4.1.2.P19

A heavy block, labeled “A”, is sitting on a table. On top of that block is a lighter block, labeled “B” as shown in the figure at the right. For the first parts of this problem you are asked to identify the direction of forces in this system under various circumstances. In this problem, we will be looking at the relationships between the various forces in the problem under various circumstances. In order to simplify the equations we write, we will not use our full "who is acting on whom" force notation, but will use the following simplifications. (Note that we have taken for granted that you understood and could use Newton's 3rd law.)

• N_F→A = F  (force of the finger pushing block A)
• N_A→B = N_B→A = N (the normal forces acting between the blocks)
• N_T→A = N_T  (the normal force of the table acting on block A)
• f_T→A = f_T  (the friction force between block A and the table)
• f_A→B = f_B→A = f  (the friction force between the two blocks)
• W_E→A = W_A  (weight of block A)
• W_E→B = W_B  (weight of block B)

We then have the seven symbols representing all the forces in the problem: F, N_T, N, f_T, f, W_A, and W_B. 

(A) If the finger is pushing but not hard enough, neither block will move. By Newton's 2^nd law, for any object that is not accelerating, the forces in each direction must balance.

1. Write the equations for the balance of the forces in the horizontal and vertical directions for block A and for block B (four equations).
2. Three of the quantities are in principle straightforward to measure (using, say, a spring scale): F, W_A, and W_B. The other four, f_T, f, N_T, and N,  are "invisible" -- that is, a little bit harder to measure directly. (Though you might be able to think of a way!) If you knew the three easily measurable quantities, could you find the other four (invisible) ones? If so, write equations to  express each invisible one in terms of the three measurable ones. If not, explain what else you would need.

(B) If the finger is pushing hard enough, the two blocks will start to speed up. Assume they speed up together without slipping. By Newton's 2^nd law, for any object that is accelerating, ma in each direction must equal the net forces in that direction.

1. Write the equations for ma in the horizontal and vertical directions for block A and for block B (four equations).
2. In addition to the three quantities that are in principle straightforward to measure (F, W_A, and W_B), the accelerations a_A and a_B can be easily measured (say using a video capture program)_. If you knew the accelerations and the three easily measurable quantities, could you find the other four (invisible) ones? If so, write equations to  express each invisible one in terms of the three known quantities. If not, explain what else you would need.
3. Compared to part (A), we have new quantities, but the same number of equations. How is this possible? What changes between the two situations?

(C) Once the blocks have sped up, the finger still has to push on it to keep it going at a constant velocity, even though N2 tells us that at a constant velocity, all forces must balance. Assume that both blocks move with the same speed and don't slide.

1. Write the equations that express the balance of forces for each block and each direction. 
2. Suppose we assume that the finger is pushing harder on the blocks in part (C) than it did in part (A) but not as hard as it was in part (B). Which of the other 6 forces have to change compared to what they were in part (B)? If they do change, do they each get bigger? small? go to zero?

Joe Redish 8/19/15 

Joe Redish 8/17/15