1. Pendulums Physics 202 Professor Lee Carkner Lecture 4 “The sweep of the pendulum had increased … As a natural consequence its velocity was also much greater.” --Edgar Allan Poe, “The Pit and the Pendulum”

2. PAL #3 SHM  Equation of motion for SHM, pulled 10m from rest, takes 2 seconds to get back to rest    = 2  /T = 0.79   How long to get ½ back    arccos(5/10)/0.79 = t =1.3 seconds

3. PAL #3 SHM (cont.)  Max speed  v = -  x m sin(  t)  v max when sin =1   Where is max v?   Max acceleration  a = -  2 x m cos(  t)   Where is max a?  The ends (max force from spring)

4. Simple Harmonic Motion  For motion with period = T and angular frequency =  = 2  /T: v=-  x m sin(  t +  )  The force is represented as:  where k=spring constant= m  2

5. SHM and Energy  A linear oscillator has a total energy E, which is the sum of the potential and kinetic energies (E=U+K)   As one goes up the other goes down 

6. SHM Energy Conservation

7. Potential Energy   From our expression for x U=½kx m 2 cos 2 (  t+  )

8. Kinetic Energy  K=½mv 2 = ½m  2 x m 2 sin 2 (  t+  )  K = ½kx m 2 sin 2 (  t+  )  The total energy E=U+K which will give: E= ½kx m 2

9. Types of SHM  Every system of SHM needs a mass to store kinetic energy and something to store the potential energy (to provide the springiness)  There are three types of systems that we will discuss:     Each system has an equivalent for k

10. Pendulums  A mass suspended from a string and set swinging will oscillate with SHM   Consider a simple pendulum of mass m and length L displaced an angle  from the vertical, which moves it a linear distance s from the equilibrium point

11. Pendulum Forces

12. Forces on a Pendulum m Gravity = mg Tension  Restoring Force = mg sin   L s

13. The Period of a Pendulum  The the restoring force is: F = -mg sin    We can replace  with s/L  Compare to Hooke’s law F=-kx   Period for SHM is T = 2  (m/k) ½ T=2  (L/g) ½

14. Pendulum and Gravity  The period of a pendulum depends only on the length and g, not on mass   A pendulum is a common method of finding the local value of g 

15. The Pendulum Clock Invented in 1656 by Christiaan Huygens, the pendulum clock was the first timekeeping device to achieve an accuracy of 1 minute per day.

16. Application of a Pendulum: Clocks  Since a pendulum has a regular period it can be used to move a clock hand  Consider a clock second hand attached to a gear   The gear is stopped by a toothed mechanism attached to a pendulum of period = 2 seconds   Since the period is 2 seconds the second hand advances once per second

17. Physical Pendulum   Properties of a physical pendulum depend on its moment of inertia (I) and the distance between the pivot point and the center of mass (h), specifically: T=2  (I/mgh) ½

18. Non-Simple Pendulum

19. Torsion Pendulum

20.   If the disk is twisted a torque is exerted to move it back due to the torsion in the wire:    We can use this to derive the expression for the period: T=2  (I/  ) ½ 