1. Harmonic Motion
Chapter 13

2. Simple Harmonic Motion
A single sequence of moves that constitutes the repeated unit in a periodic motion is called a cycle
The time it takes for a system to complete a cycle is a period (T)

3. Simple Harmonic Motion
The period is the number of units of time per cycle; the reciprocal of that—the number of cycles per unit of time—is known as the frequency (f).
The SI unit of frequency is the hertz (Hz), where 1 Hz = 1 cycle/s = 1 s-1
Amplitude (A) is the maximum displacement of an object in SHM

4. Simple Harmonic Motion
One complete orbit (one cycle) object sweeps through 2p rad
f - number of cycles per second
Number of radians it moves through per second is 2pf - that's angular speed (w)
w - angular frequency
Sinusoidal motion (harmonic) with a single frequency - known as simple harmonic motion (SHM)

5. Displacement in SHM

6. Velocity in SHM
vmax = Aw

7. Acceleration in SHM
The acceleration of a simple harmonic oscillator is proportional to its displacement

8. Example 1
A spot of light on the screen of a computer is oscillating to and fro along a horizontal straight line in SHM with a frequency of 1.5 Hz. The total length of the line traversed is 20 cm, and the spot begins the process at the far right. Determine
(a) its angular frequency,
(b) its period,
(c) the magnitude of its maximum velocity, and
(d) the magnitude of its maximum acceleration,
(e) Write an expression for x and find the location of the spot at t = 0.40 s.

9. Problem
A point at the end of a spoon whose handle is clenched between someone’s teeth vibrates in SHM at 50Hz with an amplitude of 0.50cm. Determine its acceleration at the extremes of each swing.

10. Equilibrium
The state in which an elastic or oscillating system most wants to be in if undisturbed by outside forces.

11. Elastic Restoring Force
When a system oscillates naturally it moves against a restoring force that returns it to its undisturbed equilibrium condition
A "lossless" single-frequency ideal vibrator is known as a simple harmonic oscillator.

12. An Oscillating Spring
If a spring with a mass attached to it is slightly distorted, it will oscillate in a way very closely resembling SHM.
Force exerted by an elastically stretched spring is the elastic restoring force F, = -ks.
Resulting acceleration ax = -(k/m)x
F is linear in x; a is linear in x - hallmark of SHM

13. Frequency and Period Simple harmonic oscillator
Shown every ¼ cycles for 2 cycles
Relationship between x, vx, t, and T

14. Hooke’s Law
Beyond being elastic, many materials deform in proportion to the load they support - Hooke's Law

15. Hooke’s Law
The spring constant or elastic constant k - a measure of the stiffness of the object being deformed
k has units of N/m

16. Hooke’s Law
k has units of N/m

17. Frequency and Period
w0 - the natural angular frequency, the specific frequency at which a physical system oscillates all by itself once set in motion
natural angular frequency
and since w0 = 2pf0
natural linear frequency
Since T= 1/f0
Period

18. Resonance vs. Damping
If the frequency of the disturbing force equals the natural frequency of the system, the amplitude of the oscillation will increase—RESONANCE
If the frequency of the periodic force does NOT equal the natural frequency of the system, the amplitude of the oscillation will decrease--DAMPING

19. Example 2
A 2.0-kg bag of candy is hung on a vertical, helical, steel spring that elongates 50.0 cm under the load, suspending the bag 1.00 m above the head of an expectant youngster. The candy is pulled down an additional 25.0 cm and released. How long will it take for the bag to return to a height of 1.00 m above the child?

20. The Pendulum
The period of a pendulum is independent of the mass and is determined by the square root of its length

21. Example 3
How long should a pendulum be if it is to have a period of 1.00 s at a place on Earth where the acceleration due to gravity is 9.81 m/s2?

22. Problem 2
What would the length of a pendulum need to be on Jupiter in order to keep the same time as a clock on Earth? gJupiter = 25.95m/s2

23. When “f” is known
When “f” is NOT known
vmax = Aw