A horizontal force applied on a body on a rough horizontal surface produces an acceleration 'a'. If coefficient of friction between the body & surface which is m is reduced to m/3, the accele-ration increases by 2 units. The value of m is

Options

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

Answer

To solve the problem step by step, we will analyze the forces acting on the body and use Newton's second law of motion.

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

### Step 2: Set up the equations for the two cases
1. **Case 1**: When the coefficient of friction is \( \mu = m \), the acceleration \( a \) is given by:
\[ F - f = ma \]
Substituting for \( f \):
\[ F - \mu mg = ma \quad \text{(1)} \]

2. **Case 2**: When the coefficient of friction is reduced to \( \mu = \frac{m}{3} \), the new acceleration becomes \( a + 2 \):
\[ F - f' = m(a + 2) \]
where \( f' = \frac{m}{3} mg \):
\[ F - \frac{m}{3} mg = m(a + 2) \quad \text{(2)} \]

### Step 3: Rearranging the equations
From equation (1):
\[ F = ma + \mu mg \]
From equation (2):
\[ F = m(a + 2) + \frac{m}{3} mg \]

### Step 4: Equate the two expressions for \( F \)
Setting the two expressions for \( F \) equal to each other:
\[ ma + \mu mg = m(a + 2) + \frac{m}{3} mg \]

### Step 5: Substitute \( \mu \) and simplify
Substituting \( \mu = m \):
\[ ma + m \cdot mg = m(a + 2) + \frac{m}{3} mg \]
This simplifies to:
\[ ma + m^2g = ma + 2m + \frac{m}{3}g \]
Cancelling \( ma \) from both sides:
\[ m^2g = 2m + \frac{m}{3}g \]

### Step 6: Combine like terms
Rearranging gives:
\[ m^2g - \frac{m}{3}g = 2m \]
Factoring out \( g \):
\[ g\left(m^2 - \frac{m}{3}\right) = 2m \]

### Step 7: Solve for \( m \)
Dividing through by \( m \) (assuming \( m \neq 0 \)):
\[ g\left(m - \frac{1}{3}\right) = 2 \]
Rearranging gives:
\[ m - \frac{1}{3} = \frac{2}{g} \]
Thus:
\[ m = \frac{2}{g} + \frac{1}{3} \]

### Step 8: Find a common denominator and simplify
To combine the fractions:
\[ m = \frac{6}{3g} + \frac{1}{3} = \frac{6 + 1g}{3g} \]
Thus:
\[ m = \frac{6 + g}{3g} \]

### Final Answer
To find the specific value of \( m \), we can substitute \( g \) with \( 9.8 \, \text{m/s}^2 \) or leave it in terms of \( g \) for a general solution.