A block of mass `m` slides down the plane inclined at angle `30^(@)` with an acceleration `g/4`.The value of coefficient of kinetic friction will be :

Options

A. `(2sqrt(3)+1)/(2)`
B. `(2sqrt(3)-1)/(2)`
C. `(sqrt(3))/(2)`
D. `(1)/(2sqrt(3))`

Answer

To solve the problem, we need to analyze the forces acting on the block sliding down the inclined plane. Let's go through the solution step by step.

### Step 1: Identify the forces acting on the block
The forces acting on the block of mass `m` on the inclined plane are:
1. Gravitational force acting downwards: \( F_g = mg \)
2. Normal force acting perpendicular to the surface: \( N \)
3. Frictional force acting opposite to the direction of motion: \( F_f = \mu N \)

### Step 2: Resolve the gravitational force into components
The gravitational force can be resolved into two components:
- Parallel to the incline: \( F_{\parallel} = mg \sin \theta \)
- Perpendicular to the incline: \( F_{\perpendicular} = mg \cos \theta \)

For our case, \( \theta = 30^\circ \):
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)

Thus, we have:
- \( F_{\parallel} = mg \cdot \frac{1}{2} = \frac{mg}{2} \)
- \( F_{\perpendicular} = mg \cdot \frac{\sqrt{3}}{2} = \frac{mg\sqrt{3}}{2} \)

### Step 3: Write the equation of motion
According to Newton's second law, the net force acting on the block is equal to the mass times its acceleration. The net force acting down the incline is given by:
\[ F_{\parallel} - F_f = ma \]

Substituting the expressions for the forces:
\[ mg \sin 30^\circ - \mu mg \cos 30^\circ = ma \]
\[ \frac{mg}{2} - \mu \frac{mg\sqrt{3}}{2} = ma \]

### Step 4: Substitute the acceleration
We know the acceleration \( a = \frac{g}{4} \). Substituting this into the equation gives:
\[ \frac{mg}{2} - \mu \frac{mg\sqrt{3}}{2} = m \cdot \frac{g}{4} \]

### Step 5: Simplify the equation
Dividing through by \( m \) (assuming \( m \neq 0 \)):
\[ \frac{g}{2} - \mu \frac{g\sqrt{3}}{2} = \frac{g}{4} \]

Now, we can eliminate \( g \) from the equation:
\[ \frac{1}{2} - \mu \frac{\sqrt{3}}{2} = \frac{1}{4} \]

### Step 6: Solve for the coefficient of friction \( \mu \)
Rearranging the equation:
\[ \frac{1}{2} - \frac{1}{4} = \mu \frac{\sqrt{3}}{2} \]
\[ \frac{1}{4} = \mu \frac{\sqrt{3}}{2} \]

Now, solve for \( \mu \):
\[ \mu = \frac{1/4}{\sqrt{3}/2} \]
\[ \mu = \frac{1}{4} \cdot \frac{2}{\sqrt{3}} \]
\[ \mu = \frac{1}{2\sqrt{3}} \]

### Final Answer
The value of the coefficient of kinetic friction \( \mu \) is:
\[ \mu = \frac{1}{2\sqrt{3}} \]

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