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Physics 201: Lecture 27 Oscillations

Physics 201: Lecture 27 Oscillations. Energy in Simple Harmonic Motion More Simple Harmonic Motion Examples Damped oscillations Forced oscillations and resonance. k. m. s. 0. s . L. Simple Harmonic Motion: Summary from Tuesday. k. s. 0. m. Force:. Solution:

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Physics 201: Lecture 27 Oscillations

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  1. Physics 201: Lecture 27Oscillations • Energy in Simple Harmonic Motion • More Simple Harmonic Motion Examples • Damped oscillations • Forced oscillations and resonance Physics 201, Fall 2006, UW-Madison

  2. k m s 0 s L Simple Harmonic Motion: Summary from Tuesday k s 0 m Force: Solution: s = A cos(t + ) Physics 201, Fall 2006, UW-Madison

  3. m x x=0 Analogy: marble in a bowl: PE KE PE KE PE “height” of bowl ~ x2 Potential Energy of a Spring Where x is measured fromthe equilibrium position PES x 0 Physics 201, Fall 2006, UW-Madison

  4. U K E U s -A 0 A Energy in SHM • For both the spring and the pendulum, we can derive the SHM solution by using energy conservation. • The total energy (K + U) of a system undergoing SHM will always be constant! • This is not surprising since there are only conservative forces present, hence K+U energy is conserved. Physics 201, Fall 2006, UW-Madison

  5. m x x=0 Energy Conservation Etotal = 1/2 Mv2 + 1/2 kx2 = constant KEPE KEmax = Mv2max /2 = M2A2 /2=kA2 /2 PEmax = kA2 /2 Etotal = kA2 /2 Physics 201, Fall 2006, UW-Madison

  6. Physical Pendulum General Physical Pendulum • Suppose we have some arbitrarily shaped solid of mass M hung on a fixed axis, and that we know where the CM is located and what the moment of inertia I about the axis is. • The torque about the rotation (z) axis for small  is (sin   ) = -Mgd-MgR z-axis R  x CM   d Mg where Physics 201, Fall 2006, UW-Madison

  7. Transport Tunnel • A straight tunnel is dug from Madison through the center of the Earth and out the other side. A physics 201 student jumps into the hole at noon. • What time does she get back to Madison? Physics 201, Fall 2006, UW-Madison

  8. Transport Tunnel... where MRis the mass inside radius R FG R RE MR but Physics 201, Fall 2006, UW-Madison

  9. Transport Tunnel... FG R RE MR Like a mass on a spring with Physics 201, Fall 2006, UW-Madison

  10. Transport Tunnel... Like a mass on a spring with So: FG R RE plug in g = 9.81 m/s2 and RE = 6.38 x 106 m get  = .00124 s-1 and so= 5067 s MR Physics 201, Fall 2006, UW-Madison

  11. Transport Tunnel... • So she gets back to Madison 84 minutes later, at 1:24 p.m. Physics 201, Fall 2006, UW-Madison

  12. Transport Tunnel... • Strange but true:The period of oscillation does not require that the tunnel be straight through the middle!! Any straight tunnel gives the same answer, as long as it is frictionless and the density of the Earth is constant. Physics 201, Fall 2006, UW-Madison

  13. Transport Tunnel... • More strange but true:An object orbiting the earth near the surface will have a period of the same length as that of the transport tunnel. a =2R 9.81 = 2 6.38(10)6 m  = .00124 s-1 so Physics 201, Fall 2006, UW-Madison

  14. wire   I Torsion Pendulum • Consider an object suspended by a wire attached at its CM. The wire defines the rotation axis, and the moment of inertia I about this axis is known. • The wire acts like a “rotational spring.” • When the object is rotated, the wire is twisted. This produces a torque that opposes the rotation. • In analogy with a spring, the torque produced is proportional to the displacement: = -k Physics 201, Fall 2006, UW-Madison

  15. Torsion Pendulum wire   I Torsion Pendulum... • Since = -k=Ibecomes where This is similar to the “mass on spring” except I has taken the place of m (no surprise). Physics 201, Fall 2006, UW-Madison

  16. SHM Solution... • Drawing of A cos(t ) • A = amplitude of oscillation T = 2/ A     - A Physics 201, Fall 2006, UW-Madison

  17. SHM Solution... • Drawing of A cos(t + )      - Physics 201, Fall 2006, UW-Madison

  18. SHM Solution... • Drawing of A cos(t - /2)  = /2 A     - = A sin(t)! Physics 201, Fall 2006, UW-Madison

  19. Damped Oscillations • In many real systems, nonconservative forces are present • This is no longer an ideal system • Friction is a common nonconservative force • In this case, the mechanical energy of the system diminishes in time, the motion is said to be damped • A graph for a damped oscillation • The amplitude decreases with time • The blue dashed lines represent the envelope of the motion Physics 201, Fall 2006, UW-Madison

  20. Damped Oscillation, Example • One example of damped motion occurs when an object is attached to a spring and submerged in a viscous liquid • The retarding force can be expressed as R = - b v where b is a constant • b is called the damping coefficient • The restoring force is – kx • From Newton’s Second Law SFx = -k x – bvx = max • When the retarding force is small compared to the maximum restoring force we can determine the expression for x • This occurs when b is small Physics 201, Fall 2006, UW-Madison

  21. Damping Oscillation, Example • The position can be described by • The angular frequency will be • When the retarding force is small, the oscillatory character of the motion is preserved, but the amplitude decreases exponentially with time • The motion ultimately ceases Physics 201, Fall 2006, UW-Madison

  22. Types of Damping • 0 is also called the natural frequency of the system • If Rmax = bvmax < kA, the system is said to be underdamped • When b reaches a critical value bc such that bc / 2 m = w0 , the system will not oscillate • The system is said to be critically damped • If Rmax = bvmax > kA and b/2m > w0, the system is said to be overdamped • For critically damped and overdamped there is no angular frequency Physics 201, Fall 2006, UW-Madison

  23. Forced Oscillations • It is possible to compensate for the loss of energy in a damped system by applying an external sinusoidal force (frequency ) • The amplitude of the motion remains constant if the energy input per cycle exactly equals the decrease in mechanical energy in each cycle that results from resistive forces • After a driving force on an initially stationary object begins to act, the amplitude of the oscillation will increase • After a sufficiently long period of time, Edriving = Elost to internal • Then a steady-state condition is reached • The oscillations will proceed with constant amplitude • The amplitude of a driven oscillation is • w0 is the natural frequency of the undamped oscillator Physics 201, Fall 2006, UW-Madison

  24. Resonance • When w » w0 an increase in amplitude occurs • This dramatic increase in the amplitude is called resonance • The natural frequencyw0 is also called the resonance frequency • At resonance, the applied force is in phase with the velocity and the power transferred to the oscillator is a maximum • The applied force and v are both proportional to sin (wt + f) • The power delivered is F . v • This is a maximum when F and v are in phase • Resonance (maximum peak) occurs when driving frequency equals the natural frequency • The amplitude increases with decreased damping • The curve broadens as the damping increases • The shape of the resonance curve depends on b Physics 201, Fall 2006, UW-Madison

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