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Damped and Forced SHM

Damped and Forced SHM. Physics 202 Professor Lee Carkner Lecture 5. PAL #4 Pendulums. Double amplitude (x m ) k depends only on spring, stays same v max = - w x m , increases Increase path pendulum travels v must increase (since T is constant, but path is longer) so max KE increases

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Damped and Forced SHM

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  1. Damped and Forced SHM Physics 202 Professor Lee Carkner Lecture 5

  2. PAL #4 Pendulums • Double amplitude (xm) • k depends only on spring, stays same • vmax = -wxm, increases • Increase path pendulum travels • v must increase (since T is constant, but path is longer) so max KE increases • If max KE increases, max PE increases • Clock runs slow, move mass up or down? • Since T = 2p(L/g)½, want smaller L, move weight up

  3. Uniform Circular Motion • Consider a particle moving in a circle with the origin at the center • The projection of the displacement, velocity and acceleration onto the edge-on circle are described by the SMH equations

  4. UCM and SHM

  5. Uniform Circular Motion and SHM y-axis Particle moving in circle of radius xm viewed edge-on: Particle at time t xm wt+f x-axis x(t)=xm cos (wt+f)

  6. Observing the Moons of Jupiter • He discovered the 4 inner moons of Jupiter • He (and we) saw the orbit edge on

  7. Galileo’s Sketches

  8. Apparent Motion of Callisto

  9. Application: Planet Detection • The planet cannot be seen directly, but the velocity of the star can be measured • The plot of velocity versus time is a sine curve (v=-wxmsin(wt+f)) from which we can get the period

  10. Orbits of a Star+Planet System Center of Mass Vplanet Star Planet Vstar

  11. Light Curve of 51 Peg

  12. Damped SHM • Consider a system of SHM where friction is present • The damping force is usually proportional to the velocity • If the damping force is represented by • Where b is the damping constant • Then, x = xmcos(wt+f) e(-bt/2m) x’m = xm e(-bt/2m)

  13. Energy and Frequency • The energy of the system is: E = ½kxm2 e(-bt/m) • The period will change as well: w’ = [(k/m) - (b2/4m2)]½

  14. Exponential Damping

  15. Damped Systems • Most damping comes from 2 sources: • Air resistance • Energy dissipation • Lost energy usually goes into heat

  16. Damping

  17. Forced Oscillations • If you apply an additional force to a SHM system you create forced oscillations • If this force is applied periodically then you have 2 frequencies for the system w = wd = • The amplitude of the motion will increase the fastest when w=wd

  18. Tacoma Narrows Disaster

  19. Resonance • Resonance occurs when you apply maximum driving force at the point where the system is experiencing maximum natural force • All structures have natural frequencies

  20. Summary: Simple Harmonic Motion x=xmcos(wt+f) v=-wxmsin(wt+f) a=-w2xmcos(wt+f) w=2p/T=2pf F=-kx w=(k/m)½ T=2p(m/k)½ U=½kx2 K=½mv2 E=U+K=½kxm2

  21. Summary: Types of SHM • Mass-spring T=2p(m/k)½ • Simple Pendulum T=2p(L/g)½ • Physical Pendulum T=2p(I/mgh)½ • Torsion Pendulum T=2p(I/k)½

  22. Summary: UCM, Damping and Resonance • A particle moving with uniform circular motion exhibits simple harmonic motion when viewed edge-on • The energy and amplitude of damped SHM falls off exponentially x = xundamped e(-bt/2m) • For driven oscillations resonance occurs when w=wd

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