1. Simple Harmonic Motion
SPH4U
Energy and Momentum

2. Simple Harmonic Motion
Simple Harmonic Motion is the PERIODIC motion of an object attached to a spring when it is subject to the linear elastic restoring force of the spring due to Hooke’s Law.
Think back to Newton’s 3rd Law… if it takes a force for you to push or pull the spring to that point, there must be an equal and opposite force trying to pull the spring back to where it started. So what happens if we let go after stretching a spring?
When the weight is expanded far away from the equilibrium point, the force will be larger so the weight will have a big acceleration back towards its equilibrium point.

3. SHM is closely related to wave motion
SHM is closely related to wave motion. The motion is SINUSOIDIAL in time and demonstrates a single resonant frequency as the mass oscillates back and forth.
The object experiences a restoring force that is proportional and opposite to the displacement.

5. Where is periodic motion found?
Clocks and watches
Swing sets
Sounds waves, water waves, earthquakes
The EM spectrum: everything from radio waves that carry signals to our cell phones and TVs, to gamma radiation emitted from radioactive materials, and light entering your eye, ALL EM waves exhibit periodic motion

6. When is SHM not wanted or beneficial?
Tacoma Narrows Bridge Disaster
Shocks and springs in cars and other vehicles
In tall buildings
In bridges
In these cases, the vibration needs to be damped whereby the motion is stopped.

7. Damped Motion
Damped Harmonic Motion: repeated motion where the amplitude of vibration and the energy both decrease with time.

8. SHM can be related to uniform circular motion
SHM can be related to uniform circular motion. When describing SHM mathematically, picture a reference circle, like the CD shown. The mass attached to the spring in the figure vibrates left and right with SHM. At the same time, the point shown on the CD rotates with uniform circular motion.

9. If the amplitude (max. ∆x) of the SHM equals the CD’s radius(r)and the period of vibration for the SHM exactly equals the period of rotation for the CD, then the x-coordinates of the mass and the reference point on the CD are equal at all times. Therefore, the acceleration of the mass and the centripetal acceleration of the point are also the same. An equation for the period of a mass on a spring can then be derived (see pg.198).

10. where T is the period of vibration (s) (recall f = 1/T)
m is the mass of the vibrating object (kg)
k is the spring constant (N/m)
𝑻=𝟐𝝅 𝒎 𝒌
The period (and frequency) depend only on the mass of the vibrating object and the spring constant of the material. Displacement does not effect the period (or frequency).
CHECK OUT THIS VIDEO: Simple Harmonic Motion: Crash Course Physics #16

11. Example Problem
Two people are having a belly-flop contest from a low diving board. The first person steps onto the board. His mass is 126 kg and the spring constant of the diving board is 8.1 x 103 N/m. The mass of the board is negligible.
Calculate the diving board’s:
period
frequency of vibration
period and frequency when the other man jumps (m=71 kg)
a) T = 0.78 s
b) f = 1.3 Hz
c) T = 0.59 s; f = 1.7 Hz