Question

A ball of mass m is resting on top of a vertical spring, with spring constant k and equilibrium length L. The spring is secured to the ground.

a) Draw a free-body diagram for the ball.

b) The ball rests at a height that is not equal to L. Find the equilibrium height of the ball.

c) Find an expression for the compression required to launch the ball to a height h from the ground. Write this compression in terms of distance from the shifted equilibrium in part b).

d) Lets submerge the spring-ball setup in some oil of density p and depth d. Draw a free-body diagram for the ball, assuming it lays at rest on top of the spring, and the spring is still securely fastened to the floor.

e) What compression is required to have the ball reach the surface of the oil? You may use the variable V for the ball’s volume.

f) What force did you neglect in part e) that makes your expression unrealistic? Explain how it would affect the compression required.

Answer 1

a)

b)

k = spring constant

x = compression of spring

m = mass of ball

using equilibrium of forces

kx = mg

x = mg/k

equilibrium height above the ground, h_eq = L - x = L - ( mg/k )

c)

x' = compression below the equilibrium height

using conservation of energy

spring potential energy + initial potential energy = final potential energy at height "h"

(0.5) k (x + x')² + m g (h_eq - x') = m gh

(0.5) k (x + x')² + m g (h_eq - x') = m gh

(0.5) k (x + x')² + m g (L - (mg/k) - x') = m gh