A body of mass `M` is resting on a rough horizontal plane surface the coefficient of friction being equal to `mu` At `t = 0` a horizontal force `F = F_(0) t` starts acting on it , where `F_(0)` is a constant find the time T at which the motion starts?

Options

A. `mu Mg//F_(0)`
B. ` mg// mu F_(0)`
C. `muF_(0)Mg`
D. None of these

Explanation

To find the time \( T \) at which the motion of the body starts, we can follow these steps:

### Step 1: Understand the forces acting on the body
The body of mass \( M \) is resting on a rough horizontal surface. The forces acting on the body include:
- The weight of the body, \( mg \) (acting downwards).
- The normal force \( N \) (acting upwards).
- The applied force \( F \) which is given by \( F = F_0 t \) (acting horizontally).
- The frictional force, which opposes the motion.

### Step 2: Determine the normal force
Since the body is resting on a horizontal surface and not moving vertically, the normal force \( N \) is equal to the weight of the body:
\[ N = mg \]

### Step 3: Calculate the limiting friction
The limiting friction \( f_{\text{lim}} \) can be expressed as:
\[ f_{\text{lim}} = \mu N = \mu mg \]
where \( \mu \) is the coefficient of friction.

### Step 4: Set up the condition for motion to start
For the body to start moving, the applied force \( F \) must be greater than the limiting friction. Therefore, we set up the inequality:
\[ F_0 t > \mu mg \]

### Step 5: Solve for the time \( T \)
To find the time \( T \) at which the motion starts, we set the applied force equal to the limiting friction:
\[ F_0 T = \mu mg \]
Now, we can solve for \( T \):
\[ T = \frac{\mu mg}{F_0} \]

### Conclusion
The time \( T \) at which the motion starts is given by:
\[ T = \frac{\mu mg}{F_0} \]

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