Given the figure shown below, block A has a mass of 10kg and block B has a mass of 25kg. Assume the coefficient of friction between the blocks is four times the coefficient of friction between the block and the plane. Determine the normal force between block A and B.
We can approach this problem by first considering the forces acting on each block individually.
For block A, the forces acting on it are its weight (mg) and the friction force due to the surface of the plane (f). The normal force (N) acting on block A is equal to the weight of block A plus the weight of block B (since block B is resting on top of block A), or N = (10+25)g = 35g.
For block B, the forces acting on it are its weight (mg), the normal force from block A (N), and the friction force due to the interface between block A and block B (f'). Note that the friction force between block A and the plane does not affect block B, since block B is not in contact with the plane.
Now, we are given that the coefficient of friction between the blocks is four times the coefficient of friction between block A and the plane. Let us denote the coefficient of friction between block A and the plane as µ and the coefficient of friction between the blocks as 4µ. Then, we have:
f = µN = µ(35g)
f' = 4µN = 4µ(35g)
Since the blocks are not accelerating vertically, the normal force N must be equal to the combined weight of the blocks: N = (10+25)g = 35g.
To determine the friction forces, we need to compare them to the maximum possible friction forces. The maximum possible friction force for block A is µN, while the maximum possible friction force for block B is µ'B, where µ' is the coefficient of friction between the blocks and B is the weight of block B. Thus:
f ≤ µN = µ(35g)
f' ≤ µ'B = 4µ(25g)
Note that we use 4µ instead of µ for the friction coefficient between the blocks, as given in the problem statement.
The blocks will begin to move if the friction forces exceed the maximum possible friction forces. Therefore, we have:
f + f' = (µ + 4µ)(35g) = 5µ(35g) ≤ (10+25)g
Simplifying this inequality, we get:
µ ≤ 3/(35*5)
µ ≤ 0.0171
Therefore, the normal force between block A and B is 35g, and the coefficient of friction between block A and the plane must be less than or equal to 0.0171 for the system to remain stationary.