1. Vibrations and Waves

2. Simple Harmonic Motion
Vibrations about an equilibrium position which a restoring force is proportional to the displacement from equilibrium
What does this mean?
Periodic motion is back and forth over the same path.

3. Simple Harmonic Motion
Equilibrium position, velocity reaches a maximum.
Let’s look at an example.
Spring

4. Simple Harmonic Motion
Equilibrium position is when the spring is unstretched
When a spring has a mass attached at the end

5. Simple Harmonic Motion
If the spring is stretched and released, the spring exerts a force on the mass towards the equilibrium position.

6. Simple Harmonic Motion
The spring force decrease, as the spring reaches the equilibrium position. Eventually reaching zero.

7. Simple Harmonic Motion
The mass’s acceleration also reaches zero at equilibrium.

8. Simple Harmonic Motion
However, the velocity of the mass increases as it approaches equilibrium. Reaching a maximum velocity at equilibrium.

9. Simple Harmonic Motion
At maximum displacement, spring force and acceleration reach a maximum.

10. Simple Harmonic Motion
This can occur in the opposite direction (compression).

11. Simple Harmonic Motion
Spring’s compression is equal to the distance the spring was stretched.

12. Simple Harmonic Motion
Ideally, the mass-spring system would oscillate indefinitely.

13. Simple Harmonic Motion
But friction slows, the motion and eventually the mass-spring will come to a rest. (Damping)

14. Simple Harmonic Motion
Restoring force- spring pushes and pulls a mass back to equilibrium.

15. Simple Harmonic Motion
Restoring force- directly proportional to the displacement of the mass.

16. Hooke’s Law F elastic = -kx
Spring force = - (spring constant)(displacement)
Negative sign signifies direction of the spring force opposite the direction of the mass displacement.

17. Hooke’s Law K = spring constant (always positive)
Measure of the stiffness of the spring
Greater value = stiffer the spring
SI units of K is N/m

18. Simple Harmonic Motion
All systems we will consider mechanical energy is conserved.
PE initial = KE final

19. Simple Pendulum
Pendulum consist of a mass (bob) attached to a fixed string
Disregard friction and the mass of the string

20. Simple Pendulum Is a simple pendulum, simple harmonic motion?
What is the restoring force?
Is the restoring force proportional to the displacement?
If so, then YES!

21. Simple Pendulum Is a simple pendulum, simple harmonic motion?
What is the restoring force?
Is the restoring force proportional to the displacement?
If so, then YES!

22. Simple Pendulum
Does the force acting on the bob act as the restoring force?
Force exerted by the string acts along the y-axis
Any point other than equilibrium, bob’s weight can be resolved into two components

23. Simple Pendulum
Does the force acting on the bob act as the restoring force?
Force exerted by the string and the y component of the bob’s weight are perpendicular to the motion
X component of the bob’s weight is the net force acting on the bob in the direction of motion.

24. Simple Pendulum
Does the force acting on the bob act as the restoring force?
Therefore, the X component pushes or pulls the bob towards equilibrium
It is a restoring force.

25. Simple Pendulum Restoring force of a SP is SHM
Small angles, the pendulum is SHM
Less than 15°