1. SIMPLE HARMOIC MOTION
CCHS Physics

2. Facts of SHM
SHM occurs when an object is vibrating at a single frequency and period
PERIODOIC: when a vibration or oscillation repeats itself, back and forth, over the same path
We’ll deal mainly with simple harmonic oscillations where the position of the object can be described by a sinusoidal (sin, cos) function

3. Examples of SHM Block attached to spring Motion of a pendulum
Vibrations of a stringed musical instrument
Clocks
Oscillations of houses, bridges, …

4. Meaning of Simple Harmonic Motion
SIMPLE  single frequency
HARMONIC  sinusoidal
A system in which the restoring force is proportional to the negative displacement (Hooke’s Law) will move in simple harmonic motion.

5. Spring and SHM

6. Example of Complex Harmonic Motion
Approximately simple-harmonic in this region

7. Hooke’s Law
The magnitude of the restoring force is proportional to the displacement
Where:
F = force (N)
k = spring constant (N/m)
x = displacement
Acceleration
Force and Acceleration
Proportional to x
Directed toward the equilibrium position

8. Hooke’s Law cont.
Accurate as long as there is not too much displacement
There is a negative sign because the force always acts opposite to the displacement
Note: F varies with position

9. Five Fun Definitions
DISPLACEMENT: the distance x from the equilibrium point
AMPLITUDE: the maximum displacement
CYCLE: one complete to-and-from motion
PERIOD (T): time to complete one cycle
FREQUENCY (f): number of cycles per second
Frequency is measured in s-1 = Hertz (Hz)

10. Period and Frequency
Period and Frequency are inversely related:

11. Force vs. Distance Graphs
Slope = spring constant
Area = work (or energy)

12. Energy of a Spring
Remember that potential energy is stored energy due to position
The calculation of potential energy is equivalent to calculating work
Note: usually going to neglect mass of spring

13. Example 1 Given the following graph: What is the spring constant?
If a 2 kg mass attached to this spring is displaced 3 m, what is the acceleration of the mass?
If the spring is stretched to 5 m, how much energy is stored in the spring?

14. Conservation of Energy
Mechanical Energy - the sum of all kinetic and potential energies
Now we add in elastic potential energy
We still have that energy must be conserved, thus:

15. Example 2 First, find the spring constant:
A spring stretches by 5 cm when a 100 g mass is hung from it. A 250 g mass attached to this spring and pulled back 0.70 m and released from rest. What is the speed of the block as it passes through the equilibrium point?
First, find the spring constant:
Now, use conservation of energy

16. Example 3 Note: x2 = h1 call this distance d Conservation of Energy
A 0.2-kg ball is attached to a vertical massless spring as shown. The spring constant of the spring is 28 N/m. The ball, supported initially so that the spring is neither stretched nor compressed, is released from rest. In the absence of air resistance, how far does the ball fall before being brought to a momentary stop by the spring?
Conservation of Energy
Note: x2 = h1 call this distance d

17. Velocity and Position Initially the mass is at its maximum extension A
Entirely elastic potential energy
The initial elastic potential energy converted to a combination of kinetic and eleastic potential energy
Solving for v

18. Simple Harmonic Motion vs. Uniform Circular Motion
A ball is glued to the top of a turntable
Focus on the shadow of the ball
If the turntable rotates with constant angular velocity the shadow of the ball moves in simple harmonic motion

19. SHM vs. UCM cont.
If we can prove that the velocity of the shadow varies with position like the function on the previous slides, we know the shadow moves in SHM.
Looking at the top triangle (made with velocity vectors)
Now looking at the larger triangle
Equating these two equations
Thus we have proved that the velocity of the shadow in the x direction is related to displacement in exactly the same manner as the velocity of an object undergoing SHM.

20. Frequency and Period
The velocity of the ball in the previous slide is: or
Where
A is the amplitude
T is the period
Focus on 1/4 of the trip,
Imagine the shadow is a block on a spring, in the 1/4 of a cycle the block moves from a point of all elastic potential energy to a point of all kinetic energy
Substituting for A/v0

21. Period of a Pendulum Restoring force = mgsin
Not SHM because F  sin, not 
However, for small : sin ≈ (measured in radians)
Thus, F = -mgsin ≈ mg
Also, s = L
Essentially SHM (Hooke’s Law)
Plugging into period of a spring formula with k = (mg)/L

22. Displacement Equation
Looking at reference circle again
Ball starts on the x axis at x = +A and moves through the angle  in a time t
Ball rotates at constant angular speed  (because UCM)
 = t
Displacement x of the shadow is just the projection of the radius A onto the x-axis

23. Properties of SHM Position of particle at time t: A…amplitude
w…Angular frequency
f…phase constant, phase angle
T…period
(wt + F)…phase

24. Period and Frequency (Revisited)
PERIOD: time required to complete one cycle
The value of T depends on the angular speed (frequency) 
Greater , less T
For 1 cycle,  = 2, and t = T:
Using the fact that frequency and period are inverses:

25. Velocity and Acceleration

26. More Properties of SHM
Phase of velocity differs by – p/2 or 90° from phase of displacement.
Phase of acceleration differs by – p or 180° from phase of displacement.

27. Even More Properties of SHM
Acceleration of particle is proportional to the displacement, but is in the opposite direction (a = - 2·x).
Displacement, velocity and acceleration vary sinusoidally.
The frequency and period of the motion are independent of the amplitude.

28. Summary of SHM t x v a KE Us A
-w2A
½kA2
T/4
-wA
T/2 -A
3T/4 T

29. Example
A block whose mass m is 680 g is fastened to a spring whose spring constant k is 65 N/m. The block is pulled a distance of x = 11 cm from its equilibrium position at x = 0 on a frictionless surface and released from rest at t = 0.
What force does the spring exert on the block just before the block is released?
What are the period, frequency and angular frequency?

30. Example cont. What is the amplitude of the oscillation?
The amplitude is 11 cm (that is where it is released from)
What is the maximum speed of the oscillating block?
max speed occurs whenever x = 0
What is the magnitude of the maximum acceleration of the block?
max acceleration occurs at the end points

31. Example cont. What is the phase constant  of the motion?
At t = 0, the displacement is at its maximum value A, and the velocity of the block is zero. If we put these initial conditions into the displacement and velocity equations we find:
The smallest angle that solves these two equations is  = 0. (Any multiple of 2 will also work)