The dimensional formula of couple

Options

A. `ML^(2)T^(-2)
B. `MLT^(-1)
C. `ML^(-1)T^(-1)
D. `M^(1)L^(1)T^(-2)

Answer

To find the dimensional formula of a couple, we need to understand what a couple is and how it is calculated. A couple consists of two equal and opposite forces acting on a body, which creates a rotational effect. The torque (or moment) produced by a couple is given by the product of one of the forces and the perpendicular distance between the lines of action of the forces.

### Step-by-Step Solution:

1. **Understanding the Concept of a Couple**:
- A couple consists of two equal and opposite forces acting on a rigid body, causing it to rotate without translating.

2. **Formula for Torque (Couple)**:
- The torque (or moment) due to a couple is given by:
\[ \text{Torque} = \text{Force} \times \text{Perpendicular Distance} \]
- If we denote the force as \( F \) and the perpendicular distance between the forces as \( L \), then:
\[ \text{Couple} = F \times L \]

3. **Dimensional Formula of Force**:
- The dimensional formula for force \( F \) is given by:
\[ F = m \cdot a \]
- Where \( m \) is mass and \( a \) is acceleration. The dimensional formula for acceleration \( a \) is:
\[ a = \frac{L}{T^2} \]
- Therefore, the dimensional formula for force becomes:
\[ [F] = [M][L][T^{-2}] = M^1 L^1 T^{-2} \]

4. **Dimensional Formula of Length**:
- The dimensional formula for length \( L \) is simply:
\[ [L] = L^1 \]

5. **Combining the Dimensional Formulas**:
- Now, substituting the dimensional formulas into the couple formula:
\[ [\text{Couple}] = [F] \cdot [L] = (M^1 L^1 T^{-2}) \cdot (L^1) \]
- This simplifies to:
\[ [\text{Couple}] = M^1 L^2 T^{-2} \]

### Final Result:
The dimensional formula of a couple is:
\[ [M^1 L^2 T^{-2}] \]