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Lecture 25: Chapter 13 Vibrations: Simple Harmonic Motion; Damped oscillation

Lecture 25: Chapter 13 Vibrations: Simple Harmonic Motion; Damped oscillation. Reminder: Hour Exam III, Thur , Dec. 3: 5:45 - 7 PM 165 Bascom : 302, 303, 304, 306, 312, 318, 320, 324 B-10 Ingraham : 305, 313, 317, 321, 322, 327, 328, 329, 330

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Lecture 25: Chapter 13 Vibrations: Simple Harmonic Motion; Damped oscillation

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  1. Lecture 25: Chapter 13Vibrations:Simple Harmonic Motion; Damped oscillation Reminder: • Hour Exam III, Thur, Dec. 3: 5:45 - 7 PM • 165 Bascom: 302, 303, 304, 306, 312, 318, 320, 324 B-10 Ingraham: 305, 313, 317, 321, 322, 327, 328, 329, 330 • 3650 Humanities: 307, 308, 309, 310, 311, 314, 315, 319, 323, 326 (all the same as Exam I) • Material from Chapters 9,10,11,12 inclusive • One page of notes (8.5” x 11”) allowed • 20 multiple choice questions plus test code • Scantron will be used - bring #2 HB pencils + calculator • You must know your section number (301 - 330), fill it in on the test • Alternative time signup available. Room Cham 4320 (the labs) Physics 103, Fall 2009, U. Wisconsin

  2. relaxed position FX = 0 x x=0 Springs • Hooke’s Law:The force exerted by a spring is proportional to the distance the spring is stretched or compressed from its relaxed position. • FX = -k xWhere xis the displacement from the relaxed position and k is the constant of proportionality. (often called “spring constant” or “force constant”) Physics 103, Fall 2009, U. Wisconsin

  3. Springs • Hooke’s Law:The force exerted by a spring is proportional to the distance the spring is stretched or compressed from its relaxed position. • FX = -k xWhere xis the displacement from the relaxed position and k is the constant of proportionality. (often called “spring constant” or “force constant”) relaxed position FX = -kx > 0 x • x  0 x=0 Physics 103, Fall 2009, U. Wisconsin

  4. Springs • Hooke’s Law:The force exerted by a spring is proportional to the distance the spring is stretched or compressed from its relaxed position. • FX = -k xWhere xis the displacement from the relaxed position and k is the constant of proportionality. (often called “spring constant” or “force constant”) relaxed position FX = - kx < 0 x • x > 0 x=0 Physics 103, Fall 2009, U. Wisconsin

  5. X=0 X=A; v=0; a=-amax X=0; v=-vmax; a=0 X=-A; v=0; a=amax X=0; v=vmax; a=-0 X=A; v=0; a=-amax X=-A X=A Springs and Simple Harmonic Motion F=-kx F=ma ma = -kx a= -kx/m x = -ma/k Physics 103, Fall 2009, U. Wisconsin

  6. x = R cos q =R cos (wt) sinceq = wt What does moving in a circlehave to do with moving back & forth in a straight line ?? x x 1 1 R 2 2 8 8 q R 7 3 3  0 y 7 4 6 -R 4 6 5 5 Physics 103, Fall 2009, U. Wisconsin

  7. Simple Harmonic Motion x(t) = [A]cos(t) v(t) = -[A]sin(t) a(t) = -[A2]cos(t) Maximum value xmax = A vmax = A amax = A2 Angular frequency  = 2f = 2/T Physics 103, Fall 2009, U. Wisconsin

  8. X=A; v=0; a=-amax Springs and Simple Harmonic Motion xmax = A vmax = A amax = 2A a(t) = -2x(t) F = ma = -m 2x F = -kx  2 = k/m Conclusion: for mass on spring, Physics 103, Fall 2009, U. Wisconsin

  9. m x x=0 Potential Energy of a Spring Where x is measured fromthe equilibrium position PES Analogy: marble in a bowl: PE  KE  PE  KE  PE “height” of bowl  x2 x 0 Physics 103, Fall 2009, U. Wisconsin

  10. m Same thing for a vertical spring: y y=0 Where y is measured fromthe equilibrium position PES y 0 Physics 103, Fall 2009, U. Wisconsin

  11. y m m y=0 x x=0 In either case... Etotal = 1/2 Mv2 + 1/2 kx2 = constant KEPE KEmax = Mv2max /2 = M2A2 /2=kA2 /2 PEmax = kA2 /2 Etotal = kA2 /2 Physics 103, Fall 2009, U. Wisconsin

  12. x +A CORRECT t -A Lecture 25,Preflight 5 A mass on a spring oscillates back & forth with simple harmonic motion of amplitude A. A plot of displacement (x) versus time (t) is shown below. At what points during its oscillation is the speed of the block biggest? 1. When x = +A or -A (i.e. maximum displacement) 2. When x = 0 (i.e. zero displacement) 3. The speed of the mass is constant xmax = A vmax = A x(t) = [A]cos(t)v(t) = -[A]sin(t) Physics 103, Fall 2009, U. Wisconsin

  13. x +A CORRECT t -A Lecture 25,Preflight 6 A mass on a spring oscillates back & forth with simple harmonic motion of amplitude A. A plot of displacement (x) versus time (t) is shown below. At what points during its oscillation is the magnitude of the acceleration of the block biggest? 1. When x = +A or -A (i.e. maximum displacement) 2. When x = 0 (i.e. zero displacement) 3. The acceleration of the mass is constant x(t) = [A]cos(t)v(t) = -[A]sin(t)a(t) = -[A2]cos(t) xmax = Avmax = Aamax = 2A Physics 103, Fall 2009, U. Wisconsin

  14. CORRECT Lecture 25,Preflight 7 If the amplitude of the oscillation (same block and same spring) was doubled, how would the period of the oscillation change? (The period is the time it takes to make one complete oscillation) 1. The period of the oscillation would double.2. The period of the oscillation would be halved3. The period of the oscillation would stay the same x +2A t -2A Physics 103, Fall 2009, U. Wisconsin

  15. CORRECT Lecture 25,Preflight 4 In Case 1 a mass on a spring oscillates back and forth. In Case 2, the mass is doubled but the spring and the amplitude of the oscillation is the same as in Case 1. In which case is the maximum kinetic energy of the mass the biggest? 1. Case 12. Case 23. Same KEmax = PEmax = kA2/2 = Etotal Since A and k are the same the maximum kinetic energy will also be the same. Physics 103, Fall 2009, U. Wisconsin

  16. q L x(t) mg Small Oscillations of a Pendulum • For “small oscillation”, period does not depend on • mass • amplitude Physics 103, Fall 2009, U. Wisconsin

  17. CORRECT Lecture 25,Preflight 9 A pendulum is hanging vertically from the ceiling of an elevator. Initially the elevator is at rest and the period of the pendulum is T. Now the pendulum accelerates upward. The period of the pendulum will now be 1. greater than T 2. equal to T 3. less than T “Effective g” is larger when accelerating upward (you feel heavier) Physics 103, Fall 2009, U. Wisconsin

  18. Physical Pendulum • A physical pendulum can be made from an object of any shape • The center of mass oscillates along a circular arc • The period of a physical pendulum is • I is the object’s moment of inertia • m is the object’s mass • For a simple pendulum, I = mL2 and the equation becomes that of the simple pendulum as seen before Physics 103, Fall 2009, U. Wisconsin

  19. Damped Oscillations • Only ideal systems oscillate indefinitely • In real systems, friction retards the motion • Friction reduces the total energy of the system and the oscillation is said to be damped • Example: Shock absorber: With a low viscosity fluid, the vibrating motion is preserved, but the amplitude of vibration decreases: This is known as underdamped oscillation Physics 103, Fall 2009, U. Wisconsin

  20. More Types of Damping • With a higher viscosity, the object returns rapidly to equilibrium after it is released and does not oscillate • The system is said to be critically damped • With an even higher viscosity, the piston returns to equilibrium without passing through the equilibrium position, but the time required is longer • This is said to be over damped Plot a : under damped Plot b : critically damped Plot c : over damped Physics 103, Fall 2009, U. Wisconsin

  21. CORRECT Case 1 Case 2 Lecture 25,Preflight 10 In Case 1 two people pull on the same end of a spring whose other end is attached to a wall. In Case 2, the same two people pull with the same forces, but this time on opposite ends of the spring. In which case does the spring stretch the most? 1. Case 12. Case 23. Same Physics 103, Fall 2009, U. Wisconsin

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