Transcript Power Point presentation - Physics 420 UBC Physics Demonstrations

What is oscillatory motion?
• Oscillatory motion occurs when a force acting
on a body is proportional to the displacement
of the body from equilibrium.
F x
• The Force acts towards the equilibrium
position causing a periodic back and forth
motion.
What are some examples?
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Pendulum
Spring-mass system
Vibrations on a stringed instrument
Molecules in a solid
Electromagnetic waves
AC current
Many other examples…
What do these examples have in
common?
• Time-period, T. This is the time it takes
for one oscillation.
• Amplitude, A. This is the maximum
displacement from equilibrium.
• Period and Amplitude are scalers.
Forces
• Consider a mass with two springs
attached at opposite ends…
• We want to find an equation for the
motion.
• How should we start?
• Free-body diagram!!
Free body diagram
FSpring2
FGravity
FSpring1
Fnet = ma
• Fnet = Fg + Fs1 + Fs2 = ma
• Fnet = Fhorizontal + Fvertical
• Let us assume the mass does not move
up and down  Fvertical = 0
• So, Fnet = Fhorizontal = FS-horizontal(1+2)
• Thus, ma = m(d2x/dt2) = -kx
Fnet = kx
ma = m(d2x/dt2) = -kx
• Let k/m = 
(d2x/dt2) + x = 0
• This is the second order differential
equation for a harmonic oscillator. It is
your friend. It has a unique solution…
Simple Harmonic motion
• The displacement for a simple harmonic
oscillator in one dimension is…
x(t) = Acos(t + 
 is the angular frequency. It is constant.
 is the phase constant. It depends on the
initial conditions.
• What is the velocity?
• What is the acceleration?
• Velocity: differentiate x with respect to t.
dx/dt =
v(t) = -Asin(t + )
• Acceleration: differentiate v with respect
to t. dv/dt =
a(t) = -Acos(t + )
X(t)=Acos(t + )
x

Xo
A
t
T
a(t) = -Acos(t + )
a
A
ao
T
t
t
Using data
• The accelerometer will give us all the
information we need to confirm our
analysis
• We can measure all the parameters of
this particular system and use them to
predict the results of the accelerometer.
What can we measure without the
accelerometer?
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The mass, m
Hooke’s constant, k
That’s all!
T = 2/= 2m/k)1/2 (Recall = k/m)
Everything else depends on the initial
conditions. What does this tell us?
• The time period, T, is independent of the
initial conditions!
Energy
• The system operates at a particular
frequency, v, regardless of the energy of the
system.
v = 1/T = 2(k/m)1/2
• The energy of the system is proportional to
the square of the amplitude.
E = (1/2)kA2
Proof of E=(1/2)kA2
• Kinetic Energy  = (1/2)mv2
V =  SIN(t + )
 (1/2)MASIN2 (t + )
• Elastic potential energy U=(1/2)kx2
x = Acos(t + )
U  (1/2)kA2cos2./(t + )
E=K+U=
(1/2)kA2[sin2 (t + ) + cos2 (t + )]
= (1/2)kA2
Damping
• Simple harmonic motion is really a
simplified case of oscillatory motion
where there is no friction (remember our
FBD)
• For small to medium data sets this will
not affect our results noticeably.
for the rest of class...
• We are going to find k and m and
compare to the results of the
accelerometer
Some cool oscillatory motion
websites
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http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=236
http://www.kettering.edu/~drussell/Demos/SHO/mass.html
http://farside.ph.utexas.edu/teaching/301/lectures/node136.html
http://www.physics.uoguelph.ca/tutorials/shm/Q.shm.html