6-12-98
We're going to change gears a little, and move away from systems where objects are either at rest, or are undergoing translational or rotational motion with constant acceleration, to look at simple harmonic motion (SHM). Simple harmonic motion is the oscillating motion associated with a swinging pendulum, or with a mass bouncing up and down on a spring. With simple harmonic motion, the acceleration of the object is continually changing; this makes it a little more complicated than the constant acceleration systems we've looked at up to this point. In simple harmonic motion, however, the acceleration changes in a regular, repeating, fashion, oscillating in the same manner as the object itself, which simplifies the analysis significantly.
Before jumping right in to an analysis of simple harmonic motion, and investigating the motion of an object such as a mass hanging on a spring when you pull the mass down and let it go, we should look at how things stretch (or contract) when you apply a force to them. An object such as a spring or rubber band, for example, will stretch when you pull it, and it will contract back when you let it go (unless you pull too far, that is). This is an example of elastic deformation; when you pull too far and the object doesn't return to it's original length, you have exceeded the elastic limit.
Most objects, even if they're made out of hard materials such as metal, will exhibit some elastic behavior. This is because the forces between the atoms making up the objects act like tiny springs, stretching and compressing in response to an applied force. When a force is applied, the object changes length according to the equation:
A is the cross-sectional area; L is the length; and Y is a constant known as Young's modulus, which depends on the material the object is made from.
If the force is tensile, like a tension force applied to a string (a pull, that is), the object stretches; if a compressive force (a push) is exerted, the object gets shorter. Note that to produce a given change in length, more force is required for shorter objects, objects with large cross-sectional areas, and objects with large values of Y. Metals, for example, have large Y values; rubber, has a Young's modulus which is much smaller.
There are other ways objects can be deformed in addition to simply changing in length. An object can experience shear; this is easiest to see by pushing sideways on a thick book. Objects can also change in volume, if they experience a force from all sides. A volume deformation is easiest to see when applying pressure; note that pressure is simply a force per unit area.
where S is the shear modulus, L is the thickness, and A is the area.
where V is the original volume and B is the bulk modulus.
The different kinds of deformation can all be related through the idea of stress and strain. Stress is the applied pressure, or a change in the applied pressure; strain is how the object changes shape in response to the stress. When the strain is directly proportional to the applied stress, the object behaves elastically, and will return to its original shape when the stress is removed. For elastic deformation, then, there is a linear relationship between the stress and strain. Outside of this linear region (beyond the elastic limit) the strain is no longer proportional to the applied stress.
Strain is a unitless quantity; stress has units of pressure (force / area). The pressure unit is known as the pascal. 1 Pa = 1 N / m2.
Hooke's law states that strain is proportional to stress, so it applies to elastic deformation, and to the equations concerning stretching, shear, and volume deformation. It also applies to a stretched spring.
An ideal spring is one to which Hooke's law applies; the amount an ideal spring
stretches is proportional to the applied force:
F = kx, where k is known as the spring constant.
k is a measure of how difficult it is to stretch a spring. The larger k is, the stiffer it is and the harder the spring is to stretch. Note that Hooke's law for springs is simply a modified version of the elastic deformation equation for length change.
If an object applies a force to a spring, the spring applies an equal and opposite force to the object. Therefore:
force applied by a spring : F = - kx
where x is the amount the spring is stretched. This is a restoring force, because when the spring is stretched, the force exerted by by the spring is opposite to the direction it is stretched. This accounts for the oscillating motion of a mass on a spring. If a mass hanging down from a spring is pulled down and let go, the spring exerts an upward force on the mass, moving it back to the equilibrium position, and then beyond. This compresses the spring, so the spring exerts a downward force on the mass, stopping it, and then moving it back to the equilibrium and beyond, at which point the cycle repeats. This is the kind of motion known as simple harmonic motion.
It might seem like we've started a topic that is completely unrelated to what we've done previously; however, there is a close connection between circular motion and simple harmonic motion. Consider an object experiencing uniform circular motion, such as a mass sitting on the edge of a rotating turntable. This is two-dimensional motion, and the x and y position of the object at any time can be found by applying the equations:
The motion is uniform circular motion, meaning that the angular velocity is constant, and the angular displacement is related to the angular velocity by the equation:
Plugging this in to the x and y positions makes it clear that these are the equations giving the coordinates of the object at any point in time, assuming the object was at the position x = r on the x-axis at time = 0:
How does this relate to simple harmonic motion? An object experiencing simple harmonic motion is traveling in one dimension, and its one-dimensional motion is given by an equation of the form
The amplitude is simply how far the object gets from the equilibrium position.
So, in other words, the same equation applies to the position of an object experiencing simple harmonic motion and one dimension of the position of an object experiencing uniform circular motion. Note that the 
in the SHM displacement equation is known as the angular frequency. It is related to the frequency (f) of the motion, and inversely related to the period (T):
The frequency is how many oscillations there are per second, having units of hertz (Hz); the period is how long it takes to make one oscillation.
In simple harmonic motion, the velocity constantly changes, oscillating just as the displacement does. When the displacement is maximum, however, the velocity is zero; when the displacement is zero, the velocity is maximum. It turns out that the velocity is given by:
The acceleration also oscillates in simple harmonic motion. If you consider a mass on a spring, when the displacement is zero the acceleration is also zero, because the spring applies no force. When the displacement is maximum, the acceleration is maximum, because the spring applies maximum force; the force applied by the spring is in the opposite direction as the displacement. The acceleration is given by:
For SHM, the oscillation frequency depends on the restoring force. For a mass on a spring, where the restoring force is F = -kx, this gives:
This is the net force acting, so it equals ma:
This gives a relationship between the angular velocity, the spring constant, and the mass:
Potential energy can be stored by compressing or stretching a spring.
The energy increases the more the spring's length is changed from its unstrained length:
In a perfect spring, no energy is lost; the energy is simply transferred back and forth between the kinetic energy of the mass on the spring and the potential energy of the spring (gravitational PE might be involved, too).
A simple pendulum is a pendulum with all the mass the same distance from the support point, like a ball on the end of a string. Gravity provides the restoring force (a component of the weight of the pendulum).
Summing torques, the restoring torque being the only one, gives:
For small angular displacements :
So, the torque equation becomes:
Whenever the acceleration is proportional to, and in the opposite direction as, the displacement, the motion is simple harmonic.
For a simple pendulum, with all the mass the same distance from the suspension point, the moment of inertia is:
The equation relating the angular acceleration to the angular displacement for a simple pendulum thus becomes:
This gives the angular frequency of the simple harmonic motion of the simple pendulum, because:
Note that the frequency is independent of the mass of the pendulum.