The Mass on a Spring

What does the period of oscillation of a mass on a spring depend on?

It is easy enough to do experiments with masses and springs to find out. You should discover the following.

Mass? Stiffness Amplitude?

 

We shall now derive an equation for the period of oscillation of a mass on a spring by considering the forces which act on it in different positions.

You should remember that when a force is applied to a spring its extension is proportional to this force. This means that when an object is suspended from a spring and displaced from its equilibrium position the restoring force will be proportional to its displacement. The acceleration of the mass, as we have seen, will also be proportional to its displacement but in the opposite direction.

F = - k y        where k is the spring constant            and              F = m a            so               - k y = m a      or        a = - (k/m) y

now compare these two equations. Notice how they agree with each other.      a = - (k/m) y    and    a = - ω2 y

our constant ω2 must be equal to k/m   and so     ω = 2 π f  = ( k / m )1/2  

The period     T = 1 / f = 2 π (m/k)1/2   This result is confirmed by experiment. The fact that the period is independent of the amplitude will be seen to be important later.

    

Examples

1. Calculate the period of oscillation of a 200g mass attached to a spring of stiffness 50 N/m

T = 2 π ( 0.2 / 50 )1/2 = 0.40s

2. A baby is put on a bouncer with a stiffness of 80N/m. It bounces up and down with a period of 1.9s. Calculate the mass of the baby.

Rearranging we get m =   k T2 / 4 π2    = 80 (1.9)2 / 4 π2   =  7.33 kg