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Lecture

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University of Toronto St. George

Physics

PHY354H1

Erich Poppitz

Winter

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PHY254 Lecture 7
Oscillations in Phase Space and Dynamics
▯ Readings: For Oscillation dynamics: Morin 4.1
▯ Phase space representations of oscillations, simple harmonic motion
Begin with: beats.py: show what happens when phi1 and phi2 aren’t 0.
Show what happens when the amplitudes aren’t equal. Bring tuning forks
to demonstrate beats. Also mention "carrier wave" de▯nition correction.
Phase space representation
▯ A phase space plot is a parametric plot of velocity or momentum ver-
sus position. These plots are used extensively in the study of ordinary
di▯erential equations, dynamical systems, studies of chaos, and so on.
In these notes, we want to develop techniques for calculating and in-
terpreting phase space plots.
▯ Suppose we have a particle undergoing sinusoidal motion according to
x = Acos!t. Its velocity is x _ = ▯!Asin!t. Say, for example, its a
mass on a spring. We can plot the position and velocity as a function
of time:
1.0
x/A
v/(A\omega)
0.5
0.0
0.5
1.0 1 2 3 4 5 6
t/…
1 [The script used to create the plots in this section is phase space plots.py]
▯ But this plot contains a lot of redundant information and does not
reveal a clear relationship between x and v. The phase plot takes x
and v as coordinates in the plane, and plots the parameteric curve
(x(t);v(t)) = (Acos!t;▯!Asin!t) (1)
which describes an ellipse with horizontal and vertical axes. This curve
is plotted below:
2.0
1.5
1.0
0.5
…
/ 0.0
v
0.5
1.0
1.5
2.0
2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0
x/…
▯ Notice that points where the spring is compressed or stretched maxi-
mally are points where the velocity is 0, and points of maximum velocity
occur at x=0.
▯ This curve describes the whole motion because it’s periodic. It proceeds
clockwise.
2 2 2
▯ To see why the shape is elliptical, we see that x = A cos (!t) and
_ = ! A sin (!t). Thus
x 2 _2
+ = cos (!t) + sin (!t) = 1; (2)
A 2 ! A 2
which is the equation of an ellipse in the x _-x plane.
2 ▯ We can plot phase curves for many types of motion.
▯ Example: Plot the z-z _ phase trajectory of a particle freely falling in
Earth’s gravitational ▯eld, if it is initially projected upwards from z 0
with speed v .
0
▯ Answer: The motion satis▯es z _ = v 0 gt and z = z + v 0 ▯ 0 1gt .
2
This means the z _ ▯ z curve must be a parabola. If you want proof,
solve for t in the z_ equation and plug it into the z equation:
2 2
v0 1 2 v 0 _
z = z 0 (v0▯ z _) ▯ (v 0 z _) = z +0 ▯ (3)
g 2g g 2g
This represents a parabola opening to the left, as follows:
10 Phase plot for projectile m0tion: v =10.0 m/s
0
10
z
20
30
480 70 60 50 40 30 20 10 0 10
z
This graph was made with the python program phase space plot.py
▯ Make sure you understand the relationship between the motion and the
plot: The velocity decreases from v to zero as the particle approaches
0
its maximum height (the rightmost point), then as the particle falls the
velocity becomes more negative.
▯ We will often plot systems of interest in phase space, as a way of sum-
marizing their behaviour.
3 Oscillation Dynamics
▯ Oscillations generally occur in physics because of restoring forces. Restor-
ing forces are usually dependent on the displacement from an equilib-
rium position (i.e. they usually depend on "x" somehow).
▯ For example, imagine we displace a particle on a spring a distance x
from its equilibrium. As long as the displacement isn’t too large, we
▯nd (or Hooke found) that the force is proportional to the displacement,
and in the direction opposite to the displac

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