A varying horizontal force `F =` at act on a block of mass `m` kept on a smooth horizontal surface An identical block is kept on the first block. The coefficient of friction between the blocks is `mu` . The time after which the relative sliding between the blocks prevails is

Options

A. `(a)(2 m g)/(a)`
B. `(b)(2 mu m g)/(a)`
C. `(c)(mu m g)/(a)`
D. `(d)2 mu mga`

Explanation

To solve the problem step by step, we will analyze the forces acting on the blocks and derive the time after which relative sliding occurs.

### Step 1: Understand the system
We have two blocks, each with mass \( m \). The bottom block is on a smooth horizontal surface, and a varying horizontal force \( F(t) = a \cdot t \) is applied to it. The top block is resting on the bottom block, and the coefficient of friction between the two blocks is \( \mu \).

### Step 2: Identify the forces acting on the top block
The only force acting on the top block (mass \( m \)) that will cause it to accelerate is the frictional force between the two blocks. The frictional force \( F_f \) can be expressed as:
\[ F_f = \mu \cdot N \]
where \( N \) is the normal force acting on the top block. Since the only vertical force acting on the top block is its weight, we have:
\[ N = m \cdot g \]
Thus, the frictional force becomes:
\[ F_f = \mu \cdot m \cdot g \]

### Step 3: Calculate the acceleration of the top block
The acceleration \( a_1 \) of the top block can be calculated using Newton's second law:
\[ F_f = m \cdot a_1 \]
Substituting the expression for the frictional force:
\[ \mu \cdot m \cdot g = m \cdot a_1 \]
Dividing both sides by \( m \) (assuming \( m \neq 0 \)):
\[ a_1 = \mu \cdot g \]

### Step 4: Calculate the total acceleration of the system
The total force acting on the entire system (both blocks) is the applied force \( F(t) \). The total mass of the system is \( 2m \). Therefore, the total acceleration \( a_t \) of the system can be expressed as:
\[ a_t = \frac{F(t)}{2m} \]
Substituting \( F(t) = a \cdot t \):
\[ a_t = \frac{a \cdot t}{2m} \]

### Step 5: Set the condition for relative sliding
For relative sliding to occur, the acceleration of the top block \( a_1 \) must equal the total acceleration of the system \( a_t \):
\[ a_1 = a_t \]
Substituting the expressions we derived:
\[ \mu \cdot g = \frac{a \cdot t}{2m} \]

### Step 6: Solve for time \( t \)
Rearranging the equation to solve for \( t \):
\[ t = \frac{2m \mu g}{a} \]

### Final Answer
The time after which the relative sliding between the blocks prevails is:
\[ t = \frac{2m \mu g}{a} \]