1. AP Physics Lecture Notes
Vibrations and
Waves m
AP Physics Lecture Notes
Vibrations and Waves

2. Units of Chapter 11 Simple Harmonic Motion
Energy in the Simple Harmonic Oscillator
The Period and Sinusoidal Nature of SHM
The Simple Pendulum
Vibrations and Waves

3. Simple Harmonic Motion
If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic (T). m
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 ).
X = 0
Vibrations and Waves

4. Simple Harmonic Motion
x = 0
The force exerted by the spring depends on the displacement: m x F
The minus sign on the force indicates that it is a restoring force – it is directed to restore the mass to its equilibrium position.
Vibrations and Waves

5. Simple Harmonic Motionx F
(a) (k) is the spring constant
(b) Displacement (x) is measured from the equilibrium point
(c) Amplitude (A) is the maximum displacement
(d) A cycle is a full to-and-fro motion
(e) Period (T) is the time required to complete one cycle
(f) Frequency (f) is the number of cycles completed per second
Vibrations and Waves

6. Simple Harmonic Motion
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. xo m
Equilibrium
Position mg
Vibrations and Waves

7. 11-1 Simple Harmonic Motion
Any vibrating system where the restoring force is proportional to the negative of the displacement
moves with simple harmonic motion (SHM), and is often called a simple harmonic oscillator.
Vibrations and Waves

8. Energy in the Simple Harmonic Oscillator
Potential energy of a spring is given by:
The total mechanical energy is then:
The total mechanical energy will be conserved
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9. Energy in the Simple Harmonic Oscillator
If the mass is at the limits of its motion, the energy is all potential. m A m
x = 0
vmax
If the mass is at the equilibrium point, the energy is all kinetic.
Vibrations and Waves

10. Energy in the Simple Harmonic Oscillator
The total energy is, therefore
And we can write:
This can be solved for the velocity as a function of position:
where
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11. The Period and Sinusoidal Nature of SHM
vmax
If we look at the projection onto the x axis of an object moving in a circle of radius A at a constant speed vmax, we find that the x component of its velocity varies as: v A q x
This is identical to SHM.
Vibrations and Waves

12. The Period and Sinusoidal Nature of SHM
Therefore, we can use the period and frequency of a particle moving in a circle to find the period and frequency:
Vibrations and Waves

13. The Period and Sinusoidal Nature of SHM
The acceleration can be calculated as function of displacement m x F
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14. The Simple Pendulum
A simple pendulum consists of a mass at the end of a lightweight cord. We assume that the cord does not stretch, and that its mass is negligible.
Vibrations and Waves

15. Simple Pendulum x q L k for SHM Small angles x  s m F q mg s
Vibrations and Waves