Question

Match List-I with List-II.

Choose the correct answer from the options given below :

• A. (A)-(IV), (B)-(II), (C)-(III), (D)-(I)
• B. (A)-(I), (B)-(III), (C)-(IV), (D)-(II)
• C. (A)-(IV), (B)-(II), (C)-(I), (D)-(III)
• D. (A)-(II), (B)-(III), (C)-(IV), (D)-(I)

Hint:

To find the dimensional formula of a physical quantity, express it in terms of fundamental quantities like mass (M), length (L), and time (T). Remember the definitions and basic formulas of the given quantities.

Answer 1

The Correct Option is C

Let's match the physical quantities in List-I with their corresponding dimensions in List-II.

1. Mass density is defined as mass per unit volume. The dimension formula is: 
   \([\text{Density}] = \frac{M}{L^3} = [ML^{-3}T^0]\).
2. Impulse is defined as the change in momentum. Its dimension is the same as momentum, which is: 
   \([\text{Impulse}] = MLT^{-1}\).
3. Power is defined as work done per unit time. Its dimension is: 
   \([\text{Power}] = \frac{ML^2T^{-2}}{T} = [ML^2T^{-3}]\).
4. Moment of inertia is represented as \(I = mr^2\), hence its dimension is: 
   \([\text{Moment of Inertia}] = ML^2\).

Based on the above analysis, we can match them as:

• (A) Mass density - (IV) \([ML^{-3}T^0]\)
• (B) Impulse - (II) \([MLT^{-1}]\)
• (C) Power - (I) \([ML^2T^{-3}]\)
• (D) Moment of inertia - (III) \([ML^2]\)

Thus, the correct answer is (A)-(IV), (B)-(II), (C)-(I), (D)-(III).

Answer 2

Step 1: Determine the dimensional formula for each quantity in List-I.
(A) Mass density:
Mass density (\( \rho \)) is defined as mass per unit volume. \[ \rho = \frac{\text{Mass}}{\text{Volume}} = \frac{M}{L^3} = ML^{-3}T^{0} \] So, (A) matches with (IV).
(B) Impulse:
Impulse (\( J \)) is defined as the change in momentum or the product of force and time. \[ J = \Delta p = m \Delta v = M \cdot LT^{-1} = MLT^{-1} \] Alternatively, \[ J = F \cdot t = (ma) \cdot t = (MLT^{-2}) \cdot T = MLT^{-1} \] So, (B) matches with (II).
(C) Power:
Power (\( P \)) is defined as the rate of doing work or the product of force and velocity. \[ P = \frac{\text{Work}}{\text{Time}} = \frac{F \cdot d}{t} = \frac{(MLT^{-2}) \cdot L}{T} = ML^2T^{-3} \] Alternatively, \[ P = F \cdot v = (MLT^{-2}) \cdot (LT^{-1}) = ML^2T^{-3} \] So, (C) matches with (I).
(D) Moment of inertia:
Moment of inertia (\( I \)) of a particle is given by \( mr^2 \), where \( m \) is mass and \( r \) is the distance from the axis of rotation. For a system of particles or a continuous body, it involves mass and the square of distance. \[ I = M \cdot L^2 = ML^2T^{0} \] So, (D) matches with (III).
Step 2: Match the quantities with their dimensional formulas.
(A) Mass density - \( [ML^{-3}T^{0}] \) - (IV) (B) Impulse - \( [MLT^{-1}] \) - (II) (C) Power - \( [ML^2T^{-3}] \) - (I) (D) Moment of inertia - \( [ML^2T^{0}] \) - (III)
Step 3: Choose the correct option.
The correct matching is (A)-(IV), (B)-(II), (C)-(I), (D)-(III), which corresponds to option (3).