Demo Video: Simple Harmonic Motion (SHM) of a Spring

Video Overview

When a spring is stretched, it exerts a force that tries to pull the spring back to its equilibrium length. When a spring is compressed, it exerts a force that tries to push the spring back to its equilibrium length. The magnitude of the force, , exerted by the spring is proportional to the extension of the spring, :

Here, is a constant of proportionality known as the spring constant, which is dependent on the material used to make the spring as well as the shape of the spring.

If a spring is hung vertically and a mass is attached to its end, the weight of the mass will pull downward on the spring. If the mass is allowed to descend slowly, the spring will extend until the force exerted upward by the spring is equal in magnitude to the weight of the mass. At this point, the mass-and-spring system will reach a new equilibrium, where the mass is not moving and the weight of the mass is balanced by the force of the spring.

If, however, the mass is then pulled downward by hand, extending the spring even more, the upward force of the spring will now be greater than the weight of the mass. When the mass is released, there will be a net upward force on the mass. The mass will move upward, and the spring will decrease in length.

The system of the mass and the spring tries to return to its new equilibrium position. However, it will overshoot, as by the time the mass is at the new equilibrium position, it will have an upward velocity, which will carry it past the equilibrium position.

Gravity will then pull the mass down again, but the spring and mass will overshoot the new equilibrium position again. The mass and spring will keep overshooting their equilibrium position; they will oscillate.

The period, , of these oscillations depends only on the mass, , that is attached to the end of the spring and the spring constant, :

The period of the oscillations does not depend on their amplitude. If the spring and the mass are kept the same, they will oscillate with the same period whether the oscillations have a large amplitude or a small amplitude. The period of the oscillations also does not depend on the equilibrium length of the spring.

Consider a spring that has a spring constant of 110 N/m. If a 500 g mass is attached to its end and made to oscillate, what will the period of the oscillations be?

The period of the oscillations will be as follows:

Interestingly, the amplitude of the oscillations will gradually decrease over time. The kinetic energy of the mass and the spring is gradually lost, as the movement of the spring heats it. The mass will eventually stop at its equilibrium position. If the mass is then removed, the spring will return to its original equilibrium length.