1. Elasticity and Oscillations
Lecture 3

2. Goals Review Kinetic and Potential Energy
Gain an understanding of Simple Harmonic Motion (SHM)
Understand Period and Frequency as related to SHM

3. Review of Energy
A good equation to start with when working energy problems is the following:
If there are no outside forces (gravity and spring forces are not outside forces), set Wnc equal to zero

4. Lets try an Example
Consider a sled at the top of a frictionless hill 20 meters high. Once it reaches the horizontal bottom friction is present, how far does it slide if the coefficient of kinetic friction is .1?

6. Lets look at a Spring Example
Consider a 5 kg block moving horizontally with speed 2 m/s. It collides directly with a spring that has a 100 N/m constant. How far does the spring compress?

8. Introduction to Simple Harmonic Motion (SHM)
SHM is a special type of motion that occurs whenever the restoring force is proportional to the displacement from equilibrium
Many types of natural phenomena exhibit this motion
This includes springs and pendulums to a reasonable order

9. Still Confused?
If you are still confused it may be helpful to look at the result of SHM
The displacement from equilibrium will be a sine wave, the offset is determined by the starting location

10. Lets look at the Result of the Motion

11. Period and Frequency as Related to SHM
Looking at the previous drawing we can see that the period is the time for the object to leave and then return to the start location
The frequency is the inverse of the period and tells us how many cycles occur in a second, it has the units of inverse seconds or Hertz (HZ)

12. SHM for a Spring
For a spring vibrating starting from equilibrium we know the following holds
Angular frequency is 2pi divided by the period, or multiplied by the frequency

13. Frequency of Motion for the a Spring in SHM
Due to the periodic nature of simple harmonic motion, circular motion can be used to produce theoretical results
For SHM observed for spring motion, this can give the following result

14. Velocity and Acceleration in SHM

15. Another SHM Oscillator is the Vertical Spring
A spring that is hung vertically rather than horizontally also exhibits SHM, the only difference is the equilibrium position

16. Lets See the Difference

17. Pendulums are can show SHM
To a degree, pendulums also act as Simple Harmonic Oscillators
Both the Simple and Physical Pendulum can be modeled this way if the angle is small
Review the Physical Pendulum in your book

18. Lets see why we can Approximate the Pendulum’s Motion as SHM

19. Lets Finally Derive the Period for a Simple Pendulum

20. Simple Harmonic Motion Derivation
Frestore = Fparallel = - m g sin θ
sin θ ≈ θ
The angle is also related to the displacement by θ = y/L

21. Simple Pendulum Derivation Continued
Applying Newton’s Second Law:
This is analogous to the restoring force for a spring, we can therefore replace the spring constant for SHM observed in springs to give the frequency of a simple pendulum

22. Additional Notes