1. Simple Harmonic Motion
Physics 12

2. Comprehension Check
Determine the orbital radius and speed for satellites with the following periods:
28 days
1 week
1 day
2 hours
Determine the orbital speed and period for satellites with the following altitudes:
250km
1000km
6400km

3. Comprehension Check
Determine the orbital radius and speed for satellites with the following periods:
28 days
1 week
1 day
2 hours
Determine the orbital speed and period for satellites with the following altitudes:
250km
1000km
6400km

4. Pendulum
We are familiar with a pendulum and have worked with the equation to solve for the period of the pendulum in the lab
However, we will go through the concepts of simple harmonic motion in order to develop the equation for the period:

5. Position of a Pendulum
We will consider a simple pendulum that is set into motion and oscillates between –A and A going through 0
Sketch a graph that shows the position of the pendulum as a function of time with it starting at -A

6. Position of Pendulum

7. Velocity of a Pendulum
Now that we have a graph showing the position as a function of time for a simple pendulum, we wish to consider velocity
Sketch a graph that shows the velocity of the pendulum as a function of time with it starting at -A

8. Velocity of Pendulum

9. Acceleration of a Pendulum
Now that we have a graphs showing the position and velocity as functions of time for a simple pendulum, we will consider acceleration
Sketch a graph that shows the acceleration of the pendulum as a function of time with it starting at -A

10. Acceleration of Pendulum

11. Pendulum FBD θ
mgsin θ
mgcos θ

12. Acceleration of a Pendulum
The force accelerating the pendulum in the +/-A position is proportional to the sine of the angle the pendulum makes with the vertical
Using this, we are able to see that the acceleration will vary with the position of the pendulum

13. Acceleration of a Pendulum
We have seen that the position varies as a function of time so we use the periodic nature of position
We will use A as the amplitude and ω as the angular frequency (where ω=2πf)

14. Period of a Pendulum
According to calculus, the term multiplying the cosine function in the acceleration is equal to the amplitude times the angular frequency squared
Further, the term multiplying the sine function in the velocity term is equal to the amplitude times the angular frequency

15. Mass on a Spring

16. Acceleration of a Mass on a Spring
The force accelerating the mass at the +/-A position is proportional to the amount of stretch and the spring constant
Using this, we are able to see that the acceleration will vary with the position of the mass

17. Acceleration of a Mass on a Spring
In a manner similar to how we approached the pendulum, we are able to find the acceleration as a function of time
This allows us to write the other equations of motion for the mass on a spring

18. Period of a Pendulum
Once again, using a knowledge of calculus, we are able to show that the angular frequency squared and the angular frequency can both be described using mass and the spring constant
This leads to the expression for the period of a mass on the spring

19. Practice Problems Mass on a spring Pendulum Chapter Review
Page 608 Questions 1-4
Pendulum
Page 614 Questions 5-8
Chapter Review
Page 623 Questions 22-23