1. Simple Harmonic Motion (SHM)
Hooke’s Law: Fs = -ksx (restoring force of a spring)
Potential Energy: Us=1/2 ksx2
Energy Conservation:
ETOT= ½ mv2 + ½ ksx2
At extrema: ETOT= ½ ksA2
(A = amplitude = |xMAX|)
This gives us: vMAX and v(x)
(v as a “function” of x).
Why the quotes?

2. Circle Analog to SHM
Suppose we have an object of mass (m) in uniform circular motion with angular velocity (ω) and radius (A).
Angle swept: θ(t) = ωt and: ω = 2πf = 2π/T
y-motion: y(t) = A sin(ωt) = A sin(2πft)= A sin(2πt/T)
Projection or “shadow” of vertical axis traces out an SHM with period (T).
Speed of circular motion:
v= Aω = A(2π/T)
Where v = vmax of SHM: _____
vmax=√(ks/m) A
_____
Equate to give: T=2π√(m/ks)

3. Circle Analog to SHM continued
Click here:
Circle can be used for x or y motion
y(t)= A sin(ωt) is an “equation of motion”.

4. Examples of Simple Harmonic Oscillators
Mass hanging from a spring
Simple Pendulum

5. Simple Harmonic Motion (summing up)
Dynamics defined by Hooke’s Law:
Fs = -ks y
Resulting trajectory or “Equation of Motion”
y(t) = A sin(ωt) = A sin(2πt/T)
Period of the motion for spring constant (ks)
______
T=2π√(m/ks)
Period of motion for a Simple Pendulum
________
T=2π√ (L/g)

6. LAB: Simple Harmonic Motion
Questions:
Mass on a spring:
Looking at Hooke’s Law, how can you find ks ?
Looking at the expression for the period, how can you find ks ?
* Do both and compare*
Simple Pendulum:
How does period depend on mass (m)?
How does period depend on length (L)?
How does period depend on angle (θ)?
* Verify your expectations*