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Simple Harmonic Motion
Physics 202
Professor Lee Carkner
Lecture 3
PAL #2 Archimedes
a) Iron ball removed from boat
Boat is lighter and so displaces less water

b) Iron ball thrown overboard

While sinking iron ball displaced water equal to
its volume

c) Cork ball thrown overboard
Both ball and boat still floating and so displaced
amount of water is the same

Simple Harmonic Motion

A particle that moves between 2 extremes in a
fixed period of time

Examples:
mass on a spring
pendulum
SHM Snapshots
Key Quantities
Frequency (f) -Unit=hertz (Hz) = 1 oscillation per second = s-1
Period (T) -T=1/f
Angular frequency (w) -- w = 2pf = 2p/T
Unit =
We use angular frequency because the
motion cycles
Equation of Motion
What is the position (x) of the mass at time (t)?
The displacement from the origin of a particle
undergoing simple harmonic motion is:
x(t) = xmcos(wt + f)
Amplitude (xm) -Phase angle (f) -Remember that (wt+f) is in radians
SHM Formula Reference
SHM in Action
Consider SHM with f=0:
x = xmcos(wt)

t=0, wt=0, cos (0) = 1

t=1/2T, wt=p, cos (p) = -1

t=T, wt=2p, cos (2p) = 1

Min
Rest
Max
10m
SHM Monster
Phase

The value of f relative to 2p indicates
the offset as a function of one period

It is phase shifted by 1/2 period
Amplitude, Period and Phase
Velocity
If we differentiate the equation for
displacement w.r.t. time, we get velocity:
v(t)=-wxmsin(wt + f)

Since the particle moves from +xm to -xm the
velocity must be negative (and then positive in the
other direction)

High frequency (many cycles per second) means
larger velocity
Acceleration
If we differentiate the equation for
velocity w.r.t. time, we get acceleration
a(t)=-w2xmcos(wt + f)

Making a substitution yields:
a(t)=-w2x(t)
Min
Rest
Max
10m
SHM Monster
Displacement, Velocity and
Acceleration
Consider SMH with f=0:
x = xmcos(wt)
v = -wxmsin(wt)
a = -w2xmcos(wt)

Mass is momentarily at rest, but being pulled hard
in the other direction

Mass coasts through the middle at high speed
Derivatives of SHM Equation
Force
Remember that: a=-w2x
But, F=ma so,
Since m and w are constant we can write the
expression for force as:
F=-kx

This is Hooke’s Law
Simple harmonic motion is motion where
force is proportional to displacement but
opposite in sign
Why is the sign negative?
Linear Oscillator

Example: a mass on a spring

We can thus find the angular frequency
and the period as a function of m and k
k
ω
m
m
T2π
k
Linear Oscillator
Application of the Linear
Oscillator: Mass in Free Fall

However, for a linear oscillator the
mass depends only on the period and
the spring constant:
m/k=(T/2p)2