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), like the one above, we find from energy conservation that:

1/2mv2max = 1/2kx2max


vmax = xmaxsqrt (k/m)

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

vmaxT = 2p xmax

we find that the period of a harmonic oscillator is

T = 2p sqrt (m/k)

Knowing that T = 2p / w, we find that

w2 = (k/m)

This value, w, is called the natural angular frequency of the system. We can also talk about a natural frequency, f, given by w = 2p f.

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.


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 above, 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!


Physical Pendulums

A pendulum doesn't have to consist of a mass swinging on a string. Anything that has mass and swings can be considered a pendulum. Your leg is a perfect example of the so-called "physical pendulum."

It can be shown* that for a physical pendulum, the period is given by:

T = 2 p sqrt (I / mgh)

where I is the moment of inertia of the body and h is the distance from the pivot point to the center of mass.

If we assume that a leg of length L may be approximated by a thin rod hinged at one end, with I = mL2/3, and we assume that the center of mass of a leg is at its middle h = L/2, we find that the period is given by:

T = 2 p sqrt (2L/3g)

In class we will use this information to measure our natural gait or stride.

Center of Oscillation

If we consider a physical pendulum to be a regular pendulum, we discover a strange place:

2 p sqrt (I / mgh) = 2 p sqrt (L / g)

L = I/mh

If the physical pendulum were a real pendulum it would have a length related to its moment of inertia, mass, and distance of the center of mass from the pivot point. This distance from the pivot point (call it O) is called the center of oscillation or center of percussion (call it C). This has a couple of interesting properties:

  1. If the pendulum is pivoted at C instead of O, it will oscillate with the same period as before and O will be the new center of oscillation.
  2. If the pendulum is struck along a line of action through C, there will be no reaction force on the pivot at O. For example a baseball that strikes a bat at the the center of oscillation does not produce a sting in the batter's hands.

For homework in this chapter, keep working on Page 306, Multiple Choice questions 1 - 26.

Next up, a quick look at damped harmonic oscillators!