Masses, Springs, and Pendulums (Oh my!)


Mass and Spring Systems

We can easily see the periodic motion in a mass and spring system. Consider a mass with a pen attached to it hanging vertically from a string. If a piece of paper is pulled behind the mass, a straight line will be drawn. If however, the mass is given a push upwards or downwards, a sinusoidal curve will be traced out.

If we consider a mass/spring system which obeys Hooke's law (F = -kx), we find from energy conservation that:

1/2mv2max = 1/2kx2max

or

vmax = xmaxsqrt (k/m)

Since period, T, is found in periodic (circular) motion from the equation

vmaxT = 2pxmax

we find that the period of a harmonic oscillator is

T = 2p sqrt (m/k)

Knowing that w = vmax/xmax, we find that

w2 = (k/m)

This value, w, is called the natural angular frequency. It is related to something called the natural frequency, f, by the relation

w = 2pf

As these names imply, the frequencies depend on the physical properties of the system. Since amplitude isn't involved, any push or pull will give us motion with the same period/frequency. We can draw several conclusions from this:

  1. Amplitude doesn't affect the natural frequency
  2. More massive systems are slower to respond and thus have longer periods
  3. The stiffer the spring (bigger the k value), the shorter the period.


Pendulums

Pendulum VectorsThe pendulum is another system that can perform simple harmonic motion. A simple pendulum consists of a mass suspended by a light string of constant length, L, attached to a rigid support.

When the mass is displaced to the side through an angle q with the vertical, part of the gravitational force acts as a restoring force to push the pendulum bob back toward its equilibrium position:

Restoring Force (F) = - mg sin q

The displacement is just the arc length,

S = Lq

We can approximate sin q as q for small angles. (How much error is introduced for an angle of 10°?) Thus,

F = -mgq = - (mg/L)s

This is just like the situation we described for Hooke's law (F = -kx), where k is now mg/L. It is easy to determine then that the period is given by:

T = 2p sqrt(L/g) or w2 = g/L

So we see that the period of a pendulum is independent of the mass of the pendulum bob.

Historical Note:

Christiaan Huygens in the 1600's invented the pendulum clock after Galileo observed the constancy of the pendulum by swinging the chandeliers in the cathedral of Pisa. It was the standard method of keeping time for nearly three centuries!

For practice try:

13.20, 13.23, 13.24, 13.33, 13.35

Next up, damped and forced harmonic oscillators!