Torque and Simple Harmonic Motion

1. Torque and Simple Harmonic Motion 8.01 Week 13D2 Today’s Reading Assignment Young and Freedman: 14.1-14.6

2. Problem Set 11 Due Thursday Nov 1 9 pm Sunday Tutoring in 26-152 from 1-5 pm W013D3 Reading Assignment Young and Freedman: 14.1-14.6 Announcements

3. Simple Pendulum

4. Table Problem: Simple Pendulum by the Torque Method (a) Find the equation of motion for θ(t) using the torque method. (b) Find the equation of motion if θ is always <<1.

5. Table Problem: Simple Pendulum by the Energy Method • Find an expression for the mechanical energy when the pendulum is in motion in terms of θ(t) and its derivatives, m, l, and g as needed. • Find an equation of motion for θ(t) using the energy method.

6. Simple Pendulum: Small Angle Approximation Equation of motion Angle of oscillation is small Simple harmonic oscillator Analogy to spring equation Angular frequency of oscillation Period

7. Simple Pendulum: Approximation to Exact Period Equation of motion: Approximation to exact period: Taylor Series approximation:

8. Concept Question: SHO and the Pendulum Suppose the point-like object of a simple pendulum is pulled out at by an angle q0 << 1 rad. Is the angular speed of the point-like object equal to the angular frequency of the pendulum? Yes. No. Only at bottom of the swing. Not sure.

9. Demonstration Pendulum: Amplitude Effect on Period

10. Table Problem: Torsional Oscillator A disk with moment of inertia about the center of mass rotates in a horizontal plane. It is suspended by a thin, massless rod. If the disk is rotated away from its equilibrium position by an angle , the rod exerts a restoring torque given by At t = 0 the disk is released from rest at an angular displacement of . Find the subsequent time dependence of the angular displacement .

11. Worked Example: Physical Pendulum A general physical pendulum consists of a body of mass m pivoted about a point S. The center of mass is a distancedcmfrom the pivot point. What is the period of the pendulum.

12. Concept Question: Physical Pendulum A physical pendulum consists of a uniform rod of length l and mass m pivoted at one end. A disk of mass m1 and radius a is fixed to the other end. Suppose the disk is now mounted to the rod by a frictionless bearing so that is perfectly free to spin. Does the period of the pendulum • increase? • stay the same? • decrease?

13. Physical Pendulum Rotational dynamical equation Small angle approximation Equation of motion Angular frequency Period

14. Demo: Identical Pendulums, Different Periods Single pivot: body rotates about center of mass. Double pivot: no rotation about center of mass.

15. Small Oscillations

16. Small Oscillations Potential energy function for object of mass m Motion is limited to the region Potential energy has a minimum at Small displacement from minimum, approximate potential energy by Angular frequency of small oscillation

17. Concept Question: Energy Diagram 1 A particle with total mechanical energy E has position x > 0 at t = 0 1) escapes to infinity in the – x-direction 2) approximates simple harmonic motion 3) oscillates around a 4) oscillates around b 5) periodically revisits a and b 6) not enough information

18. Concept Question: Energy Diagram 2 A particle with total mechanical energy E has position x > 0 at t = 0 1) escapes to infinity 2) approximates simple harmonic motion 3) oscillates around a 4) oscillates around b 5) periodically revisits a and b 6) not enough information

19. Concept Question: Energy Diagram 3 A particle with total mechanical energy E has position x > 0 at t = 0 1) escapes to infinity 2) approximates simple harmonic motion 3) oscillates around a 4) oscillates around b 5) periodically revisits a and b 6) not enough information

20. Concept Question: Energy Diagram 4 A particle with total mechanical energy E has position x > 0 at t = 0 1) escapes to infinity 2) approximates simple harmonic motion 3) oscillates around a 4) oscillates around b 5) periodically revisits a and b 6) not enough information

21. Concept Question: Energy Diagram 5 A particle with total mechanical energy E has position x > 0 at t = 0 1) escapes to infinity 2) approximates simple harmonic motion 3) oscillates around a 4) oscillates around b 5) periodically revisits a and b 6) not enough information

22. Table Problem: Small Oscillations • A particle of effective mass m is acted on by a potential energy given by • where and are positive constants • Sketch as a function of . • Find the points where the force on the particle is zero. Classify them as stable or unstable. Calculate the value of at these equilibrium points. • If the particle is given a small displacement from an equilibrium point, find the angular frequency of small oscillation.

23. Appendix

24. Simple Pendulum: Mechanical Energy Velocity Kinetic energy Initial energy Final energy Conservation of energy

25. Simple Pendulum: Angular Velocity Equation of Motion Angular velocity Integral form Can we integrate this to get the period?

26. Simple Pendulum: Integral Form Change of variables “Elliptic Integral” Power series approximation Solution

27. Simple Pendulum: First Order Correction Period Approximation First order correction