The dimensional formula of torque is

Options

A. `[ML^(2)T^(-2)]`
B. `[MLT^(-2)]`
C. `[ML^(-1)T^(-2)]`
D. `[ML^(-2)T^(-2)]`

Explanation

To find the dimensional formula of torque, we can follow these steps:

### Step 1: Understand the definition of torque
Torque (τ) is defined as the measure of the force that can cause an object to rotate about an axis. It is calculated as the product of the force applied and the perpendicular distance from the axis of rotation to the line of action of the force.

### Step 2: Write the formula for torque
The formula for torque can be expressed as:
\[ \tau = r \times F \]
Where:
- \( \tau \) is the torque,
- \( r \) is the perpendicular distance from the axis of rotation to the line of action of the force,
- \( F \) is the force applied.

### Step 3: Identify the dimensions of force
The dimensional formula for force (F) can be derived from Newton's second law, which states that force is mass times acceleration:
\[ F = m \cdot a \]
Where:
- Mass (m) has the dimension [M],
- Acceleration (a) has the dimension of length per time squared, which is [L][T]^{-2}.

Thus, the dimensional formula for force is:
\[ [F] = [M][L][T]^{-2} = M^1 L^1 T^{-2} \]

### Step 4: Identify the dimensions of distance
The dimensional formula for distance (r) is simply:
\[ [r] = [L] = L^1 \]

### Step 5: Combine the dimensions to find the dimensional formula for torque
Now, substituting the dimensions of force and distance into the torque formula:
\[ [\tau] = [r] \cdot [F] \]
\[ [\tau] = [L^1] \cdot [M^1 L^1 T^{-2}] \]
\[ [\tau] = M^1 L^1 T^{-2} \cdot L^1 \]
\[ [\tau] = M^1 L^{1+1} T^{-2} \]
\[ [\tau] = M^1 L^2 T^{-2} \]

### Conclusion
The dimensional formula for torque is:
\[ [\tau] = M^1 L^2 T^{-2} \]