Match List - I with List - II.
`{:("List-I","List-II"),("(a)Torque","(i)MLT"^(-1)),("(b)Impulse","(ii)MT"^(-2)),("(c )Tension","(iii)ML"^(2)"T"^(-2)),("(d)Surface Tenion","(iv)MLT"^(-2)):}`

Options

A. (a)-(ii),(b)-(i),(c )-(iv),(d)-(iii)
B. (a)-(iii),(b)-(iv),(c )-(i),(d)-(ii)
C. (a)-(iii),(b)-(i),(c )-(iv),(d)-(ii)
D. (a)-(i),(b)-(iii),(c )-(iv),(d)-(ii)

Answer

To solve the problem of matching List-I with List-II based on the dimensions of physical quantities, we will analyze each quantity one by one and derive their dimensions.

**Step 1: Analyze Torque**
- Torque (τ) is defined as the product of force (F) and the distance (r) from the pivot point, which can be expressed as:
\[ \tau = F \cdot r \]
- The unit of force (F) is given by Newton (N), which is equivalent to \( \text{kg} \cdot \text{m/s}^2 \).
- Therefore, the unit of torque can be expressed as:
\[ \text{Torque} = \text{Force} \times \text{Distance} = \text{N} \cdot \text{m} = \left(\text{kg} \cdot \text{m/s}^2\right) \cdot \text{m} = \text{kg} \cdot \text{m}^2/\text{s}^2 \]
- The dimensions of torque are:
\[ [\tau] = \text{ML}^2\text{T}^{-2} \]

**Step 2: Analyze Impulse**
- Impulse (J) is defined as the change in momentum, which can also be expressed as:
\[ J = F \cdot \Delta t \]
- The unit of impulse is the same as the unit of momentum, which is given by:
\[ \text{Momentum} = \text{mass} \times \text{velocity} = \text{kg} \cdot \left(\text{m/s}\right) = \text{kg} \cdot \text{m/s} \]
- Therefore, the dimensions of impulse are:
\[ [J] = \text{MLT}^{-1} \]

**Step 3: Analyze Tension**
- Tension (T) is a force, and thus its unit is the same as that of force:
\[ T = \text{Force} = \text{N} = \text{kg} \cdot \text{m/s}^2 \]
- The dimensions of tension are:
\[ [T] = \text{MLT}^{-2} \]

**Step 4: Analyze Surface Tension**
- Surface tension (σ) is defined as force per unit length:
\[ \sigma = \frac{\text{Force}}{\text{Length}} = \frac{\text{N}}{\text{m}} = \frac{\text{kg} \cdot \text{m/s}^2}{\text{m}} = \text{kg/s}^2 \]
- The dimensions of surface tension are:
\[ [\sigma] = \text{MT}^{-2} \]

**Step 5: Match the Lists**
Now that we have the dimensions for each quantity, we can match them with List-II:
- (a) Torque → (iii) \( \text{ML}^2\text{T}^{-2} \)
- (b) Impulse → (i) \( \text{MLT}^{-1} \)
- (c) Tension → (ii) \( \text{MLT}^{-2} \)
- (d) Surface Tension → (iv) \( \text{MT}^{-2} \)

**Final Matching:**
- (a) Torque → (iii)
- (b) Impulse → (i)
- (c) Tension → (ii)
- (d) Surface Tension → (iv)