1. Oscillations
Readings: Chapter 14

2. Oscillation: Periodic Motion
T – period of motion
- frequency

3. Oscillation: Periodic Motion: Simple Harmonic Motion
Simple Harmonic Motion – sinusoidal oscillation or
The most general expression for sinusoidal motion
- phase constant
- amplitude
If then

4. Oscillation: Periodic Motion: Simple Harmonic Motion
Example: Uniform circular motion
x- component:
- angular frequency

6. Oscillation: Periodic Motion: Simple Harmonic Motion
T – period of motion (units – s)
- amplitude
- frequency (units – hertz – Hz =1/s)
- phase constant
- angular frequency (units – rad/s)

7. - phase constant
Phase constant specifies the initial position of the oscillator

8. Simple Harmonic Motion: Position, Velocity, and Acceleration

9. Simple Harmonic Motion: object oscillating on a spring
Newton’s second law:
Hooke’s law:

10. Simple Harmonic Motion: Conservation of energy
Kinetic energy:
Potential energy:
Total energy:

12. Example: Find the relation between kinetic and potential energy at x=A/3.
The total energy is the sum of kinetic and potential energy. At x=A the kinetic energy is 0.
At x=A/3 the potential energy is
Then the kinetic energy at this point is
Then

13. Simple Harmonic Motion:
If force is proportional to displacement (it is not necessary the spring system)
then
Solution of this equation:

14. Oscillations about equilibrium position
Net force:
Oscillations about equilibrium position

15. The pendulum: small-angle approximation
The net force is the sum of two forces: tension and gravitational force.
Tangential component of the net force is
If y is the arc length then
If y<<l then
then

16. If the system has a friction then there is an energy loss
The energy is determined by the amplitude of oscillations, so the energy loss means that the amplitude is decreasing