In moving a body of mass `m` once up and down a smooth incline `0` total work done is (S is length of the plane):

Options

A. `mg sin theta xx S`
B. `mg cos theta = S`
C. `2 mu mg cos theta xx s`
D. zero

Answer

To solve the problem of calculating the total work done when moving a body of mass `m` once up and down a smooth incline at an angle `θ`, we can follow these steps:

### Step 1: Understanding Forces on the Incline
When a body is on an incline, the forces acting on it include:
- The gravitational force acting downwards, which can be resolved into two components:
- Perpendicular to the incline: \( mg \cos \theta \)
- Parallel to the incline: \( mg \sin \theta \)

### Step 2: Work Done While Moving Up the Incline
When the body is moved up the incline, the work done against gravity can be calculated. If we denote the distance moved up the incline as `S`, the work done against the gravitational force is:
\[ W_{\text{up}} = F_{\text{applied}} \cdot S - mg \sin \theta \cdot S \]
Where \( F_{\text{applied}} \) is the force applied to move the body up.

### Step 3: Work Done While Moving Down the Incline
When the body is moved back down the incline, the work done is:
\[ W_{\text{down}} = F_{\text{applied}} \cdot (-S) + mg \sin \theta \cdot (-S) \]
Here, the force applied is still \( F_{\text{applied}} \), but the direction of movement is negative.

### Step 4: Total Work Done
The total work done when moving the body up and down the incline is the sum of the work done in both directions:
\[ W_{\text{total}} = W_{\text{up}} + W_{\text{down}} \]
Substituting the expressions from Steps 2 and 3:
\[ W_{\text{total}} = (F_{\text{applied}} \cdot S - mg \sin \theta \cdot S) + (F_{\text{applied}} \cdot (-S) + mg \sin \theta \cdot (-S)) \]
This simplifies to:
\[ W_{\text{total}} = F_{\text{applied}} \cdot S - mg \sin \theta \cdot S - F_{\text{applied}} \cdot S - mg \sin \theta \cdot S \]
\[ W_{\text{total}} = -2 mg \sin \theta \cdot S \]

### Step 5: Conclusion
Since the incline is smooth (no friction), the total work done is:
\[ W_{\text{total}} = -2 mg \sin \theta \cdot S \]
This indicates that the work done is negative, reflecting that energy is lost due to the gravitational force acting on the body.