Simple Harmonic Motion Periodic (repeating) motion where the restoring (toward the equilibrium position) force is proportional to the distance from equilibrium

Examples: Pendulum. Oscillating spring (either horizontal or vertical) without friction. Point on a wheel when viewed from above in plane of wheel. Physics Classroom has a good lesson on simple harmonic motion under Waves, vibrations.

A mass is oscillating on a spring Position in equal time intervals: Restoring Force: F = k x K is the spring constant (N/m), x is distance from equilibrium.

Model: oscillation coupled to a wheel spinning at constant rate

Vertical position versus time: Period T Frequency or number of cycles per second: f = 1/T

Sinusoidal motion Time (s) Displacement (cm) Period T

Sine function: mathematically x y 2π2π π/2π3π/22π2π5π/23π3π7π/24π4π9π/25π5π 1 y=sin(x) y=cos(x)

Sine function: employed for oscillations x y π/2 π 3π/2 2π2π5π/2 3π3π 7π/24π4π 9π/2 5π5π 1 y=sin(x) Time t (s) Displacement y (m) T/2 T2T -A A y= A sin(ωt)

Sine function: employed for oscillations Time t (s) Displacement y (m) T/2 T2T -A A y= A sin(ωt) What do we need ? 1.Maximum displacement A 2.ωT = 2π 3.Initial condition

Sine function: employed for oscillations 1.Maximum displacement A 2.ωT = 2π 3.Initial condition y(t=0) Angular frequency in rad/s Amplitude A is the maximum distance from equilibrium Starting from equilibrium: y=A sin(ωt) Starting from A: y=A cos(ωt) ω = 2πf

Example 1 - find y(t) y(cm) t(s) Period? T=4 s Sine/cosine? Sine Amplitude? 15 cm Where is the mass after 12 seconds?

Example 2 – graph y(t) Amplitude? 3cm -3 3 y (cm) y(t=0)? -3cm Period? 2s t(s) When will the mass be at +3cm? 1s, 3s, 5s, … When will the mass be at 0? 0.5s, 1.5s, 2.5s, 3.5 s …

Energy in simple harmonic motion:

Useful formulas (reg.) Hookes Law for a spring: F = k x Period, T = 1/f, f is frequency in cycles/sec For a spring/mass combo : T = 2π(m/k) 1/2 For a pendulum : T = 2π (L/g) 1/2

Some useful formulas (hons.): You can solve for v max by equating the KE max at equilibrium = PE max at extremities ½ mv 2 = ½ k x max 2 you will find that this is also…. v max = A ω a max = A ω 2 For a spring/mass combo : ω = (k/m) 1/2 For a pendulum : ω = (g/L) 1/2

Summary Harmonic oscillations are sinusoidal Motion is repeated with a period T Motion occurs between a positive and negative maximum value, named Amplitude Can be described by sine/cosine function y=A sin(ωt) or y=A cos(ωt) Angular frequency ω=2π/T