1. Oscillations
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2. Periodic Motion
Periodic motion is that motion in which a body moves back and forth over a fixed path, returning to each position and velocity after a definite interval of time.
Period, T, is the time for one complete oscillation. (seconds,s)
AmplitudeA
Frequency, f, is the number of complete oscillations per second. Hertz (s-1)

3. Simple Harmonic Motion, SHM
Simple harmonic motion is periodic motion in the absence of friction and produced by a restoring force that is directly proportional to the displacement and oppositely directed. x F
A restoring force, F, acts in the direction opposite the displacement of the oscillating body.
F = -kx

4. Simple Harmonic Motion—Spring Oscillations
We assume that the surface is frictionless. There is a point where the spring is neither stretched nor compressed; this is the equilibrium position. We measure displacement from that point (x = 0 on the previous figure).

5. Simple Harmonic Motion—Spring Oscillations
The minus sign on the force indicates that it is a restoring force—it is directed to restore the mass to its equilibrium position.
k is the spring constant
The force is not constant, so the acceleration is not constant either

6. Springs are like Waves and Circles
The amplitude, A, of a wave is the same as the displacement ,x, of a spring. Both are in meters.
CREST
Equilibrium Line
Period, T, is the time for one revolution or in the case of springs the time for ONE COMPLETE oscillation (One crest and trough). Oscillations could also be called vibrations and cycles. In the wave above we have 1.75 cycles or waves or vibrations or oscillations.
Trough

7. Simple Harmonic Motion—Spring Oscillations
If the spring is hung vertically, the only change is in the equilibrium position, which is at the point where the spring force equals the gravitational force.

8. Displacement in SHM m
x = 0
x = +A
x = -A x
Displacement is positive when the position is to the right of the equilibrium position (x = 0) and negative when located to the left.
The maximum displacement is called the amplitude A.

9. Velocity in SHM m v (-) v (+)
x = 0
x = +A
x = -A
Velocity is positive when moving to the right and negative when moving to the left.
It is zero at the end points and a maximum at the midpoint in either direction (+ or -).

10. Acceleration in SHM m -x +x
x = +A
x = -A
Acceleration is in the direction of the restoring force. (a is positive when x is negative, and negative when x is positive.)
Acceleration is a maximum at the end points and it is zero at the center of oscillation.

11. Acceleration vs. Displacementx v a m
x = 0
x = +A
x = -A
Given the spring constant, the displacement, and the mass, the acceleration can be found from: or
Note: Acceleration is always opposite to displacement.

12. Energy in Simple Harmonic Motion
We already know that the potential energy of a spring is given by: U = ½ kx2 The total mechanical energy is then: The total mechanical energy will be conserved, as we are assuming the system is frictionless.

13. Energy in Simple Harmonic Motion
If the mass is at the limits of its motion, the energy is all potential. If the mass is at the equilibrium point, the energy is all kinetic. We know what the potential energy is at the turning points:
(11-4a)

14. Energy in Simple Harmonic Motion
The total energy is, therefore ½ kA2 And we can write: This can be solved for the maximum velocity which is given by making total energy equal to only Kinetic:

15. The Period and Sinusoidal Nature of SHM
If we use calculus we can find that the period of a mass and ideal spring to be:

16. 11-3 The Period and Sinusoidal Nature of SHM
We can similarly find the position as a function of time (note the diagram is 90 degrees out of phase):
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17. The Period and Sinusoidal Nature of SHM
The top curve is a graph of the previous equation. The bottom curve is the same, but shifted ¼ period so that it is a sine function rather than a cosine.

18. The Period and Sinusoidal Nature of SHM
The velocity and acceleration can be calculated as functions of time; the results are below, and are plotted at left.