Question

A child's toy consists of a m = 36 g monkey suspended from a spring of negligible mass and spring constant k. When the toy monkey is first hung on the spring and the system reaches equilibrium, the spring has stretched a distance of x = 11.7 cm. This toy is so adorable you pull the monkey down an additional d = 5.7 cm from equilibrium and release it from rest, and smile with delight as it bounces playfully up and down.
Part (a) Using the given information, determine the spring constant, k, in Newtons per meter, of the spring.

Part (b) Draw the free-body diagram that best represents the forces acting on the monkey as you are pulling ti down, immediately before you let go.

Part (c) Calculate the potential energy, E
b
o
t
t
o
m
, in joules, stored in the stretched spring immediately before you release it.

Part (d) Assume that the system has zero gravitational potential energy at the lowest point of the motion. Derive an expression for the total mechanical energy, E
e
q
u
i
l
i
b
r
i
u
m
, of the system as the monkey passes through the equilibrium position in terms of m, x, d, g, k and the speed of the monkey, v
e
.

Answer 1

Answer:

Part A - 3N/m

Part B - see attachment

Part C - 4.9 × 10-³J

Part D - E = 1/2kd² + 1/2mv² + mgh

Explanation:

This problem requires the knowledge of simple harmonic motion for cimplete solution. To find the spring constant in part A the expression relating the force applied to a spring and the resulting stretching of the spring (hooke's law) is required which is F = kx.

The free body diagram can be found in the attachment. Fp(force of pull), Ft(Force of tension) and W(weight).

The energy stored in the pring as a result of the stretching of d = 5.7cm is 1/2kd².

Part D

Three forces act on the spring-monkey system and they do work in different forms: kinetic energy 1/2mv² , elastic potential

energy due to the restoring force in the spring or the tension force 1/2kd², and the gravitational potential energy mgh of the position of the system. So the total energy of the system E = 1/2kd² + 1/2mv² + mgh.