Newton's 2nd Law:

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Newton's 2nd Law
Free Body Diagram
0
equations of motion
2
Integrate w.r.t. time
ditto
Constants from initial conditions
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Simple Harmonic Motion (SHM)
Ball in a Bowl
SF
SF
FN
Fg
Fg
FN
Stable Equilibrium
(restoring force, not constant force)
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m
k
rest position
0
x
(equation of motion)
Hookes Law
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Guess a solution
Does it work?
Yes, if
A and C can take any value
x(t)
A amplitude
t
Period 2p/B
C phase lag f
B angular freq. w w 2 p f
So usually written
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A and f from initial conditions.
Example Mass (m) on spring (k) displaced x 1
cm at t0
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When a mechanical system is at a point of stable
equilibrium, a perturbation will result in a
restoring force that drives it back to
equilibrium. The resulting equation of motion has
the following form
The solution is an oscillating trajectory at
frequency wo known as Simple Harmonic Motion.