Mastering Physics Solutions

A horizontal spring, resting on a frictionless tabletop, is stretched 18 cm from its unstretched configuration and a 1.00kg mass is attached to it. The system is released from rest. A fraction of a second later, the spring finds itself compressed 3.7 cm from its unstretched configuration.

How does its final potential energy compare to its initial potential energy? (Give your answer as a ratio, final to initial.)
Express your answer using two significant figures.

Spring potential energy is given by the formula U = 1/2kx². So we can set up a ratio by solving for the final and initial spring potential energies and then dividing. Remember that stretching or compressing doesn’t matter – the sign will disappear when the x is squared.

The final spring potential energy (3.7cm compressed):

U_f = 1/2kx_f²
U_f = 1/2k(0.037)²
U_f = 0.0006845k

And the initial spring potential energy (1.8cm stretched):

U_i = 1/2kx_f²
U_i = 1/2k(-0.18)²
U_i = 0.0162k

Now set up the ratio U_f / U_i:

U_f / U_i = 0.0006845k / 0.0162k
U_f / U_i = 0.042k / k
U_f / U_i = 0.042

Mastering physics likes things like this in scientific notation, so:

U_f / U_i = 4.2 * 10^-2