If F is the force applied , then acceleration of the body of mass M when body is on rough horizontal surface

Options

A. `f_k/M`
B. `(F-f_k)/M`
C. `F/M`
D. None of these

Answer

To find the acceleration of a body of mass \( M \) on a rough horizontal surface when a force \( F \) is applied, we can follow these steps:

### Step 1: Identify the forces acting on the body
When a force \( F \) is applied to the body on a rough surface, two main forces are acting on it:
- The applied force \( F \) (in the direction of motion)
- The frictional force \( F_k \) (acting in the opposite direction to the applied force)

### Step 2: Determine the type of friction
Since the body is moving, we will consider kinetic friction. The kinetic friction force \( F_k \) can be calculated using the formula:
\[ F_k = \mu_k \cdot N \]
where \( \mu_k \) is the coefficient of kinetic friction and \( N \) is the normal force. On a horizontal surface, the normal force \( N \) is equal to the weight of the body, which is \( M \cdot g \) (where \( g \) is the acceleration due to gravity).

Thus, we have:
\[ F_k = \mu_k \cdot (M \cdot g) \]

### Step 3: Apply Newton's second law of motion
According to Newton's second law, the net force acting on the body is equal to the mass of the body multiplied by its acceleration \( a \):
\[ F_{\text{net}} = M \cdot a \]
The net force \( F_{\text{net}} \) can be expressed as the difference between the applied force and the frictional force:
\[ F_{\text{net}} = F - F_k \]

### Step 4: Set up the equation
Combining the equations from Steps 2 and 3, we can write:
\[ F - F_k = M \cdot a \]
Substituting for \( F_k \):
\[ F - \mu_k \cdot (M \cdot g) = M \cdot a \]

### Step 5: Solve for acceleration \( a \)
Rearranging the equation to solve for acceleration \( a \):
\[ a = \frac{F - \mu_k \cdot (M \cdot g)}{M} \]

### Final Result
Thus, the acceleration \( a \) of the body of mass \( M \) on a rough horizontal surface when a force \( F \) is applied is given by:
\[ a = \frac{F - \mu_k \cdot (M \cdot g)}{M} \]