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Simple Harmonic Motion

Simple Harmonic Motion. AP Physics C. Simple Harmonic Motion What is it?. Any periodic motion that can be modeled with a sin or cosine wave function. Harmonic oscillators include: Simple pendulum – a mass swinging on a string or rod

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Simple Harmonic Motion

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  1. Simple Harmonic Motion AP Physics C

  2. Simple Harmonic Motion What is it? • Any periodic motion that can be modeled with a sin or cosine wave function. • Harmonic oscillators include: • Simple pendulum – a mass swinging on a string or rod • Mass on a spring that has been offset from its rest position and then released

  3. Reminders: Hooke’s Law and Conservation of Energy Hooke’s Law Elastic Potential Energy Conservation of Energy

  4. Waves are oscillations too!Remember Wave Characteristics • Amplitude • the maximum displacement of the medium measured from the rest position. • Wavelength • The distance between corresponding points on consecutive waves. • Frequency • the number of complete cycles (waves) that pass a given point in the medium in 1 second. • Period • the time it takes for one complete cycle to pass a given point in the medium, or the time that passes before the motion repeats itself • Wave Speed • velocity the wave travels through a medium

  5. Simple Harmonic Motion (SHM)velocity & acceleration Note: k= spring constant K=kinetic energy, T=period Position (for reference only) a=max At t=0: X = A (max disp), F=kA (toward x=0), a=kA/m (max), v=0, Uel=1/2 kA2, K=0. A v=0 v=max At t=T/4: X = 0 (no disp), F=0 (at equilibrium), a=0 (no force), v=max, Uel=0 (no stretch), K=1/2 mV2 . B a = 0 a=max At t=T/2: X = A (on other side of x=0), F=kA (toward x=0), a=kA/m (max), v=0, Uel=1/2 kA2, K=0. C v=0 v=max At t=3T/4: X = 0 (no disp), F=0 (at eq.), a=0 (no force), v=max (opp. dir.), Uel=0 (no stretch), K=1/2 mv2 . D a = 0 a=max At t=T: Back to original position so same as t=0, X = A (max disp), F=kA (toward x=0), a=kA/m (max), v=0, Uel=1/2 kA2, K=0. E v=0

  6. More velocity & acceleration in SHMWhat happens between x=0 and x=A?(t=0 to t=T/2) Note: F & a are always directed toward x=0 (eq) a=max v=0 Between t=0 and t=T/4, the mass is moving toward the equilibrium position (from x=A to x=0) with a decreasing force. a , v and a in same dir so v . Uel , and K . a v v=max x=A x=A Between t=T/4 and t=T/2, the mass is moving away from the equilibrium position (from x=0 to x=A) with an increasing force. a , v and a in opp dir, so v . Uel , and K a v a=max v=0

  7. More velocity & acceleration in SHMWhat happens between x=0 and x=A?(t=T/2 to t=T) Note: F & a are always directed toward x=0 (eq) a=max Between t=T/2 and t=3T/4, the mass is moving toward the equilibrium position (from x=A to x=0) with a decreasing force. a , v and a in same dir so v . Uel , and K . a v a=0 x=A x=A Between t=3T/4 and t=T, the mass is moving away from the equilibrium position (from x=0 to x=A) with an increasing force. a , v and a in opp dir, so v . Uel , and K . a v a=max

  8. Angular Frequency For SHM we define a quantity called angular frequency, ω (which is actually angular velocity), measured in radians per second. We use this because when modeling the SHM using a cosine function we need to be able to express frequency in terms of radians.

  9. Modeling SHM with a cosine wave. When we start an oscillation, such as a mass on a spring, we either stretch or compress a the spring a certain distance which then becomes the Amplitude of the oscillation. Where A=Amplitude, ω=angular frequency, t=time, and φ = phase shift Note: when t=0, x=A (max displacement)

  10. Modeling velocity for SHM Notice that at t=0, v=0 and this corresponds to the maximum displacement (x=A). Also…the maximum value occurs at sin(∏/2)=1, so vmax=-ωA

  11. Modeling accelerationand finding maximum acceleration Notice that at t=0, a=max in the opposite direction as the stretch and this corresponds to the maximum displacement (x=A). Also…the maximum value occurs at cos(0)=1, so amax=-ω2A

  12. Calculating acceleration for a mass on a spring This means that acceleration is a function of position. Or… more stretch means more acceleration. We will express acceleration in a general for in terms of angular frequency, ω, and position, x, as shown. Note: at x=A, a=max = -ω2A

  13. Simple Pendulum A simple pendulum is an object of mass, m, swinging in a plane of motion, suspended by a massless string or rod. In other words, it is a point mass moving in a circular path. θ l l If θ < 10°, then we can assume a small angle approximation, sin θ≈ θ, the long formula for period, which includes an infinite sine series, reduces to… Note:

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