Kinematics of Simple Harmonic Motion
Introduction;
- stable systems, vibrations oscillations, SHM examples
- restoring force
Background - Analysis of Circular Periodic Motion
- angular velocity, w
- definition and discussion of the radian
- angular velocity
- instantaneous velocity of objects orbiting with an angular velocity
- frequency and period
- angular velocity, frequency and period
- relating circular motion to sine wave, in terms of w
Graphing SHM Displacement, Velocity, Acceleration
- Basic Relationship between Displacement, Velocity and Time curves
- reference graphs handout 1
- graphing displacement vs wt (sin and cos) for condition that w=1
- graphing corresponding velocity and acceleration graphs
- determine that x=sin wt, v=coswt, a=-sinwt
- Detailed Relationship between Displacement, Velocity and Time curves
- reference graphs handout 2,3
- looking at the effect of w on the amplitude of v ant t graphs by estimation
of slope
- conclude x=sin wt, v=wcoswt, a=-w^2sinwt
- demonstrate a=-w^2x form above
- explanation of v in terms of circular motion (confirms graph)
- Explanation of Phase
- A sine wave is the same as a cosine wave if out of phase by pi/2
- another set of equations x=cos wt, v=-wsinwt, a=-w^2coswt are also
solutions to the characteristic equation for SHM (which is a=-w^2x)
Simple Harmonic Motion Force Analysis
- horizontal frictionless spring mass example
- defining shm in terms of restoring force
- discussion of stable systems and equilibrium position
- graph force vs amplitude
- graph acceleration vs amplitude
- FBD and analysis a=(-k/x) m
- from force analysis and kinematic analysis w=sqrt (k/m)
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