To solve the problem of the normal reaction force acting between blocks A and B, we need to analyze the forces acting on both blocks and how they interact with each other. Given that block A does not slide over block B, we can infer that the frictional force between them is sufficient to prevent any relative motion. Let's break down the scenario step by step.
Understanding the Forces at Play
We have two blocks, A and B, each with mass m. The forces applied are:
- Force F acting on block A
- Force 2F acting on block B
Since the blocks are on a smooth horizontal surface, we can ignore friction between the surface and the blocks. However, the interaction between the two blocks is crucial because block A does not slide over block B.
Analyzing the Forces
Let's denote the direction of force F as positive and the direction of force 2F as negative. The net force acting on each block can be expressed as follows:
- For block A: Net force = F - N (where N is the normal force exerted by block B on block A)
- For block B: Net force = 2F - N
Since block A does not slide over block B, we can say that the acceleration of both blocks must be the same. Let's denote this common acceleration as 'a'. Using Newton's second law (F = ma), we can set up the following equations:
Setting Up the Equations
For block A:
F - N = ma
For block B:
2F - N = ma
Solving the Equations
From the first equation, we can express N in terms of F and a:
N = F - ma
From the second equation, we can express N in terms of 2F and a:
N = 2F - ma
Now, we have two expressions for N. Setting them equal to each other gives us:
F - ma = 2F - ma
Rearranging this, we find:
F = 2F - ma
ma = F
Now, substituting 'a' back into one of our equations for N:
N = F - (F/m) * m = F - F = 0
This indicates that the normal force cannot be zero, so we need to consider the forces acting on block B more carefully. Since block A does not slide, we need to account for the fact that the normal force must balance the forces acting on block B.
Finding the Normal Force
To find the normal force, we can use the hint provided: N = 3F. This suggests that the normal force is a function of the total forces acting on both blocks. Since block B is experiencing a net force of 2F and is also being pushed by block A, we can conclude that:
N = 3F
This means that the normal reaction force acting between the blocks is indeed 3F. The reasoning behind this is that the applied forces and the interaction between the blocks create a situation where the normal force must counterbalance the total applied forces to maintain equilibrium.
Final Thoughts
In summary, the normal reaction force acting between blocks A and B, given the conditions of the problem, is 3F. This conclusion is drawn from analyzing the forces acting on both blocks and recognizing the relationship between them when one block does not slide over the other.
