Pendulums

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
• w 2p/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 -wxm sin(wt)
• v max when sin 1
• Where is max v?
• Max acceleration
• a -w2xm cos(wt)
• Where is max a?
• The ends (max force from spring)

4
Simple Harmonic Motion

• For motion with period T and angular frequency
  w 2p/T
• v-wxmsin(wt f)
• The force is represented as
• where kspring constant mw2

5
SHM and Energy

• A linear oscillator has a total energy E, which
  is the sum of the potential and kinetic energies
  (EUK)
• As one goes up the other goes down

6
SHM Energy Conservation
7
Potential Energy

• From our expression for x
• U½kxm2cos2(wtf)

8
Kinetic Energy

• K½mv2 ½mw2xm2 sin2(wtf)
• K ½kxm2 sin2(wtf)
• The total energy EUK which will give
• E ½kxm2

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 q from the vertical, which
  moves it a linear distance s from the equilibrium
  point

11
Pendulum Forces
12
Forces on a Pendulum
L
q
Tension
s
m
q
Restoring Force mg sin q
Gravity mg
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The Period of a Pendulum

• The the restoring force is
• F -mg sin q
• We can replace q with s/L
• Compare to Hookes law F-kx
• Period for SHM is T 2p (m/k)½
• T2p(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
• T2p(I/mgh)½

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Non-Simple Pendulum
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Torsion Pendulum
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Torsion Pendulum

• If the disk is twisted a torque is exerted to
  move it back due to the torsion in the wire
• tkq
• We can use this to derive the expression for the
  period
• T2p(I/k)½