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 determine which pairs of quantities have the same dimensional formula, we will analyze the dimensions of each quantity step by step.

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

Calculating the dimensions:
\[ KE = [M][L T^{-1}]^2 = [M][L^2 T^{-2}] = [M^1 L^2 T^{-2}] \]

### Step 2: Determine the dimensional formula for Torque
Torque (\( \tau \)) is given by the formula:
\[ \tau = r \times F \]
Where:
- \( r \) is the distance (dimension: \( [L] \))
- \( F \) is force (dimension: \( [M L T^{-2}] \))

Calculating the dimensions:
\[ \tau = [L][M L T^{-2}] = [M^1 L^2 T^{-2}] \]

### Step 3: Determine the dimensional formula for Resistance
Resistance (\( R \)) is given by Ohm's law:
\[ R = \frac{V}{I} \]
Where:
- \( V \) is voltage (dimension: \( [M L^2 T^{-3} A^{-1}] \))
- \( I \) is current (dimension: \( [A] \))

Calculating the dimensions:
\[ R = \frac{[M L^2 T^{-3} A^{-1}]}{[A]} = [M^1 L^2 T^{-3} A^{-2}] \]

### Step 4: Determine the dimensional formula for Inductance
Inductance (\( L \)) is given by:
\[ L = \frac{V}{\frac{dI}{dt}} \]
Where:
- \( \frac{dI}{dt} \) has dimensions of current change over time (dimension: \( [A T^{-1}] \))

Calculating the dimensions:
\[ L = \frac{[M L^2 T^{-3} A^{-1}]}{[A T^{-1}]} = [M^1 L^2 T^{3} A^{-2}] \]

### Step 5: Determine the dimensional formula for Young's Modulus
Young's Modulus (\( E \)) is defined as:
\[ E = \frac{\text{Stress}}{\text{Strain}} \]
Where:
- Stress has dimensions of pressure (dimension: \( [M L^{-1} T^{-2}] \))
- Strain is dimensionless.

Calculating the dimensions:
\[ E = [M L^{-1} T^{-2}] \]

### Step 6: Determine the dimensional formula for Pressure
Pressure (\( P \)) is defined as:
\[ P = \frac{F}{A} \]
Where:
- \( F \) is force (dimension: \( [M L T^{-2}] \))
- \( A \) is area (dimension: \( [L^2] \))

Calculating the dimensions:
\[ P = \frac{[M L T^{-2}]}{[L^2]} = [M^1 L^{-1} T^{-2}] \]

### Summary of Dimensional Formulas
- Kinetic Energy: \( [M^1 L^2 T^{-2}] \)
- Torque: \( [M^1 L^2 T^{-2}] \)
- Resistance: \( [M^1 L^2 T^{-3} A^{-2}] \)
- Inductance: \( [M^1 L^2 T^{3} A^{-2}] \)
- Young's Modulus: \( [M^1 L^{-1} T^{-2}] \)
- Pressure: \( [M^1 L^{-1} T^{-2}] \)

### Conclusion
From the analysis:
- Kinetic Energy and Torque have the same dimensional formula: \( [M^1 L^2 T^{-2}] \)
- Young's Modulus and Pressure have the same dimensional formula: \( [M^1 L^{-1} T^{-2}] \)

Thus, the pairs that have the same dimensional formula are:
1. Kinetic Energy and Torque
2. Young's Modulus and Pressure