The minimum force required to move a body of weight W placed on a rough horizontal surface is

Options

A. `Wsinlamda`
B. `Wcoslamda`
C. `Wtanlamda`
D. `Wcotlamda`

Answer

To determine the minimum force required to move a body of weight \( W \) placed on a rough horizontal surface, we can follow these steps:

### Step 1: Understand the Forces Acting on the Body
When a force \( P \) is applied to the body at an angle \( \theta \), it can be resolved into two components:
- Horizontal component: \( P \cos \theta \)
- Vertical component: \( P \sin \theta \)

### Step 2: Analyze the Vertical Forces
The weight of the body \( W \) acts downwards, and the normal force \( N \) acts upwards. The vertical forces can be balanced as follows:
\[ N + P \sin \theta = W \]
From this, we can express the normal force \( N \):
\[ N = W - P \sin \theta \]

### Step 3: Analyze the Horizontal Forces
The frictional force \( f \) that opposes the motion is given by:
\[ f = \mu N \]
where \( \mu \) is the coefficient of friction. The horizontal component of the applied force must overcome the frictional force:
\[ P \cos \theta = \mu N \]

### Step 4: Substitute for Normal Force
Substituting \( N \) from Step 2 into the equation for friction gives:
\[ P \cos \theta = \mu (W - P \sin \theta) \]

### Step 5: Rearranging the Equation
Rearranging the above equation to isolate \( P \):
\[ P \cos \theta + \mu P \sin \theta = \mu W \]
Factoring \( P \) out:
\[ P (\cos \theta + \mu \sin \theta) = \mu W \]
Thus, we can solve for \( P \):
\[ P = \frac{\mu W}{\cos \theta + \mu \sin \theta} \]

### Step 6: Determine the Minimum Force
To find the minimum force \( P \), we need to consider the angle \( \theta \). The minimum force occurs when the expression for \( P \) is minimized. This can be done by differentiating with respect to \( \theta \) or by recognizing that the minimum occurs when the angle is such that:
\[ \mu = \tan \lambda \]
where \( \lambda \) is the angle of repose. Substituting this back into the equation gives:
\[ P = \frac{W \tan \lambda}{\cos \theta + \tan \lambda \sin \theta} \]

### Step 7: Final Expression for Minimum Force
The minimum force required to move the body can be expressed as:
\[ P = \frac{W \sin \lambda}{\cos \theta - \lambda} \]

### Conclusion
Thus, the minimum force required to move a body of weight \( W \) on a rough horizontal surface is:
\[ P = W \sin \lambda \]