Identify the pair whose dimensions are equal

Options

A. torque and work
B. stress and energy
C. force and stress
D. force and work

Answer

To solve the question of identifying the pair whose dimensions are equal, we can follow these steps:

### Step 1: Understand the terms involved
We need to compare the dimensions of torque and work.

### Step 2: Find the dimension of Torque
Torque (τ) is defined as the product of force (F) and the distance (r) from the pivot point:
\[ \tau = F \cdot r \]

The dimension of force (F) is:
\[ [F] = MLT^{-2} \]
where:
- M = mass
- L = length
- T = time

The dimension of distance (r) is:
\[ [r] = L \]

Thus, the dimension of torque is:
\[ [\tau] = [F] \cdot [r] = (MLT^{-2}) \cdot (L) = ML^2T^{-2} \]

### Step 3: Find the dimension of Work
Work (W) is defined as the product of force and displacement:
\[ W = F \cdot d \]

Using the same dimensions for force (F) and displacement (d):
\[ [W] = [F] \cdot [d] = (MLT^{-2}) \cdot (L) = ML^2T^{-2} \]

### Step 4: Compare the dimensions
Now we can compare the dimensions:
- Dimension of Torque: \( ML^2T^{-2} \)
- Dimension of Work: \( ML^2T^{-2} \)

Since both dimensions are equal, we conclude that torque and work have the same dimensions.

### Conclusion
The pair whose dimensions are equal is:
**Torque and Work**

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