JEE Physics Units and Measurements MCQs Set 16

Multiple Choice Questions (MCQs) for JEE Physics: Units and Measurements

Access targeted multiple-choice questions for Units and Measurements designed to align with the latest JEE academic syllabus for JEE Physics. These objective practice sets help students evaluate their conceptual understanding and improve exam readiness.

Practice Units and Measurements MCQs for JEE Physics

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Multiple Choice Questions

Question 1: L, C and R represent inductance, capacitance and resistance respectively. Which of the combinations \( \dfrac1{RC} \), \( \dfrac RL \), \( \dfrac1{\sqrt{LC}} \), \( \dfrac CL \) have the dimensions of frequency?
(a) \( \dfrac1{RC} \), \( \dfrac RL \) and \( \dfrac1{\sqrt{LC}} \) only
(b) \( \dfrac RL \) and \( \dfrac CL \) only
(c) \( \dfrac1{RC} \) and \( \dfrac CL \) only
(d) All four
Show Answer & Explanation

Answer: (a) \( \dfrac1{RC} \), \( \dfrac RL \) and \( \dfrac1{\sqrt{LC}} \) only

Explanation:
1. \( RC \) and \( L/R \) are time constants, and \( \sqrt{LC} \) is the time scale of an LC oscillation. So \( 1/RC \), \( R/L \) and \( 1/\sqrt{LC} \) all have the dimension \( \mathrm{T^{-1}} \).
2. \( C/L \) has dimension \( \mathrm{M^{-2}L^{-4}T^{6}A^{4}} \), which is not \( \mathrm{T^{-1}} \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 2: In an electromagnetic-theory paragraph, [E] and [B] stand for the dimensions of the electric and magnetic fields, [\( \varepsilon_0 \)] and [\( \mu_0 \)] for those of the permittivity and permeability of free space, and [L] and [T] for those of length and time. The relation between [E] and [B] is
(a) \( [E]=[B][L][T] \)
(b) \( [E]=[B]^{-1}[L][T]^{-1} \)
(c) \( [E]=[B][L][T]^{-1} \)
(d) \( [E]=[B][L]^{-1}[T]^{-1} \)
Show Answer & Explanation

Answer: (c) \( [E]=[B][L][T]^{-1} \)

Explanation:
1. For an electromagnetic wave \( E=cB \), and \( c \) has the dimension \( [L][T]^{-1} \).
2. Hence \( [E]=[B][L][T]^{-1} \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 3: With the same notation (E, B, \( \varepsilon_0 \), \( \mu_0 \), L, T), the relation between \( [\varepsilon_0] \) and \( [\mu_0] \) is
(a) \( [\mu_0]=[\varepsilon_0][L]^2[T]^{-2} \)
(b) \( [\mu_0]=[\varepsilon_0][L]^{-2}[T]^{2} \)
(c) \( [\mu_0]=[\varepsilon_0]^{-1}[L]^{2}[T]^{-2} \)
(d) \( [\mu_0]=[\varepsilon_0]^{-1}[L]^{-2}[T]^{2} \)
Show Answer & Explanation

Answer: (d) \( [\mu_0]=[\varepsilon_0]^{-1}[L]^{-2}[T]^{2} \)

Explanation:
1. \( c^2=1/(\mu_0\varepsilon_0) \Rightarrow \mu_0 = 1/(\varepsilon_0c^2) \).
2. \( [\mu_0]=[\varepsilon_0]^{-1}[L]^{-2}[T]^{2} \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 4: To find the distance d over which a signal can be seen in foggy conditions, a railway engineer assumes that d depends on the mass density \( \rho \) of the fog, the intensity (power/area) S of the light from the signal and its frequency f. He finds that d is proportional to \( S^{1/n} \). The value of n is
(a) 3
(b) 2
(c) 4
(d) 1
Show Answer & Explanation

Answer: (a) 3

Explanation:
1. Let \( d=\rho^aS^bf^c \), with \( [\rho]=\mathrm{ML^{-3}} \), \( [S]=\mathrm{MT^{-3}} \), \( [f]=\mathrm{T^{-1}} \).
2. Mass: \( a+b=0 \). Length: \( -3a=1 \Rightarrow a=-\tfrac13 \), so \( b=\tfrac13 \).
3. So \( d\propto S^{1/3} \) and \( n=3 \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 5: The equation of state of a real gas is \( \left(p+\dfrac a{V^2}\right)(V-b)=RT \). The dimensions of the constant a are
(a) \( [\mathrm{ML^{5}T^{-2}}] \)
(b) \( [\mathrm{ML^{-1}T^{-2}}] \)
(c) \( [\mathrm{ML^{2}T^{-2}}] \)
(d) \( [\mathrm{M^{0}L^{3}T^{0}}] \)
Show Answer & Explanation

Answer: (a) \( [\mathrm{ML^{5}T^{-2}}] \)

Explanation:
1. \( a/V^2 \) must have the dimension of pressure, \( \mathrm{ML^{-1}T^{-2}} \).
2. \( [a]=\mathrm{ML^{-1}T^{-2}}\times\mathrm{L^{6}}=\mathrm{ML^{5}T^{-2}} \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 6: The dimensions of electrical conductivity are
(a) \( [\mathrm{M^{-1}L^{-3}T^{3}A^{2}}] \)
(b) \( [\mathrm{ML^{3}T^{-3}A^{-2}}] \)
(c) \( [\mathrm{M^{-1}L^{-2}T^{3}A^{2}}] \)
(d) \( [\mathrm{ML^{-3}T^{3}A^{2}}] \)
Show Answer & Explanation

Answer: (a) \( [\mathrm{M^{-1}L^{-3}T^{3}A^{2}}] \)

Explanation:
1. Conductivity \( \sigma = 1/\rho \), and \( \rho = RA/l \) has dimension \( \mathrm{ML^3T^{-3}A^{-2}} \).
2. So \( [\sigma] = \mathrm{M^{-1}L^{-3}T^{3}A^{2}} \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 7: Planck's constant has the dimensions
(a) \( [\mathrm{ML^{2}T^{-1}}] \)
(b) \( [\mathrm{ML^{2}T^{-2}}] \)
(c) \( [\mathrm{MLT^{-1}}] \)
(d) \( [\mathrm{ML^{-2}T^{-1}}] \)
Show Answer & Explanation

Answer: (a) \( [\mathrm{ML^{2}T^{-1}}] \)

Explanation:
1. \( E=h\nu \Rightarrow h=E/\nu \).
2. \( [h]=\mathrm{ML^2T^{-2}}/\mathrm{T^{-1}}=\mathrm{ML^2T^{-1}} \), the dimension of angular momentum.

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 8: The dimensions of magnetic flux and of the modulus of rigidity, in terms of mass (M), length (L), time (T) and charge (Q), are respectively
(a) \( [\mathrm{ML^{2}T^{-1}Q^{-1}}] \) and \( [\mathrm{ML^{-1}T^{-2}}] \)
(b) \( [\mathrm{ML^{2}T^{-2}Q^{-1}}] \) and \( [\mathrm{ML^{-1}T^{-2}}] \)
(c) \( [\mathrm{ML^{2}T^{-1}Q^{-1}}] \) and \( [\mathrm{ML^{2}T^{-2}}] \)
(d) \( [\mathrm{MLT^{-1}Q^{-1}}] \) and \( [\mathrm{ML^{-1}T^{-1}}] \)
Show Answer & Explanation

Answer: (a) \( [\mathrm{ML^{2}T^{-1}Q^{-1}}] \) and \( [\mathrm{ML^{-1}T^{-2}}] \)

Explanation:
1. Flux \( \Phi = BA \) and \( B = F/(qv) = \mathrm{MT^{-1}Q^{-1}} \), so \( [\Phi]=\mathrm{ML^2T^{-1}Q^{-1}} \).
2. Modulus of rigidity is stress/strain, so \( [\eta]=\mathrm{ML^{-1}T^{-2}} \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 9: A gas bubble from an explosion under water oscillates with a period T proportional to \( p^ad^bE^c \), where p is the static pressure, d is the density of water and E is the total energy of the explosion. The values of a, b and c are
(a) \( a=-\tfrac56,\ b=\tfrac12,\ c=\tfrac13 \)
(b) \( a=\tfrac56,\ b=-\tfrac12,\ c=\tfrac13 \)
(c) \( a=-\tfrac56,\ b=-\tfrac12,\ c=\tfrac13 \)
(d) \( a=-\tfrac13,\ b=\tfrac12,\ c=\tfrac56 \)
Show Answer & Explanation

Answer: (a) \( a=-\tfrac56,\ b=\tfrac12,\ c=\tfrac13 \)

Explanation:
1. Dimensions: \( \mathrm{T}=(\mathrm{ML^{-1}T^{-2}})^a(\mathrm{ML^{-3}})^b(\mathrm{ML^2T^{-2}})^c \).
2. Mass: \( a+b+c=0 \). Length: \( -a-3b+2c=0 \). Time: \( -2a-2c=1 \).
3. Solving: \( a=-\tfrac56,\ b=\tfrac12,\ c=\tfrac13 \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 10: The area of a square is 5.29 cm². The area of 7 such squares, taking into account the significant figures, is
(a) 37.030 cm²
(b) 37.0 cm²
(c) 37.03 cm²
(d) 37 cm²
Show Answer & Explanation

Answer: (b) 37.0 cm²

Explanation:
1. Exact product: \( 7\times5.29=37.03 \).
2. The measured area has 3 significant figures (the factor 7 is an exact count), so the result is quoted as 37.0 cm².

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 11: In a simple pendulum experiment for g, the time for 20 oscillations is measured with a watch of 1 s least count and the mean value is 30 s. The length is measured with a metre scale of least count 1 mm and is 55.0 cm. The percentage error in g is close to
(a) 0.7%
(b) 6.8%
(c) 3.5%
(d) 0.2%
Show Answer & Explanation

Answer: (b) 6.8%

Explanation:
1. \( g=4\pi^2L/T^2 \Rightarrow \dfrac{\Delta g}{g}=\dfrac{\Delta L}{L}+2\dfrac{\Delta T}{T} \).
2. \( \Delta L/L=0.1/55.0=0.18\% \) and \( \Delta T/T=1/30=3.33\% \).
3. \( \Delta g/g=0.18\%+6.67\%\approx6.8\% \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 12: The diameter and height of a cylinder are measured by a metre scale to be \( 12.6\pm0.1 \) cm and \( 34.2\pm0.1 \) cm. The value of its volume, in appropriate significant figures, is
(a) \( 4300\pm80\ \text{cm}^3 \)
(b) \( 4260\pm80\ \text{cm}^3 \)
(c) \( 4264.4\pm81.0\ \text{cm}^3 \)
(d) \( 4264\pm81\ \text{cm}^3 \)
Show Answer & Explanation

Answer: (b) \( 4260\pm80\ \text{cm}^3 \)

Explanation:
1. \( V=\tfrac\pi4d^2h=\tfrac\pi4(12.6)^2(34.2)=4264.4\ \text{cm}^3 \).
2. \( \Delta V/V=2(0.1/12.6)+0.1/34.2=0.0188 \), so \( \Delta V\approx80\ \text{cm}^3 \).
3. The error has one significant figure, so V is rounded to the tens place: \( 4260\pm80\ \text{cm}^3 \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 13: The density of a material in the shape of a cube is determined by measuring the three sides of the cube and its mass. If the relative errors in measuring the mass and length are 1.5% and 1% respectively, the maximum error in determining the density is
(a) 6%
(b) 2.5%
(c) 3.5%
(d) 4.5%
Show Answer & Explanation

Answer: (d) 4.5%

Explanation:
1. \( \rho=m/l^3 \Rightarrow \Delta\rho/\rho=\Delta m/m+3\,\Delta l/l \).
2. \( =1.5\%+3(1\%)=4.5\% \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 14: A person measures the depth of a well by timing a dropped stone until the sound of impact returns. The error in the time is \( \delta T=0.01 \) s and he measures the depth as L = 20 m. Take g = 10 m s⁻² and speed of sound 300 m s⁻¹. The fractional error \( \delta L/L \) is closest to
(a) 1%
(b) 5%
(c) 3%
(d) 0.2%
Show Answer & Explanation

Answer: (a) 1%

Explanation:
1. \( T=\sqrt{2L/g}+L/v \). Differentiating: \( \delta T=\left(\dfrac1{\sqrt{2gL}}+\dfrac1v\right)\delta L \).
2. \( \sqrt{2gL}=\sqrt{400}=20 \), so \( \delta T=(0.05+0.00333)\,\delta L=0.05333\,\delta L \).
3. \( \delta L=0.01/0.05333=0.1875 \) m, and \( \delta L/L=0.94\%\approx1\% \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 15: The period of a simple pendulum is \( T=2\pi\sqrt{L/g} \). The measured value of L is 20.0 cm known to 1 mm accuracy, and the time for 100 oscillations is found to be 90 s using a wrist watch of 1 s resolution. The accuracy in the determination of g is
(a) 3%
(b) 2%
(c) 1%
(d) 5%
Show Answer & Explanation

Answer: (a) 3%

Explanation:
1. \( \Delta g/g=\Delta L/L+2\,\Delta T/T \).
2. \( \Delta L/L=0.1/20=0.5\% \); \( \Delta T/T=1/90=1.11\% \).
3. \( \Delta g/g=0.5\%+2.22\%=2.7\%\approx3\% \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 16: The current-voltage relation of a diode is \( I=(e^{1000V/T}-1) \) mA, where V is in volt and T is in kelvin. A student makes an error of \( \pm0.01 \) V in measuring V, when the current is 5 mA at 300 K. The error in the value of the current is
(a) 0.2 mA
(b) 0.02 mA
(c) 0.5 mA
(d) 0.05 mA
Show Answer & Explanation

Answer: (a) 0.2 mA

Explanation:
1. \( I+1=e^{1000V/T}=6 \) at 5 mA.
2. \( dI=e^{1000V/T}\cdot\dfrac{1000}{T}\,dV=6\times\dfrac{1000}{300}\times0.01 \).
3. \( dI=0.2 \) mA.

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 17: A wire has mass \( (0.3\pm0.003) \) g, radius \( (0.5\pm0.005) \) mm and length \( (6\pm0.06) \) cm. The maximum percentage error in the measurement of its density is
(a) 1
(b) 2
(c) 3
(d) 4
Show Answer & Explanation

Answer: (d) 4

Explanation:
1. \( \rho=\dfrac m{\pi r^2l} \Rightarrow \dfrac{\Delta\rho}\rho=\dfrac{\Delta m}m+2\dfrac{\Delta r}r+\dfrac{\Delta l}l \).
2. Each relative error is 1%, so the total is \( 1+2+1=4\% \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 18: A cube has a side of length \( 1.2\times10^{-2} \) m. Its volume, to the correct number of significant figures, is
(a) \( 1.7\times10^{-6}\ \text{m}^3 \)
(b) \( 1.73\times10^{-6}\ \text{m}^3 \)
(c) \( 1.70\times10^{-6}\ \text{m}^3 \)
(d) \( 1.732\times10^{-6}\ \text{m}^3 \)
Show Answer & Explanation

Answer: (a) \( 1.7\times10^{-6}\ \text{m}^3 \)

Explanation:
1. \( V=(1.2\times10^{-2})^3=1.728\times10^{-6}\ \text{m}^3 \).
2. The side has 2 significant figures, so \( V=1.7\times10^{-6}\ \text{m}^3 \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 19: The energy of a system as a function of time is \( E(t)=A^2\exp(-\alpha t) \), where \( \alpha=0.2\ \text{s}^{-1} \). The measurement of A has an error of 1.25%. If the error in the measurement of time is 1.50%, the percentage error in the value of E(t) at t = 5 s is
(a) 4%
(b) 2%
(c) 6%
(d) 1.5%
Show Answer & Explanation

Answer: (a) 4%

Explanation:
1. \( \ln E=2\ln A-\alpha t \Rightarrow \dfrac{\Delta E}E=2\dfrac{\Delta A}A+\alpha t\dfrac{\Delta t}t \).
2. \( \alpha t=0.2\times5=1 \), so \( \Delta E/E=2(1.25\%)+1\times1.5\%=4\% \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 20: The least count of the main scale of a screw gauge is 1 mm. The minimum number of divisions on its circular scale required to measure a 5 \( \mu \)m diameter of a wire is
(a) 50
(b) 200
(c) 500
(d) 100
Show Answer & Explanation

Answer: (b) 200

Explanation:
1. Least count \( = \) pitch / N \( = 1\ \text{mm}/N \).
2. For \( 5\ \mu\text{m}=0.005 \) mm: \( N=1/0.005=200 \).

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

Question 21: The pitch and number of divisions on the circular scale of a screw gauge are 0.5 mm and 100. When fully tightened without any object, the zero of its circular scale lies 3 divisions below the mean line. The readings for a thin sheet are 5.5 mm on the main scale and 48 on the circular scale. The thickness of the sheet is
(a) 5.950 mm
(b) 5.725 mm
(c) 5.755 mm
(d) 5.740 mm
Show Answer & Explanation

Answer: (c) 5.755 mm

Explanation:
1. Least count \( =0.5/100=0.005 \) mm. The zero lies below the mean line, so the zero error is negative (\( -3 \) divisions) and the correction is \( +3\times0.005 \).
2. Thickness \( =5.5+48(0.005)+3(0.005)=5.755 \) mm.

Teacher's Note:
1. Check the dimensions or units term by term before choosing an option.
2. Work with the standard base quantities so that each step can be verified.

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