1. Simple Harmonic Motion
AP Physics C Mrs. Coyle

2. Periodic Motion
A motion of an object that repeats with a constant period.

3. Simple Harmonic Motion
It is a periodic motion.
AND
It has a restoring force that acts to restore the oscillator to equilibrium. The restoring force is given by:
Hooke’s Law F=-kx
x is the displacement from equilibrium and k is the force constant (spring constant).
Examples of SHM: Pendulum, Spring-Mass, Object sliding back and forth in a frictionless vertical circular track, etc.

4. Simple Harmonic Motion Simulations
-SHM
-Particle oscillating in SHM

5. The force constant, k, is the slope of a F vs x graph.

6. Remember: For springs in series: 1/keff = 1/k1 + 1/k2
For springs in parallel:
keff = k1 + k2

7. Simple Harmonic Motion
Velocity:
maximum as it passes through equilibrium
zero as it passes through the extreme positions in its oscillation.
Acceleration: a=F/m = -kx/m
-maximum at extreme points
-zero at equilibrium

8. The acceleration in SHM is not constant. It varies with x over time.
When solving a problem where you have to prove if a motion is SHM, you must show that the acceleration is proportional to –x.
Then the coefficient of x will be w2.

9. Ex. #1
A ball dropped from a height of 4.00m makes a perfectly elastic collision with the ground. Assuming no mechanical energy is lost due to air resistance,
Show that the ensuing motion is periodic.
Determine the period of the motion.
Is the motion SHM? Explain
Ans: a)yes, b)1.82s, c) no

10. How does SHM relate to angles and trig functions?

11. Equations of Motion for Displacement

12. Characteristic Quantities of Simple Harmonic Motion

13. Period and Frequency
NOTE: The period of SHM oscillator does not depend on the amplitude.

14. Example What is the amplitude? What is the period?
c) What total distance does the particle travel in one period?
-0.5m, Y=-0.5sin(pi times t) because 2pi is in a period of 2s and -0.5 is the amplitude, 2m

15. Example Write the equation of motion for the above oscillator. Answer:
x=-0.5sin(πt)
-0.5m, Y=-0.5sin(pi times t) because 2pi is in a period of 2s and -0.5 is the amplitude, 2m

16. Note
Since, the acceleration in SHM is not constant, regular kinematics equations cannot be used.
There are two options to finding v and a:
To find v take the first derivative of x over time (for x use an equation of motion, ex. x=Acos(wt +f)). Then take the second derivative to find a.
Find v at a given position using conservation of mechanical energy.
E= ½ mv2 + ½ kx2 = constant

17. When taking the derivative of x over time or v over time, remember:
AND
Remember to use the chain rule when differentiating.

18. Examples:

19. Graphs of SHM
For any graph of SHM:
When x=0, v is at max.
x is 900 out of phase with v at any time.
When v=0 , a is at max.
The acceleration is 1800 out of phase with position.

20. Example 1: f = 0
At t = 0:
x (0)= A
v (0) = 0
f = 0
amax = ± w2A
vmax = ± wA

21. At t = 0: x (0)=0 v (0) = vi f = - p/2 Example 2: f = - p /2
The graph is shifted one-quarter cycle to the right compared to the graph of x (0) = A

22. Period of A Spring Mass Oscillator
____
T=2p√m/k
m mass
k spring constant
T does not depend on g
The period is smaller for a stiffer spring (large values of k).

23. Period of a Pendulum Period ___ T=2p√L/g L=length of string
Period
___
T=2p√L/g
L=length of string
T depends on g L

24. Question
If you had a spring-mass system on the moon, would the period be the same or different than that of this system on the earth?
What if it were a pendulum system?