2. BA is not correct but R is correct
3. CA is correct but R is not correct
4. DBoth A and R are correct but R is not the correct explanation of A
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Question 21 · Easy
In the measured value 0.002050 m, how many significant figures are there?
1. A7
2. B4
3. C6
4. D3
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Question 22 · Medium
Given below are two statements: Statement I: An equation that is dimensionally correct must also be physically correct. Statement II: The method of dimensions cannot give the value of a dimensionless constant in a relation. Choose the correct answer.
1. ABoth Statement I and Statement II are incorrect
2. BStatement I is correct but Statement II is incorrect
3. CStatement I is incorrect but Statement II is correct
4. DBoth Statement I and Statement II are correct
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Question 23 · Medium
The velocity of a particle is given by v = at + b/(t + c), where t is time. What are the dimensions of b?
1. A[T]
2. B[LT⁻¹]
3. C[L]
4. D[LT⁻²]
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Question 24 · Medium
Match List-I with List-II. List-I (Physical quantity) A. Impulse B. Power C. Pressure gradient (pressure per unit length) D. Rate of change of angular momentum List-II (Dimensional formula) I. [ML²T⁻²] II. [MLT⁻¹] III. [ML⁻²T⁻²] IV. [ML²T⁻³] Choose the correct answer from the options given below.
1. AA-IV, B-II, C-III, D-I
2. BA-II, B-I, C-III, D-IV
3. CA-II, B-IV, C-I, D-III
4. DA-II, B-IV, C-III, D-I
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Question 25 · Hard
Using only Planck's constant h, the speed of light c and the gravitational constant G, which combination has the dimension of length?
1. A√(hc/G)
2. B√(hG/c⁵)
3. ChG/c³
4. D√(hG/c³)
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6 more Units & Dimensions questions

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Units & Dimensions

Suppose force F, acceleration A and time T are chosen as the base quantities of a system of units. In this system, what are the dimensions of angular momentum?

Match List-I with List-II. List-I (quantity) A. Young's modulus B. Coefficient of viscosity C. Torque D. Planck's constant List-II (dimensional formula) I. [ML²T⁻¹] II. [ML⁻¹T⁻¹] III. [ML⁻¹T⁻²] IV. [ML²T⁻²]

A screw gauge has a pitch of 0.5 mm and 50 circular divisions. With the jaws closed and nothing between them, it already reads 0.04 mm. For a wire, the pitch scale shows 1.5 mm and circular division 28 is on the reference line. What is the true diameter of the wire?

An empty beaker has a mass of (50.0 ± 0.1) g. With a liquid in it, the mass is (90.0 ± 0.1) g. The volume of the liquid is (20.0 ± 0.1) cm³. How should the density of the liquid be reported?

Which pair of physical quantities has the same dimensional formula?

Twenty divisions on the sliding scale of a measuring instrument span the same length as nineteen main scale divisions of 1 mm each. Find the least count of the instrument.

A student records the mass of a metal pellet as 0.0050200 kg. How many significant figures does this reading have?

Arrange the following quantities in increasing order of the power of length L in their dimensional formulae. A. Pressure B. Impulse C. Angular momentum D. Surface tension

From Newton's law of gravitation, what is the dimensional formula of the gravitational constant G?

To find g with a simple pendulum, a student measures the length as (80.0 ± 0.2) cm and the time for 40 oscillations as (64.0 ± 0.4) s. Using g = 4π²L/T², what is the percentage error in g?

Small drops of a liquid oscillate in shape when disturbed. Assume the frequency f of these oscillations depends only on the surface tension S, the density ρ and the radius r of the drop. Dimensional analysis then gives f proportional to

Match List-I with List-II. List-I (Physical quantity) A. Coefficient of viscosity B. Surface tension C. Universal gravitational constant D. Universal gas constant List-II (SI unit) I. N m² kg⁻² II. J mol⁻¹ K⁻¹ III. Pa s IV. N m⁻¹ Choose the correct answer from the options given below.

The kinetic energy of a body is found from K = p²/2m. If the momentum p is measured with an error of 2% and the mass m with an error of 1%, what is the maximum percentage error in K?

Three rods of lengths 12.4 cm, 3.26 cm and 0.587 cm are laid end to end. Reported with the correct number of significant figures, what is their total length?

Planck's constant, h, appears in the relation E = hν. Which quantity has the same dimensional formula as h?

In a new system of units, the unit of mass is 5 kg, the unit of length is 10 m and the unit of time is 2 s. What is the numerical value of an energy of 100 J in this system?

Pressure is force per unit area. Which of the following has the same dimensional formula as pressure?

A student finds the density of a small metal sphere. Its mass is measured as 50.0 g with an error of ± 0.5 g, and its diameter as 2.00 cm using a vernier calliper of least count 0.01 cm. Taking the error in the diameter as one least count, what is the maximum percentage error in the density?

In an experiment the potential difference across a resistor is measured as (100 ± 5) V and the current through it as (10 ± 0.2) A. How should the resistance R = V/I be reported?

Assertion A: An equation that is dimensionally consistent must be physically correct. Reason R: An equation that is dimensionally inconsistent must be wrong. Choose the correct answer.

In the measured value 0.002050 m, how many significant figures are there?

Given below are two statements: Statement I: An equation that is dimensionally correct must also be physically correct. Statement II: The method of dimensions cannot give the value of a dimensionless constant in a relation. Choose the correct answer.

The velocity of a particle is given by v = at + b/(t + c), where t is time. What are the dimensions of b?

Using only Planck's constant h, the speed of light c and the gravitational constant G, which combination has the dimension of length?