Question about force?

Question

An inventive child wants to reach an apple in a tree without climbing the tree. Sitting in a chair connected to a rope that passes over a frictionless pulley (Fig. P4.79), the child pulls on the loose end of the rope with such a force that the spring scale reads 320 N. The child's true weight is 260 N, and the chair weighs 160 N.
Find the force the child exerts on the seat of the chair.

I know that the acceleration is 5.13 m/s^2 but how do I find the force the child exerts on the seat?? Thanks !

Answer 1

So the end of the rope in child's hand provides a lift force of 320N. The other end on the chair gives a lift force of 320N. The total weight of the child plus the chair is (260+160)N = 420N. The acceleration is a = (640N - 420N)*g/420N = (11/21)g. Taken g = 9.81m/s^2, a = 5.14m/s^2.
Assume the force the child exerts on the seat of the chair is W(N). Thus we know the child's acceleration is a, which can also be calculated as:
{320 - (260 - W)}*g/(260 - W) = a = (11/21)g
or: 21(60 + W) = 11(260 - W)
32W = 11*260 - 21*60 = 1600
W = 50 (N)
This is very reasonable, since both ends of the rope provide a lift force of 320N, lift a mass of 210N/g, and cause the same acceleration.