1. Lecture Outline Chapter 13 Physics, 4th Edition James S. Walker
Copyright © 2010 Pearson Education, Inc.

2. Oscillations about Equilibrium
Chapter 13
Oscillations about Equilibrium

3. Units of Chapter 13 Periodic Motion Simple Harmonic Motion
Connections between Uniform Circular Motion and Simple Harmonic Motion
The Period of a Mass on a Spring
Energy Conservation in Oscillatory Motion

4. Units of Chapter 13 The Pendulum Damped Oscillations
Driven Oscillations and Resonance

5. 13-1 Periodic Motion
Period: time required for one cycle of periodic motion
Frequency: number of oscillations per unit time
This unit is called the Hertz:

6. 13-2 Simple Harmonic Motion
A spring exerts a restoring force that is proportional to the displacement from equilibrium:

7. 13-2 Simple Harmonic Motion
A mass on a spring has a displacement as a function of time that is a sine or cosine curve:
Here, A is called the amplitude of the motion.

8. 13-2 Simple Harmonic Motion
If we call the period of the motion T – this is the time to complete one full cycle – we can write the position as a function of time:
It is then straightforward to show that the position at time t + T is the same as the position at time t, as we would expect.

9. 13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion
An object in simple harmonic motion has the same motion as one component of an object in uniform circular motion:

10. 13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion
Here, the object in circular motion has an angular speed of
where T is the period of motion of the object in simple harmonic motion.

11. 13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion
The position as a function of time:
The angular frequency:

12. 13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion
The velocity as a function of time:
And the acceleration:
Both of these are found by taking components of the circular motion quantities.

13. 13-4 The Period of a Mass on a Spring
Since the force on a mass on a spring is proportional to the displacement, and also to the acceleration, we find that .
Substituting the time dependencies of a and x gives

14. 13-4 The Period of a Mass on a Spring
Therefore, the period is

15. 13-5 Energy Conservation in Oscillatory Motion
In an ideal system with no nonconservative forces, the total mechanical energy is conserved. For a mass on a spring:
Since we know the position and velocity as functions of time, we can find the maximum kinetic and potential energies:

16. 13-5 Energy Conservation in Oscillatory Motion
As a function of time,
So the total energy is constant; as the kinetic energy increases, the potential energy decreases, and vice versa.

17. 13-5 Energy Conservation in Oscillatory Motion
This diagram shows how the energy transforms from potential to kinetic and back, while the total energy remains the same.

18. 13-6 The Pendulum
A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass).
The angle it makes with the vertical varies with time as a sine or cosine.

19. 13-6 The Pendulum
Looking at the forces on the pendulum bob, we see that the restoring force is proportional to sin θ, whereas the restoring force for a spring is proportional to the displacement (which is θ in this case).

20. 13-6 The Pendulum
However, for small angles, sin θ and θ are approximately equal.

21. 13-6 The Pendulum
Substituting θ for sin θ allows us to treat the pendulum in a mathematically identical way to the mass on a spring. Therefore, we find that the period of a pendulum depends only on the length of the string:

22. 13-6 The Pendulum
A physical pendulum is a solid mass that oscillates around its center of mass, but cannot be modeled as a point mass suspended by a massless string. Examples:

23. 13-6 The Pendulum
In this case, it can be shown that the period depends on the moment of inertia:
Substituting the moment of inertia of a point mass a distance l from the axis of rotation gives, as expected,

24. 13-7 Damped Oscillations
In most physical situations, there is a nonconservative force of some sort, which will tend to decrease the amplitude of the oscillation, and which is typically proportional to the speed:
This causes the amplitude to decrease exponentially with time:

25. 13-7 Damped Oscillations
This exponential decrease is shown in the figure:

26. 13-7 Damped Oscillations
The previous image shows a system that is underdamped – it goes through multiple oscillations before coming to rest. A critically damped system is one that relaxes back to the equilibrium position without oscillating and in minimum time; an overdamped system will also not oscillate but is damped so heavily that it takes longer to reach equilibrium.

27. 13-8 Driven Oscillations and Resonance
An oscillation can be driven by an oscillating driving force; the frequency of the driving force may or may not be the same as the natural frequency of the system.

28. 13-8 Driven Oscillations and Resonance
If the driving frequency is close to the natural frequency, the amplitude can become quite large, especially if the damping is small. This is called resonance.

29. Summary of Chapter 13
Period: time required for a motion to go through a complete cycle
Frequency: number of oscillations per unit time
Angular frequency:
Simple harmonic motion occurs when the restoring force is proportional to the displacement from equilibrium.

30. Summary of Chapter 13
The amplitude is the maximum displacement from equilibrium.
Position as a function of time:
Velocity as a function of time:

31. Summary of Chapter 13 Acceleration as a function of time:
Period of a mass on a spring:
Total energy in simple harmonic motion:

32. Summary of Chapter 13 Potential energy as a function of time:
Kinetic energy as a function of time:
A simple pendulum with small amplitude exhibits simple harmonic motion

33. Summary of Chapter 13 Period of a simple pendulum:
Period of a physical pendulum:

34. Summary of Chapter 13
Oscillations where there is a nonconservative force are called damped.
Underdamped: the amplitude decreases exponentially with time:
Critically damped: no oscillations; system relaxes back to equilibrium in minimum time
Overdamped: also no oscillations, but slower than critical damping

35. Summary of Chapter 13
An oscillating system may be driven by an external force
This force may replace energy lost to friction, or may cause the amplitude to increase greatly at resonance
Resonance occurs when the driving frequency is equal to the natural frequency of the system