1. Oscillations
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2. Periodic Motion
Periodic motion is that motion in which a body moves back and forth over a fixed path, returning to each position and velocity after a definite interval of time.
Period, T, is the time for one complete oscillation. (seconds,s)
AmplitudeA
Frequency, f, is the number of complete oscillations per second. Hertz (s-1)

3. Simple Harmonic Motion, SHM
Simple harmonic motion is periodic motion in the absence of friction and produced by a restoring force that is directly proportional to the displacement and oppositely directed. x F
A restoring force, F, acts in the direction opposite the displacement of the oscillating body.
F = -kx

4. Simple Harmonic Motion (SHM)
The restoring force is one that gets progressively larger with displacement from the equilibrium position.  For example, the more you stretch a spring the larger the force trying to get the spring back to its original shape.

5. Position VS. Time graph
The displacement of the oscillating mass varies sinusoidally as a function of time. 

6. Comparison To Heartbeat
Heartbeat Oscillating mass on a spring
Periodic Motion Simple Harmonic Motion

7. Position vs. Time
The amplitude, A, is the maximum displacement x of a spring. Both are in meters.
CREST
Equilibrium Line
Period, T, is the time for ONE COMPLETE oscillation (One crest and trough). Oscillations could also be called vibrations or cycles.
Trough

8. Displacement in SHM m
x = 0
x = +A
x = -A x
Displacement is positive when the position is to the right of the equilibrium position (x = 0) and negative when located to the left.
The maximum displacement is called the amplitude A.

9. Velocity in SHM m v (-) v (+)
x = 0
x = +A
x = -A
Velocity is positive when moving to the right and negative when moving to the left.
It is zero at the end points and a maximum at the midpoint in either direction (+ or -).

10. Acceleration in SHM m -x +x
x = +A
x = -A
Acceleration is in the direction of the restoring force. (a is positive when x is negative, and negative when x is positive.)
Acceleration is a maximum at the end points and it is zero at the center of oscillation.

11. Acceleration vs. Displacementx v a m
x = 0
x = +A
x = -A
Given the spring constant, the displacement, and the mass, the acceleration can be found from: or
Note: Acceleration is always opposite to displacement.

12. Amplitude
Amplitude is the magnitude of the maximum displacement.

13. Period, T
For any object in simple harmonic motion, the time required to complete one cycle is the period T.

14. Frequency, f
The frequency f of the simple harmonic motion is the number of cycles of the motion per second.

15. Exercise on Simple Harmonic Motion
Q1. What is the amplitude? Q2. What is the period?
Q3. What is the frequency?

16. Graphical Treatment
Equations of SHM

18. If an object moving with constant speed in a circular path is observed from a distant point (in the plane of the motion), it will appear to be oscillating with SHM. The shadow of a pendulum bob moves with s.h.m. when the pendulum itself is either oscillating (through a small angle) or moving in a circle with constant speed, as shown in the diagram.

19. Reference Circle

20. An object moving is simple harmonic motion can be located using:
A is amplitude
f is frequency
x is displacement from equilibrium
ω is angular velocity
Displacement, x, against time
start point at max amplitude
** Set Calculator in Radians.

21. Velocity in SHM
The velocity (v) of an oscillating body at any instant is the horizontal component of its tangential velocity (vT).
vT = wR = wA; w = 2f
v = -vT sin  ;  = wt
v = -w A sin w t
v = -2f A sin 2f t

22. Velocity against time v = vocoswt
Starting where?
Midpoint = max velocity.

23. Acceleration Reference Circle
The acceleration (a) of an oscillating body at any instant is the horizontal component of its centripetal acceleration (ac).
a = -ac cos q = -ac cos(wt)
R = A
a = -w2A cos(wt)

24. Equations of Graphs x = xo cos wt x = xo sin wt
v = -vo sin wt v = vo cos wt
a = -aocos wt ao sin wt
Released from Released equilibrium.
top.

25. Displacement
Velocity
Acceleration

26. Ex 1. A mass on a spring is oscillating with f = 0. 2 Hz and xo = 3 cm
Ex 1. A mass on a spring is oscillating with f = 0.2 Hz and xo = 3 cm. What is the displacement of the mass s after its release from the top?
x = Acos wt A = 3 cm
w = 2pf = 0.4 p Hz =1.26 rad/s.
t = s
x = 0.03 cos (1.26 x 10.66) = m
You must use radians on calculator.

27. Example 2: The frictionless system shown below has a 2-kg mass attached to a spring (k = 400 N/m). The mass is displaced a distance of 20 cm to the right and released. What is the frequency of the motion? m
x = 0
x = +0.2 m x v a
x = -0.2 m
f = 2.25 Hz

28. Example 2 (Cont.): Suppose the 2-kg mass of the previous problem is displaced 20 cm and released (k = 400 N/m). What is the maximum acceleration? (f = 2.25 Hz) m
x = 0
x = +0.2 m x v a
x = -0.2 m
Acceleration is a maximum when x =  A
a =  40 m/s2

29. Example 2: The 2-kg mass of the previous example is displaced initially at x = 20 cm and released. What is the velocity 2.69 s after release? (Recall that f = 2.25 Hz.) m
x = 0
x = +0.2 m x v a
x = -0.2 m
v = -2f A sin 2f t
(Note: q in radians)
v = m/s
The minus sign means it is moving to the left.

30. Example 3: At what time will the 2-kg mass be located 12 cm to the left of x = 0? (A = 20 cm, f = 2.25 Hz)
-0.12 m m
x = 0
x = +0.2 m x v a
x = -0.2 m
t = s

31. The negative sign shows Fnet & accl direction opposite displacement.
For any displacement:
a = -w²x
amax = -w²A
The negative sign shows Fnet & accl direction opposite displacement.

32. a = -w²x w = 2pf = p a = -(p)2 (0.02 m) = -0.197 m/s2. left.
Ex 4. A pendulum swings with f = 0.5 Hz. What is the size & direction of the acceleration when the bob has displacement of 2 cm right?
a = -w²x
w = 2pf = p
a = -(p)2 (0.02 m) = m/s2. left.

33. Ex 5: A mass is bobbing on a spring with a period of 0. 20 seconds
Ex 5: A mass is bobbing on a spring with a period of 0.20 seconds. What is its angular acceleration at a point where its displacement is 1.5 cm?
w = 2p/T
.w = 31 rad/s
a = -w²x
a = (31rad/s)(1.5 cm) = cm/s2.
15 m/s2.

34. Energy in Simple Harmonic Motion
We already know that the potential energy of a spring is given by: U = ½ kx2 The total mechanical energy is then: The total mechanical energy will be conserved, as we are assuming the system is frictionless.

35. Energy in Simple Harmonic Motion
If the mass is at the limits of its motion, the energy is all potential. If the mass is at the equilibrium point, the energy is all kinetic. We know what the potential energy is at the turning points:
(11-4a)

36. When the mass is at equilibrium, x = 0, and velocity is maximum:
To find the velocity of an oscillating mass or pendulum at any displacement:
When the mass is at equilibrium, x = 0, and velocity is maximum:
vo = ± wxo.
Derivation on H pg 77.

37. Ex 6. A pendulum swings with f = 1 Hz and amplitude 3 cm
Ex 6. A pendulum swings with f = 1 Hz and amplitude 3 cm. At what position will be its maximum velocity &what is the velocity?
At max velocity vo = wxo.
w = 2pf = 2p(1) = 2p rad/s
vo = (2p rad/s)(0.03)
vo = m/s
vo = 0.2 m/s

38. Energy in Simple Harmonic Motion
The total energy is, therefore ½ kA2 And we can write: This can be solved for the maximum velocity which is given by making total energy equal to only Kinetic:

39. The Period and Sinusoidal Nature of SHM
If we use calculus we can find that the period of a mass and ideal spring to be:

40. 11-3 The Period and Sinusoidal Nature of SHM
We can similarly find the position as a function of time (note the diagram is 90 degrees out of phase):
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41. The Period and Sinusoidal Nature of SHM
The top curve is a graph of the previous equation. The bottom curve is the same, but shifted ¼ period so that it is a sine function rather than a cosine.

42. The Period and Sinusoidal Nature of SHM
The velocity and acceleration can be calculated as functions of time; the results are below, and are plotted at left.