A weight `W` rests on a rough horizontal plane,If the angle of friction is `theta`,the least force that can move the body along the plane will be

Options

A. `W cos theta`
B. `W tan theta`
C. `W cot theta`
D. `W sin theta`

Explanation

To solve the problem of finding the least force that can move a weight \( W \) resting on a rough horizontal plane at an angle of friction \( \theta \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Forces Acting on the Weight:**
- The weight \( W \) exerts a downward force due to gravity. This force can be represented as \( W = mg \), where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity.
- The normal force \( N \) acts upward, balancing the weight.

2. **Identifying the Frictional Force:**
- The frictional force \( f \) opposing the motion can be expressed as \( f = \mu N \), where \( \mu \) is the coefficient of friction. In this case, since the angle of friction is \( \theta \), we can relate \( \mu \) to \( \theta \) using the equation \( \mu = \tan(\theta) \).

3. **Setting Up the Equations:**
- The normal force \( N \) is equal to the weight of the object, so \( N = W \).
- Therefore, the frictional force becomes \( f = \mu W = \tan(\theta) W \).

4. **Applying the Force to Move the Weight:**
- To move the weight along the plane, we need to apply a force \( F \) that overcomes the frictional force. Thus, we have:
\[ F = f = \tan(\theta) W \]

5. **Conclusion:**
- The least force \( F \) that can move the body along the plane is given by:
\[ F = W \tan(\theta) \]

### Final Answer:
The least force that can move the body along the plane is \( F = W \tan(\theta) \).

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