1. Oscillations and Harmonic Motion
Notes Ch.15
Oscillations and Harmonic Motion

2. Periodic Motion
Regular vibrations or oscillations repeat the same movement on either side of the equilibrium position f times per second (f is the frequency)
Displacement is the distance from the equilibrium position
Amplitude is the maximum displacement
Period (T) is the time for one cycle or or 1 complete oscillation

3. Producing time traces
2 ways of producing a voltage analogue of the motion of an oscillating system

4. The conical pendulum
The vertical component of the tension (Tcosθ) supports the weight (mg)
The horizontal component of tension (Tsinθ) provides the centripetal force

5. Time traces

6. 15.1 Simple Harmonic Motion
Simple harmonic motion: Motion that follows a repetitive pattern, caused by a restoring force that is proportional to displacement from the equilibrium position.
Simple harmonic motion is linear motion in which the acceleration is proportional to the displacement from an equilibrium position and directed toward that position
Period is independent of amplitude
Same time for a large swing and a small swing
For a pendulum this only works for angles of deflection up to about 20º
Simple harmonic motion

7. 15.2 Simple harmonic motion: graph and equation
Gradient of displacement v. time graph gives a velocity v. time graph
Max veloc at x = 0
Zero veloc at x = max
Graphing simple harmonic motion

8. Simple Harmonic Motion cont…
Acceleration v. time graph is produced from the gradient of a velocity v. time graph
Max a at V = zero
Zero a at v = max

9. Simple Harmonic Motion cont…
Displacement and acceleration are out of phase
a is proportional to – x
Hence the minus

10. Simple Harmonic Motion cont…
a = -ω²x equation defines SHM
T = 2π/ω
F = -kx eg. a trolley tethered between two springs

11. Circular Motion and SHM
The peg following a circular path casts a shadow which follows SHM
This gives a mathematical connection between the period T and the angular velocity of the rotating peg

12. The Pendulum
Pendulum – an object suspended so that it can swing back and forth about an axis.
Rules of simple pendulums
Period is independent of mass
Period is independent of amplitude as long as the arc is small (<20º)
Period is directly proportional to the square root of the length.
Period is inversely proportional square root of the acceleration of gravity