Problem Statement:
Two blocks, A and B, with masses 5 kg and 3 kg respectively, rest on a smooth horizontal surface. Block B is placed on top of block A. The coefficient of friction between block A and B is 0.5. We need to determine the maximum horizontal force that can be applied to block A without causing relative slipping between the blocks.

Solution:

1. Free Body Diagrams:
To solve this problem, let's start by drawing the free body diagrams for both blocks A and B.

For block A:
- There is a gravitational force acting downwards (mg).
- There is a normal force exerted by block B (N).
- There is a frictional force between block A and B (f).

For block B:
- There is a gravitational force acting downwards (mg).
- There is a normal force exerted by the surface (N).

2. Equations of Motion:
Now, let's analyze the forces acting on block A.

Forces in the vertical direction:
- N - mg = 0 (Since block A is not moving vertically)

Forces in the horizontal direction:
- f - F = ma (Newton's second law, where F is the applied force and a is the acceleration of block A)

For block B, since it is not moving horizontally, the sum of forces in the horizontal direction is zero.

3. Frictional Force:
The frictional force (f) between block A and B can be calculated using the equation:

f = μN

Where μ is the coefficient of friction and N is the normal force. Since the normal force (N) is equal to the weight of block B (mg), we can rewrite the equation as:

f = μmg

4. Maximum Applied Force:
To find the maximum force (F) that can be applied to block A without causing relative slipping between the blocks, we need to consider the maximum frictional force that can be exerted between the blocks.

The maximum frictional force (f_max) is given by:

f_max = μN = μmg

Therefore, the maximum force (F) that can be applied to block A is equal to the maximum frictional force:

F = f_max = μmg

Substituting the given values, we have:

F = 0.5 * 5 kg * 9.8 m/s^2

Simplifying the equation, we find:

F = 24.5 N

Therefore, the maximum horizontal force that can be applied to block A without causing relative slipping between the blocks is 24.5 Newtons.