1. Physics 243
... Vibrations and Waves ...

2. Phys 243: Vibrations and Waves
Main Ideas
Hooke’s Law
Energy Conservation applied to a spring
Spring as an exemplar of other systems
Simple Harmonic Motion
Damped versus Free Motion
Phys 243: Vibrations and Waves

3. Phys 243: Vibrations and Waves
Hooke’s Law becomes:
The function x(t) is the displacement-time function
for the spring-mass oscillator
Phys 243: Vibrations and Waves

4. Phys 243: Vibrations and Waves
Our intuition tells us that x(t) looks like:
Which can be described by the equation...
angular frequency
amplitude
How can we show that this really is so?
Phys 243: Vibrations and Waves

5. Phys 243: Vibrations and Waves
Feed
into
How can we get the left and right hand sides of the equation to match?
Phys 243: Vibrations and Waves

6. “grind out the derivatives and equate both sides”
Phys 243: Vibrations and Waves

7. Velocity and acceleration
How can we derive the following?
Phys 243: Vibrations and Waves

8. Velocity and acceleration
Phys 243: Vibrations and Waves

9. The Simple (and not so simple) Pendulum
A pendulum can, under appropriate circumstances act as a simple harmonic oscillator
The sine component of W provides a restoring torque
Phys 243: Vibrations and Waves

10. Phys 243: Vibrations and Waves
This can be expressed...
Recall
but, if L = length of string then:
Phys 243: Vibrations and Waves

11. Phys 243: Vibrations and Waves
Re-arrange to give:
Now, notice the similarity with
Phys 243: Vibrations and Waves

12. Phys 243: Vibrations and Waves
Important step...
If the angle is small - less than about 0.1 radians (7 degrees) then
We can re-write to get:
Phys 243: Vibrations and Waves

13. Phys 243: Vibrations and Waves
From this we can
Find an expression for the period of a simple pendulum... Try it!
Phys 243: Vibrations and Waves

14. General equation for SHM...
Whenever the acceleration in the system is directly proportional and opposed to the displacement, SHM will result:
Acceleration = - (Displacement)
Some constant
“opposed to”
Phys 243: Vibrations and Waves