"Comprehensive MCQ Test on Units and Measurement: Mastering Dimensional Analysis, Significant Figures, and Their Applications in Physics for CBSE, JEE, and NEET Exams—Key Concepts for Understanding Measurement Errors, Precision, and the Interrelation of Physical Quantities in Competitive Assessments."

"Cracking Physics: Key MCQs on Units and Dimensions"

Time Left: 50:00

Total Marks: 80, Obtained Marks:

1. The density of material in CGS system of units is 4 g/cm³. In a system of units in which the unit of length is 10 cm and the unit of mass is 100 g, the value of density of material will be:

A. 4 × 10⁻³ units

B. 4 × 10⁻⁴ units

C. 4 × 10⁻¹ units

D. 40 units

The correct answer is 40 units. To convert the density from the CGS system to the new system, we use the conversion factor based on the new unit of length (10 cm) and the new unit of mass (100 g). The density becomes: \[ \rho_{\text{new}} = 4 \times \frac{1000}{100} = 40 \, \text{units} \]

2. The time period of a body under S.H.M. is represented by: \( T = P^a D^b S^c \), where \( P \) is pressure, \( D \) is density, and \( S \) is surface tension. What are the values of \( a \), \( b \), and \( c \)?

A. \( a = -1, b = 1, c = 2 \)

B. \( a = 1, b = 1, c = -2 \)

C. \( a = -\frac{3}{2}, b = \frac{1}{2}, c = 1 \)

D. \( a = -2, b = 1, c = \frac{1}{2} \)

The correct answer is \( a = -\frac{3}{2}, b = \frac{1}{2}, c = 1 \). We apply dimensional analysis to solve this: - Pressure \( P \) has dimensions \( [M^1 L^{-1} T^{-2}] \), - Density \( D \) has dimensions \( [M^1 L^{-3}] \), - Surface tension \( S \) has dimensions \( [M^1 T^{-2}] \). By equating the dimensions of mass, length, and time on both sides of \( T = P^a D^b S^c \), we form three equations and solve for \( a \), \( b \), and \( c \), leading to \( a = -\frac{3}{2}, b = \frac{1}{2}, c = 1 \).

3. The respective number of significant figures for the numbers 23.023, 0.0003, and 2.1 × 10^–3 are:

A. 4, 2, 3

B. 5, 1, 3

C. 5, 1, 2

D. 5, 2, 2

The correct answer is C: 5, 1, 2. The significant figures for each number are determined as follows: - **23.023** has 5 significant figures (all digits are significant). - **0.0003** has 1 significant figure (only the 3 is significant). - **2.1 × 10^–3** has 2 significant figures (the digits 2 and 1 are significant).

4. Young’s modulus of a material has the same unit as that of:

A. Density

B. Stress

C. Pressure

D. Strain

The correct answer is C: Pressure. Young's modulus is defined as the ratio of stress to strain. The unit of stress is Pascal (Pa), which is the same as the unit of pressure. Therefore, Young’s modulus has the same unit as pressure.

5. Of the following quantities, which one has dimensions different from the remaining three?

A. Product of voltage and charge per unit volume

B. Angular momentum

C. Energy per unit volume

D. Force per unit area

The correct answer is B: Angular momentum. The dimensions of angular momentum are \([M L^2 T^{-1}]\), while the other three quantities have dimensions of \([M L^{-1} T^{-2}]\).

6. The pressure on a square plate is measured by measuring the force on the plate and length of the sides of the plate using the formula \( P = \frac{F}{l^2} \). If the maximum errors in the measurement of force and length are 4% and 2% respectively, what is the maximum error in the measurement of pressure?

A. 8%

B. 6%

C. 10%

D. 5%

To find the maximum error in pressure, we apply the formula for pressure \( P = \frac{F}{l^2} \). The maximum errors in force and length are given as 4% and 2% respectively.

The relative error in pressure is calculated as:
\[ \frac{\Delta P}{P} = \frac{\Delta F}{F} + 2 \cdot \frac{\Delta l}{l} \]
Substituting the relative errors:
\[ \frac{\Delta P}{P} = 0.04 + 2 \cdot 0.02 = 0.04 + 0.04 = 0.08 \]
Thus, the maximum error in pressure as a percentage is:
\[ \text{Maximum error in pressure} = 0.08 \times 100\% = 8\% \]
Therefore, the maximum error in the measurement of pressure is **8%**.

7. The siemen is the SI unit of:

A. Resistance

B. Conductance

C. Capacitance

D. Inductance

The correct answer is B: Conductance. The siemen (S) is the SI unit of conductance, which measures how easily electricity flows through a component. It is defined as the reciprocal of resistance, with 1 siemen equal to 1 ampere per volt (A/V).

8. An object is moving through a liquid. The viscous damping force acting on it is proportional to the velocity. What are the dimensions of the constant of proportionality?

A. \([M L^{-1} T^{-2}]\)

B. \([M T^{-1}]\)

C. \([L^2 T^{-2}]\)

D. \([M^2 T^{-2}]\)

The correct answer is B: \([M T^{-1}]\). The viscous damping force \( F_d \) acting on an object moving through a liquid is proportional to its velocity \( v \), described by the equation:
\[ F_d = -k v \]
Here, \( k \) is the constant of proportionality. To find the dimensions of \( k \), we rearrange the equation:
\[ k = -\frac{F_d}{v} \]
The dimensions of force (\( F_d \)) are \([M L T^{-2}]\) and the dimensions of velocity (\( v \)) are \([L T^{-1}]\). Substituting these dimensions into the equation gives:
\[ [k] = \frac{[F_d]}{[v]} = \frac{[M L T^{-2}]}{[L T^{-1}]} \]
Simplifying this expression, we find:
\[ [k] = [M T^{-1}] \]
Thus, the dimensions of the constant of proportionality \( k \) are \([M T^{-1}]\).

9. The least count of a stopwatch is 0.2 seconds. The time of 20 oscillations of a pendulum is measured to be 25 seconds. What is the percentage error in the measurement of time?

A. 0.8%

B. 1.2%

C. 0.5%

D. 1.0%

The correct answer is A: 0.8%. The least count of the stopwatch is 0.2 seconds, which represents the absolute error in the measurement.

The formula for percentage error is given by:
\[ \text{Percentage Error} = \left( \frac{\text{Absolute Error}}{\text{Measured Value}} \right) \times 100 \]
Here, the measured value is the time for 20 oscillations, which is 25 seconds. Substituting the values:
\[ \text{Percentage Error} = \left( \frac{0.2}{25} \right) \times 100 = 0.8\% \]
Therefore, the percentage error in the measurement of time is **0.8%**.

10. The physical quantity which has the dimensional formula \([M^1 T^{-3}]\) is:

A. Density

B. Pressure

C. Energy density

D. Mass density

The correct answer is D: Mass density. The dimensional formula \([M^1 T^{-3}]\) corresponds to a quantity that is a mass per unit volume over time. In this case, it represents mass density, which is mass divided by volume.

11. The density of a cube is measured by measuring its mass and the length of its sides. If the maximum errors in the measurement of mass and length are 4% and 3% respectively, what is the maximum error in the measurement of density?

A. 10%

B. 12%

C. 13%

D. 14%

The correct answer is C: 13%. The density (\( \rho \)) of a cube is given by the formula:
\[ \rho = \frac{m}{l^3} \]
The relative error in density can be calculated using:
\[ \frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 3 \cdot \frac{\Delta l}{l} \]
Given the maximum errors: - Maximum error in mass: \( 4\% = 0.04 \) - Maximum error in length: \( 3\% = 0.03 \)
Substituting these values, we find:
\[ \frac{\Delta \rho}{\rho} = 0.04 + 3 \cdot 0.03 = 0.04 + 0.09 = 0.13 \]
Therefore, the maximum error in the measurement of density is \( 0.13 \times 100\% = 13\% \).

12. If the error in the measurement of the volume of a sphere is 6%, then the error in the measurement of its surface area will be:

A. 2%

B. 4%

C. 6%

D. 8%

The correct answer is B: 4%. The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
The surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]
The relative error in volume can be expressed as:
\[ \frac{\Delta V}{V} = 3 \cdot \frac{\Delta r}{r} \]
Given that the error in volume is 6%:
\[ 0.06 = 3 \cdot \frac{\Delta r}{r} \]
This implies:
\[ \frac{\Delta r}{r} = \frac{0.06}{3} = 0.02 \]
For surface area, the relative error is given by:
\[ \frac{\Delta A}{A} = 2 \cdot \frac{\Delta r}{r} = 2 \cdot 0.02 = 0.04 \]
Therefore, the error in the measurement of surface area is: \[ 0.04 \times 100\% = 4\% \]

13. Multiply 107.88 by 0.610 and express the result with correct number of significant figures.

A. 65.9

B. 65.8

C. 66.0

D. 65.7

The correct answer is B: 65.8. The result is expressed with three significant figures.

14. Which of the following is a dimensional constant?

A. Gravitational constant (G)

B. Acceleration (a)

C. Temperature (T)

D. Mass (m)

The correct answer is A: Gravitational constant (G). It has specific dimensions and remains constant in physical equations.

15. If E, m, J, and G represent energy, mass, angular momentum, and gravitational constant respectively, then the dimensional formula of EJ²/m⁵G² is the same as that of:

A. A dimensionless quantity

B. Velocity

C. Acceleration

D. Force

The correct answer is A: A dimensionless quantity. The dimensional formula of the expression simplifies to 1, indicating it is dimensionless.

16. The refractive index of water measured by the relation m = real depth / apparent depth is found to have values of 1.34, 1.38, 1.32, and 1.36; what is the mean value of refractive index with percentage error?

A. 1.34, 1.5%

B. 1.35, 1.48%

C. 1.36, 1.4%

D. 1.38, 1.6%

The correct answer is B: 1.35, 1.48%. The mean value is calculated from the given refractive indices and the percentage error is determined based on the average absolute error.

17. If e is the charge, V the potential difference, and T the temperature, then the units of eV/T are the same as that of:

A. Pressure (Pa)

B. Electric field (N/C)

C. Heat capacity (J/K)

D. Work (J)

The correct answer is C: Heat capacity (J/K). The units of eV/T simplify to J/K, which measures energy per unit temperature.

18. Two quantities A and B have different dimensions. Which mathematical operation given below is physically meaningful?

A. A + B

B. A × B

C. K(A+B)

D. A - B

The correct answers are B: A × B and C: A / B. Multiplication and division are meaningful operations, whereas addition and subtraction require the same dimensions.

19. The velocity of water waves (v) may depend on their wavelength (l), the density of water (ρ), and the acceleration due to gravity (g). Using dimensional analysis, the relation between these quantities is:

A. v = k l ρ g

B. v = k √(l g)

C. v = k l² / g

D. v = k g / l

To derive the relation, we start with identifying the dimensions of each quantity:

1. **Velocity (v)**:
    Dimensions: [L¹ T^-1]

2. **Wavelength (l)**:
    Dimensions: [L¹]

3. **Density (ρ)**:
    Dimensions: [M¹ L^-3]

4. **Acceleration due to gravity (g)**:
    Dimensions: [L¹ T^-2]

Assuming a relation:
    v = k l^a ρ^b g^c

Next, we set up the equation with dimensions:
[L¹ T^-1] = [L¹]^a [M¹ L^-3]^b [L¹ T^-2]^c

Expanding this gives:
[L¹ T^-1] = [L^{a - 3b + c} M^b T^-2c]

By equating dimensions, we get the following equations:

• For L: 1 = a - 3b + c
• For M: 0 = b
• For T: -1 = -2c ⇒ c = 1/2

Substituting b = 0 into the first equation yields:
1 = a + c ⇒ 1 = a + 1/2 ⇒ a = 1/2

Therefore, we find:
- a = 1/2
- b = 0
- c = 1/2

The relation is:
v = k √(l g)

This indicates that the velocity of water waves is proportional to the square root of the product of wavelength and acceleration due to gravity.

20. The unit of impulse is the same as that of:

A. Momentum

B. Energy

C. Power

D. Force

The correct answer is A: Momentum. Both impulse and momentum have the same unit, which is Newton-second (Ns). Impulse is defined as the change in momentum when a force is applied over a period of time.