The minimum value of acceleration of wedge for which the block start sliding on the wedge, is:

Options

A. `g((costheta+musintheta)/(sintheta-mucostheta))`
B. `g((sintheta+mucostheta)/(costheta-musintheta))`
C. `g((sintheta-mucostheta)/(sintheta+mucostheta))`
D. `g((costheta-musintheta)/(sintheta+mucostheta))`

Explanation

To solve the problem of finding the minimum value of acceleration of the wedge for which the block starts sliding on the wedge, we can follow these steps:

### Step 1: Understand the System
We have a wedge inclined at an angle \( \theta \) with a block resting on it. The wedge is accelerating horizontally, and we need to determine the minimum acceleration \( a \) of the wedge that will cause the block to start sliding down.

### Step 2: Draw the Free Body Diagram (FBD)
Draw the FBD of the block on the wedge. The forces acting on the block are:
- The gravitational force \( mg \) acting downwards.
- The normal force \( N \) acting perpendicular to the surface of the wedge.
- The frictional force \( f \) acting parallel to the surface of the wedge, opposing the motion.

### Step 3: Resolve Forces
Resolve the gravitational force \( mg \) into two components:
- Perpendicular to the incline: \( mg \cos \theta \)
- Parallel to the incline: \( mg \sin \theta \)

### Step 4: Apply Newton's Second Law
For the block to start sliding down, the net force acting on it along the incline must overcome the frictional force. The frictional force can be expressed as:
\[ f = \mu N \]
where \( \mu \) is the coefficient of friction.

The equations of motion can be set up as follows:
1. In the direction perpendicular to the incline:
\[ N = mg \cos \theta - ma \sin \theta \]
2. In the direction parallel to the incline:
\[ ma \cos \theta = mg \sin \theta + f \]

### Step 5: Substitute the Frictional Force
Substituting the expression for the frictional force \( f \):
\[ ma \cos \theta = mg \sin \theta + \mu (mg \cos \theta - ma \sin \theta) \]

### Step 6: Rearranging the Equation
Rearranging the equation gives:
\[ ma \cos \theta + \mu ma \sin \theta = mg \sin \theta + \mu mg \cos \theta \]
Factoring out \( ma \) on the left side:
\[ ma (\cos \theta + \mu \sin \theta) = mg (\sin \theta + \mu \cos \theta) \]

### Step 7: Solve for Acceleration \( a \)
Now, we can solve for \( a \):
\[ a = \frac{g (\sin \theta + \mu \cos \theta)}{\cos \theta + \mu \sin \theta} \]

### Conclusion
Thus, the minimum value of acceleration \( a \) of the wedge for which the block starts sliding is:
\[ a = \frac{g (\sin \theta + \mu \cos \theta)}{\cos \theta + \mu \sin \theta} \]

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