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 will follow these steps:

### Step 1: Understand the concept of a couple
A couple consists of two equal and opposite forces acting on a body, which creates a rotational effect without translating the body.

### Step 2: Define the couple mathematically
The torque (or moment) created by a couple is given by the formula:
\[ \text{Couple} = F \times L \]
where \( F \) is the magnitude of one of the forces and \( L \) is the perpendicular distance between the lines of action of the forces.

### Step 3: Express force in terms of mass and acceleration
Force \( F \) can be expressed using Newton's second law:
\[ F = m \cdot a \]
where \( m \) is mass and \( a \) is acceleration.

### Step 4: Substitute acceleration in terms of dimensions
Acceleration \( a \) can be expressed as:
\[ a = \frac{L}{t^2} \]
where \( L \) is distance and \( t \) is time.

### Step 5: Combine the expressions
Now substituting \( a \) into the force equation:
\[ F = m \cdot \frac{L}{t^2} \]

### Step 6: Substitute back into the couple formula
Now, substituting \( F \) back into the couple equation:
\[ \text{Couple} = F \times L = \left(m \cdot \frac{L}{t^2}\right) \times L \]

### Step 7: Simplify the expression
This simplifies to:
\[ \text{Couple} = m \cdot \frac{L^2}{t^2} \]

### Step 8: Write the dimensional formula
Thus, the dimensional formula for a couple is:
\[ [\text{Couple}] = M L^2 T^{-2} \]

### Conclusion
The dimensional formula for a couple is \( M L^2 T^{-2} \).