# The Big Essay

There are four equations that you get to play with. You must recognize them, right? Here they are:

[pic]This baby relates wavelength and frequency to the velocity of the wave. Very important equation, not only now, but later on as well.

[pic]This one relates the period of a wave to the frequency – you can flip it (do a bit of cross multiplying) as well and get frequency equal to the inverse of the period.

[pic]This third one finds the period for an oscillating mass on a spring. The period is directly proportional to the mass and inversely proportional to the spring constant. Increase the mass, increase the period. Increase the spring constant value, decrease the period.

[pic]This fourth one is the period for a pendulum. The period is a function of the length of the thing. Increase the length and you increase the period. So the period is directly proportional to the length of the pendulum.

Here are the things that you are required to be able to do – turns out that there is a lot of stuff in this unit.

Oscillations

1. You should understand the kinematics of simple harmonic motion so you can:

a. Sketch or identify a graph of displacement as a function of time, and determine from such a graph the amplitude, period, and frequency of the motion.

This is not too difficult. We’ve done several of these type things in the previous units.

b. Identify points in the motion where the velocity is zero or achieves its maximum positive or negative value.

This is also a fairly simple task. Just examine the graph or drawing and pick off the required values.

c. State qualitatively the relation between acceleration and displacement.

The relationship is simple. The maximum acceleration occurs at the points of maximum displacement. The acceleration is least when the displacement is zero. The maximum acceleration coincides with the maximum restoring force.

d. Identify points in the motion where the acceleration is zero or achieves its greatest positive or negative value.

This is pretty much the same stuff as the thing above. Again, the maximum acceleration occurs at the points of maximum displacement. The acceleration is least when the displacement is zero.

e. State and apply the relation between frequency and period.

The relationship between frequency and period is given by:[pic]The nice thing is that the equation will be provided you. You can see that as the frequency gets larger, the period gets smaller and vice versa.

f. State how the total energy of an oscillating system depends on the amplitude of the motion, sketch or identify a graph of kinetic or potential energy as a function of time, and identify points in the motion where this energy is all potential or all kinetic.

The total energy of the system is set by the amplitude. This determines the potential energy the system has. When the mass is released, the potential energy is converted to kinetic energy. At zero displacement all the energy is kinetic. When the mass is at the maximum displacement (the amplitude) all the energy is kinetic. Between the amplitude and zero displacement the energy will be a combination of kinetic and potential energy.

A graph of kinetic energy, potential energy, and displacement Vs time looks like this:

Picking off the points where the energies are zero should be pie for a superior student such as yourself.

g. Calculate the kinetic and potential energies of an oscillating system as functions of time, sketch or identify graphs of these functions, and prove that the sum of kinetic energy and potential energy is constant.

You can use the equations for the potential energy of a spring and kinetic energy to work things out.

The potential energy of the spring is given by: [pic]

When the displacement is...

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