Which of the following pairs of physical quantities have same dimensions?

Options

A. Force and power
B. Torque and energy
C. Torque and power
D. Force and torque.

Explanation

To determine which pairs of physical quantities have the same dimensions, we will calculate the dimensions of each quantity step by step.

### Step 1: Calculate the dimensions of Force
- **Formula for Force**: \( F = m \cdot a \)
- **Where**:
- \( m \) (mass) has dimensions \( [M] \)
- \( a \) (acceleration) has dimensions \( [L][T^{-2}] \)

- **Dimensions of Force**:
\[ [F] = [M][L][T^{-2}] = [L^1 M^1 T^{-2}] \]

### Step 2: Calculate the dimensions of Torque
- **Formula for Torque**: \( \tau = F \cdot r \)
- **Where**:
- \( F \) (force) has dimensions \( [L^1 M^1 T^{-2}] \)
- \( r \) (arm length) has dimensions \( [L] \)

- **Dimensions of Torque**:
\[ [\tau] = [L^1 M^1 T^{-2}] \cdot [L^1] = [L^2 M^1 T^{-2}] \]

### Step 3: Calculate the dimensions of Energy
- **Formula for Energy**: \( E = m \cdot v^2 \)
- **Where**:
- \( m \) (mass) has dimensions \( [M] \)
- \( v \) (velocity) has dimensions \( [L][T^{-1}] \)

- **Dimensions of Energy**:
\[ [E] = [M] \cdot [L^1 T^{-1}]^2 = [M^1 L^2 T^{-2}] \]

### Step 4: Calculate the dimensions of Power
- **Formula for Power**: \( P = \frac{W}{t} \)
- **Where**:
- \( W \) (work done) is equivalent to energy, which has dimensions \( [M^1 L^2 T^{-2}] \)
- \( t \) (time) has dimensions \( [T] \)

- **Dimensions of Power**:
\[ [P] = \frac{[M^1 L^2 T^{-2}]}{[T^1]} = [M^1 L^2 T^{-3}] \]

### Summary of Dimensions
- Force: \( [L^1 M^1 T^{-2}] \)
- Torque: \( [L^2 M^1 T^{-2}] \)
- Energy: \( [M^1 L^2 T^{-2}] \)
- Power: \( [M^1 L^2 T^{-3}] \)

### Conclusion
From the calculations, we can see that:
- Torque and Energy both have dimensions \( [M^1 L^2 T^{-2}] \).

Thus, the answer is that the pair of physical quantities with the same dimensions is **Torque and Energy**.

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