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Damped and Forced SHM

Damped and Forced SHM. Physics 202 Professor Lee Carkner Lecture 5. If the amplitude of a linear oscillator is doubled, what happens to the period?. Quartered Halved Stays the same Doubled Quadrupled.

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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. If the amplitude of a linear oscillator is doubled, what happens to the period? • Quartered • Halved • Stays the same • Doubled • Quadrupled

  3. If the amplitude of a linear oscillator is doubled, what happens to the spring constant? • Quartered • Halved • Stays the same • Doubled • Quadrupled

  4. If the amplitude of a linear oscillator is doubled, what happens to the total energy? • Quartered • Halved • Stays the same • Doubled • Quadrupled

  5. If the amplitude of a linear oscillator is doubled, what happens to the maximum velocity? • Quartered • Halved • Stays the same • Doubled • Quadrupled

  6. If the amplitude of a linear oscillator is doubled, what happens to the maximum acceleration? • Quartered • Halved • Stays the same • Doubled • Quadrupled

  7. If you have a pendulum of fixed mass and length and you increase the length of the path the mass travels, what happens to the period? • Increase • Decrease • Stays the same

  8. If you have a pendulum of fixed mass and length and you increase the length of the path the mass travels, what happens to the maximum velocity? • Increase • Decrease • Stays the same

  9. If you have a pendulum of fixed mass and length and you increase the length of the path the mass travels, what happens to the maximum acceleration? • Increase • Decrease • Stays the same

  10. The pendulum for a clock has a weight that can be adjusted up or down on the pendulum shaft. If your clock runs slow, what should you do? • Move weight up • Move weight down • You can’t fix the clock by moving the weight

  11. PAL #4 Pendulums • The initial kinetic energy is just the kinetic energy of the bullet • ½mv2 = (0.5)(0.01 kg)(500 m/s)2 = • The initial velocity of the block comes from the kinetic energy • KE = ½mv2 • v = (2KE/m)½ = ([(2)(1250)]/(5))½ = • Amplitude =xm, can get from total energy • Initial KE = max KE = total E = ½kxm • xm =(2E/k)½ = ([(2)(1250)]/(5000))½ = • Equation of motion = x(t) = xmcos(wt) • k = mw2 • w = (k/m)½ = [(5000/(5)]½ = 31.6 rad/s

  12. Uniform Circular Motion • Simple harmonic motion is uniform circular motion seen edge on • 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

  13. UCM and SHM

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

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

  16. Galileo’s Sketches

  17. Apparent Motion of Callisto

  18. 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

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

  20. Light Curve of 51 Peg

  21. 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 Fd = -bv • Then, x = xmcos(wt+f) e(-bt/2m) • e(-bt/2m) is called the damping factor and tells you by what factor the amplitude has dropped for a given time or: x’m = xm e(-bt/2m)

  22. 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

  23. Exponential Damping

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

  25. Damping

  26. 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 wd = the frequency of the driving force • The amplitude of the motion will increase the fastest when w=wd

  27. Resonance • Resonance occurs when you apply maximum driving force at the point where the system is experiencing maximum natural force • Example: pushing a swing when it is all the way up • All structures have natural frequencies

  28. Next Time • Read: 16.1-16.5 • Homework: Ch 15, P: 95, Ch 16, P: 1, 2, 6

  29. 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

  30. 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

  31. 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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