1. Oscillations and Simple Harmonic Motion:
Mechanics C

2. Oscillatory Motion is repetitive back and forth motion about an equilibrium position
Oscillatory Motion is periodic.
Swinging motion and vibrations are forms of Oscillatory Motion.
Objects that undergo Oscillatory Motion are called Oscillators.
Oscillatory Motion

3. Simple Harmonic Motion
The time to complete one full cycle of oscillation is a Period.
The amount of oscillations per second is called frequency and is measured in Hertz.
Simple Harmonic Motion

4. Heinrich Hertz produced the first artificial radio waves back in 1887!
What is the oscillation period for the broadcast of a 100MHz FM radio station?
Heinrich Hertz produced the first artificial radio waves back in 1887!

5. Heinrich Hertz produced the first artificial radio waves back in 1887!
What is the oscillation period for the broadcast of a 100MHz FM radio station?
Heinrich Hertz produced the first artificial radio waves back in 1887!

6. Simple Harmonic Motion
An objects maximum displacement from its equilibrium position is called the Amplitude (A) of the motion.
Simple Harmonic Motion

7. What shape will a velocity-time graph have for SHM?
Everywhere that the slope (first derivative) of the position graph is zero, the velocity graph crosses through zero.
What shape will a velocity-time graph have for SHM?
Either sine or cosine. The choice is up to where you call time = 0

8. We need a position function to describe the motion above.

9. Mathematical Models of SHM
x(t) to symbolize position as a function of time
A=xmax=xmin
When t=T,
cos(2π)=cos(0)
x(t)=A
Mathematical Models of SHM

10. Mathematical Models of SHM
In this context we will call omega Angular Frequency
What is the physical meaning of the product (Aω)?
The maximum speed of an oscillation!
Mathematical Models of SHM

11. Recall: Hooke’s Law
Here is what we want to do: DERIVE AN EXPRESSION THAT DEFINES THE DISPLACEMENT FROM EQUILIBRIUM OF THE SPRING IN TERMS OF TIME.
WHAT DOES THIS MEAN? THE SECOND DERIVATIVE OF A FUNCTION
THAT IS ADDED TO A CONSTANT TIMES ITSELF IS EQUAL TO ZERO.
What kind of function will ALWAYS do this?

12. Recall: Hooke’s Law
Here is what we want to do: DERIVE AN EXPRESSION THAT DEFINES THE DISPLACEMENT FROM EQUILIBRIUM OF THE SPRING IN TERMS OF TIME.
WHAT DOES THIS MEAN? THE SECOND DERIVATIVE OF A FUNCTION
THAT IS ADDED TO A CONSTANT TIMES ITSELF IS EQUAL TO ZERO.
What kind of function will ALWAYS do this?

14. What is the period of oscillation?
An airtrack glider is attached to a spring, pulled 20cm to the right, and released at t=0s. It makes 15 oscillations in 10 seconds.
What is the period of oscillation?
Example:

15. What is the period of oscillation?
An airtrack glider is attached to a spring, pulled 20cm to the right, and released at t=0s. It makes 15 oscillations in 10 seconds.
What is the period of oscillation?
Example:

16. What is the object’s maximum speed?
An airtrack glider is attached to a spring, pulled 20cm to the right, and released at t=0s. It makes 15 oscillations in 10 seconds.
What is the object’s maximum speed?
Example:

17. What is the object’s maximum speed?
An airtrack glider is attached to a spring, pulled 20cm to the right, and released at t=0s. It makes 15 oscillations in 10 seconds.
What is the object’s maximum speed?
Example:

18. What are the position and velocity at t=0.8s?
An airtrack glider is attached to a spring, pulled 20cm to the right, and released at t=0s. It makes 15 oscillations in 10 seconds.
What are the position and velocity at t=0.8s?
Example:

19. What are the position and velocity at t=0.8s?
An airtrack glider is attached to a spring, pulled 20cm to the right, and released at t=0s. It makes 15 oscillations in 10 seconds.
What are the position and velocity at t=0.8s?
Example:

20. A mass oscillating in SHM starts at x=A and has period T
A mass oscillating in SHM starts at x=A and has period T. At what time, as a fraction of T, does the object first pass through 0.5A?
Example:

21. A mass oscillating in SHM starts at x=A and has period T
A mass oscillating in SHM starts at x=A and has period T. At what time, as a fraction of T, does the object first pass through 0.5A?
Example:

22. When collecting lab data of SHM your mathematical model was:
What if your clock didn’t start at x=A or x=-A?
This value represents our initial conditions.
We call it the phase angle:
Model of SHM

23. SHM and Circular Motion
Uniform circular motion projected onto one dimension is simple harmonic motion.
SHM and Circular Motion

24. SHM and Circular Motion
Start with the x-component of position of the particle in UCM
Notice it started at angle zero
End with the same result as the spring in SHM!
SHM and Circular Motion

25. We will not always start our clocks at one amplitude.
Initial conditions:

26. Phi is called the phase of the oscillation
Phi naught is called the phase constant or phase shift. This value specifies the initial conditions.
Different values of the phase constant correspond to different starting points on the circle and thus to different initial conditions
The Phase Constant:

27. Phase Shifts:
x(t) = Acos(wt +ϕ)

28. An object on a spring oscillates with a period of 0
An object on a spring oscillates with a period of 0.8s and an amplitude of 10cm. At t=0s, it is 5cm to the left of equilibrium and moving to the left. What are its position and direction of motion at t=2s?
Initial conditions:
From the period we get:

29. An object on a spring oscillates with a period of 0
An object on a spring oscillates with a period of 0.8s and an amplitude of 10cm. At t=0s, it is 5cm to the left of equilibrium and moving to the left. What are its position and direction of motion at t=2s?
Initial conditions:
From the period we get:

30. An object on a spring oscillates with a period of 0
An object on a spring oscillates with a period of 0.8s and an amplitude of 10cm. At t=0s, it is 5cm to the left of equilibrium and moving to the left. What are its position and direction of motion at t=2s?