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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

w= 2p/T = 0.79

How long to get ½ back


arccos(5/10)/0.79 = t =1.3 seconds
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)
Simple Harmonic Motion
For motion with period = T and
angular frequency = w = 2p/T:
v=-wxmsin(wt + f)
The force is represented as:
where k=spring constant= mw2
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

2
1
UFdxkxdx  kx
2
From our expression for x
U=½kxm2cos2(wt+f)
Kinetic Energy

K=½mv2 = ½mw2xm2 sin2(wt+f)

K = ½kxm2 sin2(wt+f)
The total energy E=U+K which will
give:
E= ½kxm2
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
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
Pendulum Forces
Forces on a Pendulum
L
q
Tension
m
q
s
Restoring Force = mg sin q
Gravity = mg
The Period of a Pendulum
The the restoring force is:
F = -mg sin q

We can replace q with s/L
Compare to Hooke’s law F=-kx

Period for SHM is T = 2p (m/k)½
T=2p(L/g)½
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

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.
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
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=2p(I/mgh)½
Non-Simple Pendulum
Torsion Pendulum
Torsion Pendulum

If the disk is twisted a torque is exerted to
move it back due to the torsion in the wire:
tkq

We can use this to derive the expression for
the period:
T=2p(I/k)½
