A body is moving up an inclined plane of angle `theta` with an initial kinetic energy E. The coefficient of friction between the plane and body is `mu`. The work done against friction before the body comes to rest is

Options

A. `(muE costheta)/(costheta+sintheta)`
B. E
C. `(muE costheta)/(mucostheta-sintheta)`
D. `(muE costheta)/(mucostheta+sintheta)`

Answer

To solve the problem, we need to determine the work done against friction when a body moves up an inclined plane and comes to rest. Here’s a step-by-step solution:

### Step 1: Understand the Forces Acting on the Body
When the body is moving up the inclined plane, the forces acting on it include:
- The gravitational force, which can be broken down into two components:
- Parallel to the incline: \( mg \sin \theta \)
- Perpendicular to the incline: \( mg \cos \theta \)
- The normal force \( N \) acting perpendicular to the incline.
- The frictional force \( F_f \) acting opposite to the direction of motion, given by \( F_f = \mu N \).

### Step 2: Calculate the Normal Force
The normal force \( N \) can be expressed as:
\[ N = mg \cos \theta \]

### Step 3: Calculate the Frictional Force
Using the normal force, the frictional force can be calculated as:
\[ F_f = \mu N = \mu (mg \cos \theta) \]

### Step 4: Apply the Work-Energy Theorem
According to the work-energy theorem:
\[ \text{Work done} = \Delta KE \]
Where \( \Delta KE \) is the change in kinetic energy. The initial kinetic energy is \( E \) and the final kinetic energy is \( 0 \) (since the body comes to rest). Therefore:
\[ \text{Work done} = 0 - E = -E \]

### Step 5: Write the Expression for Work Done
The total work done on the body can be expressed as the sum of the work done against gravity and the work done against friction:
\[ W_{\text{gravity}} + W_{\text{friction}} = -E \]

### Step 6: Calculate Work Done by Gravity
The work done by gravity while the body moves a distance \( x \) up the incline is:
\[ W_{\text{gravity}} = -mg \sin \theta \cdot x \]

### Step 7: Write the Expression for Work Done Against Friction
The work done against friction can be expressed as:
\[ W_{\text{friction}} = -F_f \cdot x = -(\mu mg \cos \theta) \cdot x \]

### Step 8: Combine the Expressions
Combining the expressions for work done:
\[ -mg \sin \theta \cdot x - \mu mg \cos \theta \cdot x = -E \]
Factoring out \( -x \):
\[ -x (mg \sin \theta + \mu mg \cos \theta) = -E \]

### Step 9: Solve for Distance \( x \)
From the equation above, we can solve for \( x \):
\[ x = \frac{E}{mg \sin \theta + \mu mg \cos \theta} \]

### Step 10: Calculate the Work Done Against Friction
Substituting \( x \) back into the expression for work done against friction:
\[ W_{\text{friction}} = -(\mu mg \cos \theta) \cdot \left(\frac{E}{mg \sin \theta + \mu mg \cos \theta}\right) \]
This simplifies to:
\[ W_{\text{friction}} = -\frac{\mu mg \cos \theta \cdot E}{mg \sin \theta + \mu mg \cos \theta} \]

### Final Result
Thus, the work done against friction before the body comes to rest is:
\[ W_{\text{friction}} = -\frac{\mu E \cos \theta}{\sin \theta + \mu \cos \theta} \]