A horizontal force is applied on a body on a rough horizontal surface produces an acceleration `a`.If coefficient of friction between the body and surface which is `a` is reduced to `mu//3`,the acceleration increses by `2 units`.The value of `mu` is

Options

A. `2//3g`
B. `3//2g`
C. `3//g`
D. `1//g`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the forces acting on the body
When a horizontal force \( F \) is applied to a body on a rough horizontal surface, the frictional force \( f \) opposing the motion can be expressed as:
\[ f = \mu mg \]
where \( \mu \) is the coefficient of friction, \( m \) is the mass of the body, and \( g \) is the acceleration due to gravity.

### Step 2: Write the equation of motion for the initial condition
Using Newton's second law, the net force acting on the body can be expressed as:
\[ F - f = ma \]
Substituting the expression for friction:
\[ F - \mu mg = ma \quad \text{(1)} \]

### Step 3: Write the equation of motion for the modified condition
When the coefficient of friction is reduced to \( \frac{\mu}{3} \), the new frictional force becomes:
\[ f' = \frac{\mu}{3} mg \]
The new acceleration is \( a + 2 \). Therefore, the equation of motion becomes:
\[ F - f' = m(a + 2) \]
Substituting the new frictional force:
\[ F - \frac{\mu}{3} mg = m(a + 2) \quad \text{(2)} \]

### Step 4: Set up the equations to eliminate \( F \)
From equation (1):
\[ F = ma + \mu mg \]
Substituting this expression for \( F \) into equation (2):
\[ ma + \mu mg - \frac{\mu}{3} mg = m(a + 2) \]

### Step 5: Simplify the equation
Rearranging the equation gives:
\[ ma + \mu mg - \frac{\mu}{3} mg = ma + 2m \]
Subtracting \( ma \) from both sides:
\[ \mu mg - \frac{\mu}{3} mg = 2m \]
Factoring out \( mg \):
\[ mg \left( \mu - \frac{\mu}{3} \right) = 2m \]
This simplifies to:
\[ mg \left( \frac{2\mu}{3} \right) = 2m \]

### Step 6: Solve for \( \mu \)
Dividing both sides by \( m \) (assuming \( m \neq 0 \)):
\[ g \left( \frac{2\mu}{3} \right) = 2 \]
Multiplying both sides by \( \frac{3}{2g} \):
\[ \mu = \frac{2 \cdot 3}{2g} = \frac{3}{g} \]

### Final Answer
The value of \( \mu \) is:
\[ \mu = \frac{3}{g} \]