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?  x = 5m   arccos(5/10)/0.79 = t =1.3 seconds

3. Simple Harmonic Motion  For motion with period = T and angular frequency =  = 2  /T: x=x m cos(  t +  ) v=-  x m sin(  t +  ) a=-  2 x m cos(  t +  )

4. 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 

5. Potential Energy  Potential energy is the integral of force  From our expression for x U=½kx m 2 cos 2 (  t+  )

6. 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

7. 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

8. The Period of a Pendulum  The the restoring force is:  For small angles sin   We can replace  with s/L  Compare to Hooke’s law F=-kx  k for a pendulum is (mg/L)  T=2  (L/g) ½  The period of a pendulum depends only on the length and g, not on mass

9. Damped SHM   The mass will slow down over time   The faster it is moving, the more energy it loses  F d = -bv  Where b is the damping constant  Then, x = x m cos(  t+  ) e (-bt/2m)  e (-bt/2m) is called the damping factor and tells you by what factor the amplitude has dropped for a given time or: x’ m = x m e (-bt/2m)

10. Energy and Frequency  The energy of the system is: E = ½kx m 2 e (-bt/m)   The period will change as well:  ’ = [(k/m) - (b 2 /4m 2 )] ½ 

11. Next Time  Read: 16.1-16.5  Homework: Ch 15, P: 35, 57, 95, Ch 16, P: 1, 2

12. Summary: Simple Harmonic Motion x=x m cos(  t+  ) v=-  x m sin(  t+  ) a=-  2 x m cos(  t+  )  =2  /T=2  f F=-kx  =(k/m) ½ T=2  (m/k) ½ U=½kx 2 K=½mv 2 E=U+K=½kx m 2

13. Summary: Types of SHM  Mass-spring T=2  (m/k) ½  Simple Pendulum T=2  (L/g) ½  The energy and amplitude of damped SHM falls off exponentially x = x undamped e (-bt/2m)

14. Consider SHM with no phase shift, when the mass is moving fastest, a)It is at the end (x m ) and acceleration is maximum (a m ) b)It is at the end (x m ) and acceleration is zero c)It is at the middle (x=0) and acceleration is maximum (a m ) d)It is at the middle (x=0) and acceleration is zero e)It is half way between the end and the middle and the acceleration is zero

15. Consider SHM with no phase shift, when the mass has the most acceleration, a)It is at the end (x m ) and velocity is maximum (v m ) b)It is at the end (x m ) and velocity is zero c)It is at the middle (x=0) and velocity is maximum (v m ) d)It is at the middle (x=0) and velocity is zero e)It is half way between the end and the middle and the velocity is zero

16. Consider SHM with no phase shift, when t=0, a)x=0, v=0, a=0 b)x=x m, v=v m, a=a m c)x=0, v=v m, a=-a m d)x=x m, v=0, a=-a m e)x=-x m, v=0, a=a m

17. Consider SHM with no phase shift, when t=(1/2)T, a)x=0, v=0, a=0 b)x=x m, v=v m, a=a m c)x=0, v=v m, a=-a m d)x=x m, v=0, a=-a m e)x=-x m, v=0, a=a m