1. Simple Harmonic Motion
Examples of Periodic Motion
Swinging on a playground
Guitar string
Metal block bobbing up and down
Pendulum

2. Periodic Motion One position where Fnet = 0 = equilibrium
Whenever the object is out of equilibrium, the Fnet pulls back
If F that restores the equilibrium is directly proportional to displacement of object
SIMPLE HARMONIC MOTION

3. At equilibrium, speed is MAX
See next Figure,
A) spring is stretched away from its unstretched (equilibrium) position (x=0)
Spring exerts a force on the mass towards the equilibrium position
Spring force and acceleration decreases as the mass moves towards eq. position
B) Zero force at eq.; mass acceleration is zero
Yet speed is increasin towrds eq. andreaches MAXIMUM at eq.
Fsp=0; mass inertia causes it to overshoot eq. and compress spring

5. At max displacement, Fsp and a reach a MAX
C) Mass moves past eq, Fsp and s increase
Direction of Fsp and a (towards eq.) is opposite the mass’s dorection of motion
Mass slows down
When spring compression equals the distance of original stretch (displacement)
Mass is at max displacement
Fsp and a are at a max
Mass speed = o , force causes mass to change direction
Repeat oscillation (ignore resistance forces)
Damping: real world oscillation (resistance forces)

6. Hooke’s Law In SHM, restoring force is proportional to displacement.
Fsp is called restoring force since it is always returning the system to equilibrium (Fnet =0)
F elastic = -kx
Spring force = - (spring constant x displacement)
(-) sign signifies that the direction of the spring
force is always opposite the displacement
Higher k means a stiffer spring
K has units N/m
At Eq, Fnet = 0 = F elastic + Fg

7. Sample Problem
If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant?
Given: m=0.55 kg x=-2.0cm= m
G = -9.8m/s/s Unknown is k=?
Felastic Fg

8. Simple Harmonic Motion is described by 2 traits
Period (T): the time needed to repeat one complete cycle of motion
Amplitude: the maximum distance moved from equilibrium

9. Mass in a Spring
Period of oscillation depends on the mass of the block and strength of the spring
Fsp = F g in equilibrium

10. Simple Pendulum: bob (mass) attached to a string
A pendulum moves with SHM if the angle is less than 15 degrees.
Fnet is opposite and linear to displacement
Period of a pendulum does not depend on mass or amplitude FT
Fnet Fg

11. Restoring Force of a Pendulum is a component of the bob’s weight
Force of string along the y axis
Force of gravity can be resolved into comonents
Net force if the x component of Fg
Since Fgx is towards the equilibrium position it is the restoring force
Frestoring Fg,x = Fgsin theta
Restoring force is zweo at eq.
Small anfles this acts in simple harmonic motion FT
F gx
Fgy

12. See SMH Figure

13. Measuring SHM Term Example Definition SI unit Amplitude
Maximum displacement from equilibrium
Radian, rad
Meter, m
Period , T
Time that it takes to complete a full cycle
Secons, s
Frequency, f
Number of cycles or vibrations per unit of time
Hertz, Hz (Hz=s-1)

14. Formulas for SHM f = 1/T or T = 1/ f
Ex. If one complete cycle takes 20 s
T = 20 s
frequency: 1/20 cycles /s or 0.05 cycles /s
0.05 Hz

15. Formulas and sample problem
Period of a simple Pendulum in SHM
You need to know the height of a tower, but darkness obscures the ceiling. You note that a pendulum extedning from the ceiling almost touches the floor and that its period is 12 s. How tall is the tower?
(answer L=36 m)

16. Mechanical Resonance
Small forces are applied at regular intervals to a vibrating or oscillating object to increase the amplitude
Examples
Jumping on a diving board
Jumping on a trampoline

17. SHM: CFU
What is the length of a pendulum with a period of 1.00 s? (0.248 m)
2. Would it be practical to make a pendulum with a period of 10.0 s? Calculate the length and explain.(24.8 m; No, this is over 75 ft long!)
3. On a planet with an unknown value of g, the period of a 0.65 m long pendulum is 2.8 s. What is the g for this planet? (3.3 m/s2)