A box weighing 69 N rests on a table. A rope tied to the box runs vertically upward over a pulley and a weight is hung from the other end.

Determine the force that the table exerts on the box if the weight hanging on the other side of the pulley weighs one of the following:
a)29N
b)56N
c)96N

If the table is smooth, it should be N=69N normal to the table in all cases.

If the table is rough, with a coefficient of kinetic or static friction of μ, then it falls into two cases:
There is no motion, T<μN, T=tension in rope.
Then the force exerted on the box by the table is the normal reaction + static friction, equal to T.
If there is motion, the forces would be the normal reaction N, added (vectorially) to the kinetic friction force μN.

To determine the force that the table exerts on the box in each case, we need to understand the concept of equilibrium. In equilibrium, the sum of all the forces acting on an object is zero. This means that the upward force on the box (from the tension in the rope) must be equal to the downward force on the box (due to its weight) for it to remain stationary.

Let's analyze each case separately:

a) Weight hanging on the other side of the pulley weighs 29N:
In this case, the downward force on the box is 69N (its weight). According to the equilibrium condition, the upward force on the box must also be 69N. Since the tension in the rope is the only upward force acting on the box, it must be equal to 69N. Therefore, the force that the table exerts on the box is also 69N.

b) Weight hanging on the other side of the pulley weighs 56N:
Similarly, the downward force on the box is still 69N (its weight). According to equilibrium, the upward force on the box must be equal to 69N. However, in this case, the tension in the rope is not sufficient to balance the downward force fully. Therefore, the table must provide an additional upward force to maintain equilibrium. To find the force that the table exerts, we subtract the tension in the rope from the total force needed:
Force by the table = Total force needed - Tension in the rope
= 69N - 56N
= 13N

Therefore, the force that the table exerts on the box is 13N.

c) Weight hanging on the other side of the pulley weighs 96N:
The downward force on the box is still 69N (its weight). According to equilibrium, the upward force on the box must be equal to 69N. However, in this case, the tension in the rope is not sufficient to balance the downward force. Hence, the table needs to provide a greater upward force to maintain equilibrium. We can again calculate the force that the table exerts:
Force by the table = Total force needed - Tension in the rope
= 69N - 96N
= -27N

In this case, the calculated force by the table is negative, indicating that the table cannot provide enough upward force to maintain equilibrium. It implies that the box will not remain stationary and will instead move downward.

Therefore, the force that the table exerts on the box when the weight hanging on the other side of the pulley weighs 96N is -27N.