Which of the following pairs have same dimensional formula for both the quantities ?
`(i)` kinetic energy and torque
`(ii)` resistance and inductance
`(iii)` Young's Modulus and Pressure

Options

A. `(i)` only
B. `(ii)` only
C. `(i)` and `(iii)` only
D. all the above

Answer

To solve the problem of identifying which pairs of quantities have the same dimensional formula, we will analyze each pair step by step.

### Step 1: Analyze Kinetic Energy and Torque

**Kinetic Energy (KE)** is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
Where:
- \( m \) is mass (dimension \( [M] \))
- \( v \) is velocity (dimension \( [L T^{-1}] \))

The dimensional formula for kinetic energy is:
\[ [KE] = [M][L^2 T^{-2}] = [M L^2 T^{-2}] \]

**Torque (τ)** is given by the formula:
\[ τ = r \times F \]
Where:
- \( r \) is the distance (dimension \( [L] \))
- \( F \) is force (dimension \( [M L T^{-2}] \))

The dimensional formula for torque is:
\[ [τ] = [L][M L T^{-2}] = [M L^2 T^{-2}] \]

**Conclusion for Step 1:**
Both kinetic energy and torque have the same dimensional formula:
\[ [M L^2 T^{-2}] \]

### Step 2: Analyze Resistance and Inductance

**Resistance (R)** is defined by Ohm's law:
\[ R = \frac{V}{I} \]
Where:
- \( V \) is voltage (dimension \( [M L^2 T^{-3} I^{-1}] \))
- \( I \) is current (dimension \( [I] \))

The dimensional formula for resistance is:
\[ [R] = \frac{[M L^2 T^{-3} I^{-1}]}{[I]} = [M L^2 T^{-3} I^{-2}] \]

**Inductance (L)** is given by:
\[ L = \frac{V}{\frac{dI}{dt}} \]
Where:
- \( \frac{dI}{dt} \) has the dimension of current change over time, which is \( [I T^{-1}] \)

The dimensional formula for inductance is:
\[ [L] = \frac{[M L^2 T^{-3} I^{-1}]}{[I T^{-1}]} = [M L^2 T^{-2} I^{-2}] \]

**Conclusion for Step 2:**
Resistance and inductance have different dimensional formulas:
- Resistance: \( [M L^2 T^{-3} I^{-2}] \)
- Inductance: \( [M L^2 T^{-2} I^{-2}] \)

### Step 3: Analyze Young's Modulus and Pressure

**Young's Modulus (Y)** is defined as:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]
Where:
- Stress is defined as force per unit area: \( \text{Stress} = \frac{F}{A} \)
- Strain is dimensionless.

Thus, the dimensional formula for Young's Modulus is:
\[ [Y] = \frac{[F][A^{-1}]}{1} = \frac{[M L T^{-2}]}{[L^2]} = [M L^{-1} T^{-2}] \]

**Pressure (P)** is defined as:
\[ P = \frac{F}{A} \]
Where:
- Force is \( [M L T^{-2}] \) and area is \( [L^2] \).

The dimensional formula for pressure is:
\[ [P] = \frac{[F]}{[A]} = \frac{[M L T^{-2}]}{[L^2]} = [M L^{-1} T^{-2}] \]

**Conclusion for Step 3:**
Both Young's Modulus and Pressure have the same dimensional formula:
\[ [M L^{-1} T^{-2}] \]

### Final Conclusion
The pairs with the same dimensional formulas are:
- (i) Kinetic Energy and Torque: Same dimensional formula \( [M L^2 T^{-2}] \)
- (iii) Young's Modulus and Pressure: Same dimensional formula \( [M L^{-1} T^{-2}] \)

Thus, the correct options are (i) and (iii).