Two block A and B of masses 5 kg. and 3 kg. respectively rest on smooth horizontal surface with B over A. The coefficient of friction between A and B. 0.5 the maximum horizontal force that can be applied to A so that there will be motion of A and B without separation is :-

Options

A. 15
B. 25
C. 40
D. 50

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the System
We have two blocks, A (5 kg) and B (3 kg), resting on a smooth horizontal surface, with block B on top of block A. The coefficient of friction between the two blocks is 0.5. We need to find the maximum horizontal force \( F \) that can be applied to block A so that both blocks move together without separation.

### Step 2: Calculate the Normal Force on Block B
The normal force acting on block B is equal to its weight, which is given by:
\[ N_B = m_B \cdot g \]
where:
- \( m_B = 3 \, \text{kg} \) (mass of block B)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)

Calculating \( N_B \):
\[ N_B = 3 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 30 \, \text{N} \]

### Step 3: Calculate the Maximum Frictional Force Between A and B
The maximum frictional force \( F_{friction} \) that can act between blocks A and B is given by:
\[ F_{friction} = \mu \cdot N_B \]
where:
- \( \mu = 0.5 \) (coefficient of friction)

Calculating \( F_{friction} \):
\[ F_{friction} = 0.5 \cdot 30 \, \text{N} = 15 \, \text{N} \]

### Step 4: Determine the Maximum Acceleration of Block B
Using Newton's second law, the maximum acceleration \( a_B \) of block B can be calculated as:
\[ a_B = \frac{F_{friction}}{m_B} \]
Calculating \( a_B \):
\[ a_B = \frac{15 \, \text{N}}{3 \, \text{kg}} = 5 \, \text{m/s}^2 \]

### Step 5: Calculate the Total Mass of the System
The total mass \( m_{total} \) of the system (blocks A and B) is:
\[ m_{total} = m_A + m_B = 5 \, \text{kg} + 3 \, \text{kg} = 8 \, \text{kg} \]

### Step 6: Calculate the Maximum Force \( F \) Applied to Block A
To ensure that both blocks move together, the force \( F \) applied to block A must produce an acceleration equal to \( a_B \):
\[ F = m_{total} \cdot a_B \]
Calculating \( F \):
\[ F = 8 \, \text{kg} \cdot 5 \, \text{m/s}^2 = 40 \, \text{N} \]

### Final Answer
The maximum horizontal force that can be applied to block A so that both blocks A and B move together without separation is:
\[ \boxed{40 \, \text{N}} \]