### Introduction

We can describe an oscillating mass in terms of its position, velocity, and acceleration as a function of time. We can also describe the system from an energy perspective. In this experiment, you will measure the position and velocity as a function of time for an oscillating mass and spring system, and from those data, plot the kinetic and potential energies of the system.

Energy is present in three forms for the mass and spring system. The mass, m, with velocity, v, can have kinetic energy KE

$KE = \frac{1}{2}m{v^2}$

The spring can hold elastic potential energy, or PEelastic. We calculate PEelastic by using

$P{E_{elastic}} = \frac{1}{2}k{y^2}$

where k is the spring constant and y is the extension or compression of the spring measured from the equilibrium position.

The mass and spring system also has gravitational potential energy (PEgravitational = mgy), but we do not have to include the gravitational potential energy term if we measure the spring length from the hanging equilibrium position. We can then concentrate on the exchange of energy between kinetic energy and elastic potential energy.

If there are no other forces experienced by the system, then the principle of conservation of energy tells us that the sum ΔKE + ΔPEelastic = 0, which we can test experimentally.

### Objectives

• Examine the energies involved in simple harmonic motion.
• Test the principle of conservation of energy.