How much energy (in J) is stored in the spring under this circumstance?

Question

A 38 kg child uses a pogo stick to bounce up and down. The spring constant, k, of the toy equals 9,300 N/m.



(a) By how much would the spring be compressed by the child if she simply balanced herself vertically on the pedals of the stick? (Give your answer in cm.)

(b) How much energy (in J) is stored in the spring under this circumstance?

Answer 1

(a) Hooke's Law for springs states that F = k * x, where F is the Force acting on the spring, k is the spring constant, and x is the distance that the spring moves.

Start with F = k * x

To find the compression distance, solve for x: divide both sides by k. Then we see:

x = F / k = F * 1/k

The force that compresses the spring is the child's weight. Therefore F = F_{g, child} = mg. Substitute in for F:

x = m*g * 1/k

The mass of the child is given as m=38 kg. We know g=9.8 m/s² because the child is on Earth. The spring constant is given as k = 9300 N/m. Substitute these in:

x = 38 * 9.8 * 1/9300 = 0.040 meters.

Convert from meters to centimeters using dimensional analysis:

0.040 meters * 100 cm / 1 meter = 4.0 cm. Therefore x = 4.0 cm.

(b) The elastic potential energy stored in a spring is equal to E_sp = 1/2 * k * x², where k is the given spring constant and x is the compression distance we found in part a.

W = 1/2 * 9300 * (4.0)² = 74,400 J