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Physics 202
Professor Vogel
(Professor Carkner’s
notes, ed)
Lecture 3
Amplitude, Period and Phase
Phase
The phase of SHM is the quantity in
parentheses, i.e. cos(phase)
The difference in phase between 2 SHM
curves indicates how far out of phase
the motion is
The difference/2p is the offset as a
fraction of one period
Example: SHO’s f=p & f=0 are offset 1/2
period
They are phase shifted by 1/2 period
SHM and Energy
A linear oscillator has a total energy E,
which is the sum of the potential and
kinetic energies (E=U+K)
U and K change as the mass oscillates
As one increases the other decreases
Energy must be conserved
SHM Energy Conservation
Potential Energy
Potential energy is the integral of force
2
1
UFdxkxdx  kx
2
From our expression for x
U=½kxm2cos2(wt+f)
Kinetic Energy
Kinetic energy depends on the velocity,
K=½mv2 = ½mw2xm2 sin2(wt+f)
Since w2=k/m,
K = ½kxm2 sin2(wt+f)
The total energy E=U+K which will
give:
E= ½kxm2
Pendulums
A mass suspended from a string and
set swinging will oscillate with SHM
We will first consider a simple pendulum
where all the mass is concentrated in the
mass at the end of the string
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
The Period of a Pendulum
 The the restoring force is:
F = -mg sin q
 For small angles sin q  q
 We can replace q with s/L
F=-(mg/L)s
 Compare to Hooke’s law F=-kx
 k for a pendulum is (mg/L)
 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 heavier mass requires more force to
move, but is acted on by a larger
gravitational force
A pendulum is a common method of
finding the local value of g
Friction and air resistance need to be taken
into account
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 attached to weights that try
to turn it
The gear is stopped by a toothed
mechanism attached to a pendulum of
period = 2 seconds
The mechanism disengages when the
pendulum is in the equilibrium
position and so allows the second hand
to move twice per cycle
Since the period is 2 seconds the second
hand advances once per second
Physical Pendulum
Real pendulums do not have all of their
mass at one point
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
Uniform Circular Motion
Simple harmonic motion is uniform circular
motion seen edge on
Consider a particle moving in a circle with
the origin at the center
Viewed edge-on the particle seems to be moving
back and forth between 2 extremes around the
origin
The projection of the displacement, velocity
and acceleration onto the edge-on circle are
described by the SMH equations
Uniform Circular Motion and SHM
y-axis
Particle moving
in circle
of radius xm
viewed edge-on:
Particle at time t
xm
angle = wt+f
x-axis
cos (wt+f)=x/xm
x=xm cos (wt+f)
x(t)=xm cos (wt+f)
Observing the Moons of
Jupiter
Galileo was the first person to observe
the sky with a telescope in a serious
way
He discovered the 4 inner moons of
Jupiter
Today known as the Galilean moons
He (and we) saw the orbit edge-on
Jupiter and Moons
Apparent Motion of Callisto