To analyze the situation you've described, we need to consider the forces acting on the blocks and how they interact with each other. The arrangement involves three blocks: one block of mass m suspended and two blocks in contact on a smooth surface, where one has mass m and the other has mass 2m. The key point here is that the block of mass m on top is stationary with respect to the block of mass 2m, which means they are moving together as a system. Let's break this down step by step.

Understanding the Forces at Play

First, we need to identify the forces acting on each block:

• The suspended block (mass m) experiences a downward gravitational force of mg, where g is the acceleration due to gravity.
• The block of mass m on top of the block of mass 2m also has a gravitational force acting on it, which is mg.
• The block of mass 2m has its own weight, which is 2mg.

Analyzing the System

Since the ground is smooth, we can ignore any friction between the blocks and the ground. The only friction we need to consider is between the two blocks (m and 2m). The block of mass m on top will exert a force on the block of mass 2m due to gravity, which is mg. This force will create a normal force between the two blocks, which is equal to the weight of the top block, mg.

The frictional force (f) that prevents the top block from sliding off the bottom block can be calculated using the formula:

f = μN

Here, μ is the coefficient of friction between the two blocks, and N is the normal force. In this case, the normal force N is equal to the weight of the top block, which is mg. Therefore, we can express the frictional force as:

f = μ(mg)

Considering the Motion of the System

Since the block of mass m is stationary with respect to the block of mass 2m, we can analyze the entire system's motion. The total weight acting downward is:

Total Weight = mg + 2mg = 3mg

This total weight will create a tension (T) in the string that is equal to the weight of the suspended block (m). Therefore, we can say:

T = mg

Relating Tension and Friction

Now, the tension in the string must also balance the frictional force acting on the block of mass 2m. Since the block of mass 2m is not accelerating horizontally (it is stationary with respect to the block of mass m), the frictional force must equal the tension:

f = T

Substituting the expressions we derived earlier, we have:

μ(mg) = mg

From this equation, we can see that the frictional force is equal to the weight of the suspended block. Therefore, the force of friction between the blocks of masses m and 2m is:

f = mg

Final Thoughts

In summary, the force of friction between the blocks of masses m and 2m, given the conditions of the problem, is equal to the weight of the suspended block, which is mg. This analysis illustrates the importance of understanding the forces at play in a system and how they interact to maintain equilibrium.