Simple Harmonic Motion
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Transcript Simple Harmonic Motion
Simple Harmonic Motion
Physics 202
Professor Vogel
(Professor Carkner’s
notes, ed)
Lecture 2
Simple Harmonic Motion
Any motion that repeats itself in a sinusoidal fashion
e.g. a mass on a spring
A mass that moves between +xm and -xm
with period T
Properties vary from a positive maximum to a
negative minimum
Position (x)
Velocity (v)
Acceleration (a)
The system undergoing simple harmonic motion
(SHM) is a simple harmonic oscillator (SHO)
SHM Snapshots
Key Quantities
Frequency (f) -- number of complete
oscillations per unit time
Unit=hertz (Hz) = 1 oscillation per second = s-1
Period (T) -- time for one complete oscillation
T=1/f
Angular frequency (w) -- w = 2pf = 2p/T
Unit = radians per second (360 degrees = 2p
radians)
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) -- the maximum displacement
from the center
Phase angle (f) -- offset due to not starting at
x=xm (“start” means t=0)
Remember that (wt+f) is in radians
SHM Formula Reference
SHM in Action
Consider SHM with f=0:
x = xmcos(wt)
Remember w=2p/T
t=0, wt=0, cos (0) = 1
x=xm
t=1/2T, wt=p, cos (p) = -1
x=-xm
t=T, wt=2p, cos (2p) = 1
x=xm
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
Amplitude, Period and Phase
Velocity
If we differentiate the equation for
displacement w.r.t. time, we get velocity:
v(t)=-wxmsin(wt + f)
Why is velocity negative?
Since the particle moves from +xm to -xm the
velocity must be negative (and then positive in the
other direction)
Velocity is proportional to w
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)
This equation is similar to the equation
for displacement
Making a substitution yields:
a(t)=-w2x(t)
x, v and a
Consider SMH with f=0:
x = xmcos(wt)
v = -wxmsin(wt) = -vmsin(wt)
a = -w2xmcos(wt) = -amcos(wt)
When displacement is greatest
(cos(wt)=1), velocity is zero and
acceleration is maximum
Mass is momentarily at rest, but
being pulled hard in the other
direction
When displacement is zero
(cos(wt)=0), velocity is
maximum and acceleration is
zero
Mass coasts through the middle at
high speed
Force
Remember that: a=-w2x
But, F=ma so,
F=-mw2x
Since m and w are constant we can write the
expression for force as:
F=-kx
Where k=mw2 is the spring constant
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
A simple 1-dimensional SHM system is
called a linear oscillator
Example: a mass on a spring
In such a system, k=mw2
We can thus find the angular frequency
and the period as a function of m and k
k
ω
m
m
T2π
k
Linear Oscillator
Application of the Linear
Oscillator: Mass in Free Fall
A normal spring scale does not work in
the absence of gravity
However, for a linear oscillator the
mass depends only on the period and
the spring constant:
T=2p(m/k)0.5
m/k=(T/2p)2
m=T2k/4p2
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
UFdxkxdx 1kx 2
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
Next Time
Read: 15.4-15.6