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Announcements

Announcements. Quiz #10 (last one!) this Friday Units 19 (Angular Momentum) + 21 (Simple Harmonic Motion ). Lecture 22: Simple and Physical Pendula. Today’s Concept: Simple Harmonic Motion: Motion of a Pendulum. Torsion Pendulum. wire. τ. θ. I.

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Announcements

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  1. Announcements • Quiz #10 (last one!) this Friday Units 19 (Angular Momentum) + 21 (Simple Harmonic Motion)

  2. Lecture 22: Simple and Physical Pendula Today’s Concept: Simple Harmonic Motion: Motion of a Pendulum

  3. Torsion Pendulum wire τ θ I Q: In the prelecture the equation for restoring torque is given as T=-k*theta in clockwise direction..so if the restoring torque is in counter clockwise directions then would T be positive?

  4. CheckPoint: Torsion Pendulum Clock A torsion pendulum is used as the timing element in a clock as shown. The speed of the clock is adjusted by changing the distance of two small disks from the rotation axis of the pendulum. If we adjust the disks so that they are closer to the rotation axis, the clock runs: A) Faster B) Slower Small disks Moving the disks inward decreases I, so ωwill increase (period will decrease)

  5. Pendulum RCM θ Forsmall θ XCM Mg RCM θ XCM arc-length=RCMθ

  6. The Simple Pendulum The simple case θ L pivot RCM θ CM The general case I is always proportional to M, so… Frequency is ALWAYS independent of mass

  7. CheckPoint: Pendulum Clock A simple pendulum is used as the timing element in a clock as shown. An adjustment screw is used to make the pendulum shorter (longer) by moving the weight up (down) along the shaft that connects it to the pivot. If the clock is running too fast, the weight needs to be moved: A) Up B) Down Adjustment screw L is the distance between the pivot and the center of mass. Moving the weight down increases L, which decreases ω(period will increase).

  8. The Stick Pendulum pivot RCM θ CM M Same period

  9. CheckPoint: Two Pendula Case 1 Case 2 In Case 1 a stick of mass mand length Lis pivoted at one end and used as a pendulum. In Case 2 a point particle of mass mis attached to the center of the same stick. In which case is the period of the pendulum the longest? A) Case 1 B) Case 2 C) Same m m m C is not the right answer. Lets work through it

  10. ACT Case 1 Case 2 In Case 1 a stick of mass mand length Lis pivoted at one end and used as a pendulum. In Case 2 a point particle of mass mis attached to a string of length L/2? In which case is the period of the pendulum longest? A) Case 1 B) Case 2 C) Same m

  11. m m ACT Suppose you start with 2 different pendula, one having period T1 and the other having period T2. T2 T1>T2 T1 m Now suppose you make a new pendulum by hanging the first two from the same pivot and gluing them together. What is the period of the new pendulum? A) T1B) T2C) In between

  12. CheckPoint Revote: Two Pendula Case 1 Case 2 In Case 1 a stick of mass mand length Lis pivoted at one end and used as a pendulum. In Case 2 a point particle of mass mis attached to the center of the same stick. In which case is the period of the pendulum the longest? A) Case 1 B) Case 2 C) Same m m m Now lets work through it in detail

  13. Case 2 Case 1 m m Lets compare for each case. m

  14. ACT Case 2 Case 1 m m Lets compare for each case. m (A) (B) (C)

  15. So we can work out Case 2 Case 1 m m m In which case is the period longest? A) Case 1 B) Case 2 C) They are the same

  16. The Small Angle Approximation - Exact expression % difference between θand sinθ RCM θ Angle (degrees) XCM arc-length=RCMθ

  17. ACT A pendulum is made by hanging a thin hoola-hoop of diameter Don a small nail. What is the angular frequency of oscillation of the hoop for small displacements? (ICM= mR2for a hoop) Pivot A) B) C) D

  18. The angular frequency of oscillation of the hoop for small displacements will be given by Use parallel axis theorem: I = ICM+ mR2 = mR2+ mR2=2mR2 pivot (nail) R X CM So m

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