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?

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)

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”.

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

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)

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*