page 1 of 2

Page 1

Page 2

Now that we have established the theory and equations behind harmonic motion, we will examine various physical situations in which objects move in simple harmonic motion. Previously, we worked with a mass-spring system, and will examine other harmonic oscillators in a similar manner. Finally, after establishing these applications, we can examine the similarity between simple harmonic motion and uniform circular motion.

Consider a circular disk suspended from a wire fixed to a ceiling. If the disk is rotated, the wire will twist. When the disk is released, the twisted wire exerts a restoring force on the disk, causing it to rotate past its equilibrium point, twisting the wire the other direction, as shown below. This system is called a torsional oscillator.

Figure %: A torsional oscillator. The point P oscillates between the lines Q
and R with a maximum angular displacement of *θ*_{o}.

τ = - κθ |

where

- *κθ* = *Iα* = *I*

If we substitute θ = θ_{m}cos(σt) |

where

From our expression for angular frequency we can derive that

T = 2Π |

This equation for the period of a torsional oscillator has a significant experimental use. Suppose a body of unknown moment of inertia is placed on a wire of known constant

From our examination of the torsional oscillator we have derived that its motion
is simple harmonic. This oscillator can almost be seen as the rotational
analogue of the mass-spring system: just as with the mass-spring we substituted
*θ* for *x*, *I* for *m* and *κ* for *k*. Not all simple harmonic oscillators
have such close correlation.

Another common oscillation is that of the simple pendulum. The classic pendulum consists of a particle suspended from a light cord. When the particle is pulled to one side and released, it swings back past the equilibrium point and oscillates between two maximum angular displacements. It is clear that the motion is periodic--we want to see if it is simple harmonic.

We do so by drawing a free body diagram and examining the forces on the pendulum at any given time.

Figure %: A simple pendulum with cord of length *L*, shown with free body diagram
at a displacement of *θ* from the equilibrium point

F = - mg sinθ |

In this case the restoring force is

F = - mg = - x |

Thus we have an equation in the same form as our mass-spring equation; in this case

pendulum

T = 2Π = 2Π |

Note that the period, and thus the frequency, of the pendulum is independent of the mass of the particle on the cord. It only depends on the length of the pendulum and the gravitational constant. Keep in mind, also, that this is only an approximation. If the angle exceeds more than fifteen degrees or so, the approximation breaks down.

Page 1

Page 2

Take a Study Break!