A body of mass m slides down a rough plane of inclination `alpha` . If `mu` is the coefficient of friction, then acceleration of the body will be

Options

A. `gsin alpha`
B. `mucos alpha`
C. `g(sin alpha-mucos alpha)`
D. `g(cos alpha-mu sin alpha)`

Answer

To solve the problem of a body of mass \( m \) sliding down a rough inclined plane at an angle \( \alpha \) with a coefficient of friction \( \mu \), we will follow these steps:

### Step 1: Identify the Forces Acting on the Body
The forces acting on the body are:
- The gravitational force \( mg \) acting downward.
- The normal force \( N \) acting perpendicular to the inclined surface.
- The frictional force \( f \) acting opposite to the direction of motion (up the incline).

### Step 2: Resolve the Gravitational Force
We need to resolve the gravitational force into two components:
- The component parallel to the incline: \( F_{\parallel} = mg \sin \alpha \)
- The component perpendicular to the incline: \( F_{\perpendicular} = mg \cos \alpha \)

### Step 3: Determine the Normal Force
The normal force \( N \) is equal to the perpendicular component of the gravitational force, since there is no vertical motion:
\[ N = mg \cos \alpha \]

### Step 4: Calculate the Frictional Force
The frictional force \( f \) can be expressed in terms of the normal force and the coefficient of friction:
\[ f = \mu N = \mu (mg \cos \alpha) \]

### Step 5: Apply Newton's Second Law
According to Newton's second law, the net force acting on the body along the incline is equal to the mass times its acceleration \( a \):
\[ F_{\text{net}} = ma \]

The net force along the incline can be expressed as:
\[ F_{\text{net}} = F_{\parallel} - f = mg \sin \alpha - \mu (mg \cos \alpha) \]

### Step 6: Set Up the Equation
Substituting the expressions for the forces into the equation gives:
\[ mg \sin \alpha - \mu (mg \cos \alpha) = ma \]

### Step 7: Simplify the Equation
We can factor out \( m \) from the left side:
\[ m(g \sin \alpha - \mu g \cos \alpha) = ma \]

### Step 8: Solve for Acceleration
Dividing both sides by \( m \) (assuming \( m \neq 0 \)):
\[ g \sin \alpha - \mu g \cos \alpha = a \]

Thus, the acceleration \( a \) of the body down the incline is:
\[ a = g \sin \alpha - \mu g \cos \alpha \]

### Final Answer
The acceleration of the body sliding down the rough inclined plane is:
\[ a = g (\sin \alpha - \mu \cos \alpha) \]

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