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Simple Harmonic Motion

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.

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Simple Harmonic Motion

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

  2. 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.

  3. 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.

  4. Model: oscillation coupled to a wheel spinning at constant rate

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

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

  7. 5π/2 3π 2π 3π/2 π 7π/2 4π 9π/2 5π π/2 Sine function: mathematically y=sin(x) y=cos(x) y 1 x -1 2π

  8. y 1 y=sin(x) x 3π 5π T/2 3π/2 π 9π/2 2π π/2 2T 7π/2 T 5π/2 4π -1 Sine function: employed for oscillations Displacement y (m) A y= A sin(ωt) Time t (s) -A

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

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

  11. t(s) Example 1 - find y(t) y(cm) 30 15 10 5 Period? T=4 s Sine/cosine? Sine Amplitude? 15 cm Where is the mass after 12 seconds?

  12. Example 2 – graph y(t) Amplitude? 3cm y(t=0)? -3cm When will the mass be at +3cm? 1s, 3s, 5s, … Period? 2s When will the mass be at 0? 0.5s, 1.5s, 2.5s, 3.5 s … y (cm) 3 8 6 2 4 t(s) -3

  13. Energy in simple harmonic motion:

  14. 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

  15. Some useful formulas (hons.): • You can solve for vmax by equating the KEmaxat equilibrium = PEmax at extremities • ½ mv2 = ½ k xmax2 you will find that this is also…. • vmax = A ω • amax = A ω2 • For a spring/mass combo : ω = (k/m)1/2 • For a pendulum : ω = (g/L)1/2

  16. 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

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