1. Oscillatory motion (chapter twelve)
Motion of a particle on a spring
Simple harmonic motion
Energy in SHM
Simple pendulum
Physical pendulum
Damped oscillations
Forced oscillations

3. Particle attached to a spring
We can model oscillatory motion as a mass attached to a spring (linear restoring force)
Causes displaced mass to to be restored to the equilibrium position.
Potential energy  Kinetic energy.
At equilibrium – large KE but force is now zero.
Newton’s first law - keeps moving.

4. Particle attached to a spring
We can use Newton’s 2nd law to quantitatively describe the motion
Acceleration proportional to displacement.
Opposite direction.

5. Simple Harmonic Motion
Defining the ratio k/m2, the equation of motion becomes (in one dimension)
This equation has the solution

6. SHM http://webphysics.davidson.edu/Applets/java10_Archive.html
A: amplitude of the motion (maximum displacement)
: =(k/m)½ – angular frequency of the motion
: phase – where the motion starts
A and  are set by the initial conditions,  is fixed by the mass and spring constant

7. SHM Period of one full cycle of motion:
Maximum velocity and acceleration:

8. Energy in SHM Kinetic energy: Potential energy:
Total energy of the system:
Total energy is constant!

9. Energy in SHM
Oscillation is repeated conversion of kinetic to potential energy and back.
Using the expression for the total energy, we can find the velocity as a function of position

10. The simple pendulum  L T Fg
Small angle approximation - sin

11. The simple pendulum
This equation has the same form as that for the motion of the mass attached to a spring. If we define
we get the exact same differential equation, and so the system will undergo the same oscillatory motion as we saw earlier.
Note – the frequency (and period) of the pendulum are independent of the mass!

12. The Physical Pendulum
An object hanging from a point other than its COM d 
COM

13. Damped Oscillations If we add in a velocity dependent resistive force
The solution to this DE when the resistive force is weak
This describes an underdamped oscillator

14. Damped Oscillations

15. Damped Oscillations The frequency of oscillation is
In other words, some natural frequency plus a change due to the damping
When b=2m, the system is critically damped (returns to equilibrium)
For b>2m, the system is overdamped – also returns to equilibrium (slower rate).

16. Forced Oscillations
If we try to drive an oscillator with a sinusoidally varying force:
The steady-state solution is
where 0=(k/m)½ is the natural frequency of the system.
The amplitude has a large increase near 0 - resonance

17. Forced Oscillations