Most system which have an equilibrium position execute simple harmonic motion about this position when they are displaced from equilibrium, as long as the displacements are small.  The restoring forces approximately obey Hooke's law.  However, for larger displacements the systems become anharmonic oscillators, i.e. the restoring forces are no longer proportional to the displacements.  The period then depends on the amplitude.  A familiar example of such a system is the simple pendulum.

An ideal simple pendulum consists of a point mass m suspended from a support by a massless string of length L.  (A good approximation is a small mass, for example a sphere with a diameter much smaller than L, suspended from a light string.)  The equilibrium position of the mass is a distance L below the support.

If the mass is displaced from its equilibrium position while keeping the string taut, it exhibits periodic motion, moving in a vertical plane along a circular arc.

Is this periodic motion simple harmonic motion?

When the string makes an angle q with the vertical, then the displacement of the mass from its equilibrium position along the circular arc is s = Lq.  The forces acting on the mass are gravity and the tension in the string.  Only gravity provides a restoring force towards the equilibrium position.  The magnitude of this force is mgsinq.  The equation of motion, F = ma, therefore yields

md²s/dt² = mgsinq,

or

d²q/dt² = -(g/L)sinq.

The solution to this equation describes periodic, but not simple harmonic motion.  However, when the displacement from equilibrium is small, then sinq » q and the equation of motion becomes

d²q/dt² = -(g/L)q.

The solution to this equation is q(t) = q_maxcos(wt+f), with w² = g/L.

For small oscillations the period of a simple pendulum is therefore given by .

It is independent of the mass m of the bob.  It depends only on the strength of the gravitational acceleration g and the length of the string L.  By measuring the length and the period of a simple pendulum we can determine g.

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Exercise (You can earn up to 5 points extra credit by completing this exercise.)

A physical pendulum is an object suspended in a uniform gravitational field from a point other than its center of mass.  The object can rotate about an axis through the suspension point.  When the CM is displaced from its stable equilibrium point under the support, the gravitational force exerts a torque about the support, resulting in angular acceleration.  The CM accelerates towards its equilibrium position.  The object exhibits periodic motion.  For small displacements, when sinq » q, the motion is simple harmonic, q(t) = q_maxcos(wt+f), with w² = (mgd)/I.  Here I is the moment of inertia of the object about the axis of rotation through the support and d is the perpendicular distance of the CM from the axis of rotation through the support.

Damped Oscillations

Friction will damp out the oscillations of a macroscopic system, unless the oscillator is driven.  If the speed of a mass on a spring is low, then the drag force R due to air resistance is approximately proportional to the speed, R = -bv.

The total force on the object then is

F = -kx - bv.

The equation of motion, F = ma, becomes

-kx - bdx/dt = md²x/dt².

The solution to this differential equation is

x(t) = Aexp(-bt/2m)cos(wt+f),   with w² = k/m - (b/2m)²,

as long as b² < 4mk, i.e. as long as the drag force is not too large.  The oscillatory character of the motion is preserved, but the amplitude decreases with time.

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The driven oscillator

A driving force with the natural resonance frequency of the oscillator can efficiently pump energy into the system.  Assume a driving force F = F₀coswt.  The total force on the object then is F = F₀coswt - kx - Rv.  The equation of motion, F = ma, becomes

F₀coswt - kx - bdx/dt = md²x/dt².

The solution to this differential equation is

x(t) = Acos(wt+f).

Here w is the angular frequency of the driving force.  The amplitude of the motion is given by

,

where w₀² = k/m, the natural frequency of the undamped oscillator.  When the frequency of the driving force is very close to the natural frequency, and the drag force is small, the denominator in the above expression becomes very small, and the amplitude becomes very large.  This dramatic increase in amplitude is called resonance, w = w₀ is called the resonance frequency.

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If we push a child on a swing once during each period, and we always push in the same direction, we will increase the amplitude of the swing, until the positive work we do on the swing equals the negative work done on the swing by the frictional forces.
If we drive a harmonic oscillator with a driving force with the natural resonance frequency of the oscillator, then the amplitude can increase enormously, even if the work done during each cycle is very small.  The amplitude of an ideal harmonic oscillator increases forever.  Friction limits the maximum amplitude of a real oscillator.  The amplitude may become large enough for the system to become an anharmonic oscillator.  Then the driving force is no longer in phase with the oscillations, and it sometimes does negative work and reduces the amplitude.  Often, however the amount of energy put into the system is large enough to damage or break the system.

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