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”

PAL #3 SHM  Equation of motion for SHM, pulled 10m from rest, takes 2 seconds to get back to rest  x m = 10 meters, T = (4)(2) = 8 seconds   = 2  /T = 0.79   How long to get ½ back?  x = 5m  5 = 10 cos (0.79t) 

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

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

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

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

Simple Harmonic Motion  x=x m cos(  t +  ) v=-  x m sin(  t +  ) a=-  2 x m cos(  t +  )  The force is represented as:  where k=spring constant= m  2

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 

SHM Energy Conservation

Potential Energy   U=½kx m 2 cos 2 (  t+  )

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

Types of SHM   There are three types of systems that we will discuss:    Torsion Pendulum (torsion in a wire)  Each system has an equivalent for k

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

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

Pendulum and Gravity   A heavier mass requires more force to move, but is acted on by a larger gravitational force   Friction and air resistance need to be taken into account

Application of a Pendulum: Clocks  Since a pendulum has a regular period it can be used to move a clock hand   The gear is attached to weights that try to turn it   The mechanism disengages when the pendulum is in the equilibrium position and so allows the second hand to move twice per cycle 

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

Non-Simple Pendulum

Torsion Pendulum

  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 

Next Time  Read:  Homework: Ch 15, P: 35, 57, 89 (+1 extra not in book)