1. Lecture 37 Physics 111 Chapter 13: Simple Harmonic Motion
springs & pendulums
travelling waves
Friday, December 4, 1998
Physics 111

2. 4

3. Let’s instead look for the moment at a mass-
Our wealthy socialite has been playing with his spring in a gravitational field…(that complicates our problem--although it leads to the same answer).
Let’s instead look for the moment at a mass-
spring system that’s oriented horizontally on
a frictionless surface--so we can take gravity
out of the picture.
Fixed
end

4. If we measure x as the distance from the
Let’s use an energy
conservation
argument to explain
the motion.
Fixed
end x
If we measure x as the distance from the
equilibrium (natural length) of the spring,
what will the energy of the system be when
the mass reaches its maximum displacement
to the right at x = A?

5. And what will the energy of the system be
Let’s use an energy
conservation
argument to explain
the motion.
Fixed
end x
And what will the energy of the system be
when the mass passes through its
equilibrium position at x = 0?

6. Conservation of energy says that
Let’s use an energy
conservation
argument to explain
the motion.
Fixed
end x
Conservation of energy says that
Which implies that

7. We've Done It! Let’s use an energy conservation argument to explain
the motion.
Fixed
end x
We've Done It!
Recall that the period is given by

8. So... Let’s use an energy conservation argument to explain the motion.
Fixed
end x
So...
And recalling our definitions of frequency
and angular frequency, we get:

9. Let’s look at one more characteristic of our mass-spring system:
Let’s use an energy
conservation
argument to explain
the motion.
Fixed
end x
Let’s look at one more characteristic of our
mass-spring system:
How does the velocity of the mass relate
to its displacement from the equilibrium
position?
Using the point of maximum potential energy...

10. The total energy at any other point in the
Let’s use an energy
conservation
argument to explain
the motion.
Fixed
end x
The total energy at any other point in the
motion of the block is a mixture of kinetic
and potential given by: So

11. - Let’s use an energy conservation argument to explain the motion.
Fixed
end x
Solving for v, we find +
Note the
two solns -

12. In Lab... We also looked at pendulum motion.FT Fg s q
We said that the motion of a bob at the end of the pendulum can be thought of as acting under a restoring force, just like the mass on a spring. But this time, the agent responsible for the restoring force was gravity.

13. Here, the component of the
gravitational force that acts
perpendicularly to the string
supporting our green ball
results in the motion of the
pendulum. FT Fg s q
Using geometry, we find
that force to be

14. Notice that as the angle of
the pendulum changes through
the course of its motion, so
does the magnitude of this
force. FT Fg s q
The force has a minimum value
(0) when the angle is 0o, and
a maximum value at the extremes
of the motion.
This force acts just like the
restoring force in our mass-
spring system.

15. As long as the amplitude of the
oscillations remains relatively
small, we can use the small angle
approximation for sin q and
get FT Fg s q L
Which now has the form F = -kx

16. We now can simply use our results
from the motion of the mass-spring
system to describe the motion of
the pendulum. Most importantly,
we make the substitution FT Fg s q L
And we find the period of our
pendulum to be
Good for SMALL
amplitude
oscillations!

17. Wave Motion Transverse Waves Longitudinal Waves
What are waves?
Demo Time!
TRAVELLING WAVES:
Transverse Waves
Longitudinal
Waves

18. ENERGY Notice that while pieces of the slinky moved
up and down or left and right (depending
on your perspective) as the wave passed
by, those pieces ended up right back in
(nearly) the same places they started out!
Therefore, we note that waves do NOT
ultimately transport matter. They only
temporarily displace the matter in which
they move.
ENERGY
So what do waves carry?

19. Let’s try to plot the position of a small
piece of the slinky as a function of time
as a series of waves pass through it.
What kind of plot do you get?
Sinusoidal again! Just like the motion
of the mass-spring system and the
pendulum system!

20. Position of Piece of Slinky vs Time
Equilibrium
position
Amplitude
period
Position of Piece of Slinky vs Time

21. Now, draw a picture of the shape of the
slinky at ONE INSTANT in time. That is
make an x-y plot of the shape of the slinky
at a specific instant--as if your eye were
taking a photograph of the slinky.
Next, let the movie advance 1 frame in
your head. Sketch the shape of the slinky
at this next instant in time.
What kind of plots do you have?
Still sinusoidal, right?

22. Shape of Slinky at Time 1 crest Y-position trough wavelength
Equilibrium
position
Amplitude
wavelength
Shape of Slinky at Time 1
crest
Y-position
trough
X-position

23. Time 1 Time 2 Notice the wave has “traveled” to the right. Y-position
X-position
Time 2
Y-position
X-position

24. Okay, it’s relatively easy for me to measure
the wavelength of a wave, if I have a camera.
And I can certainly record the frequency
of the wave as well, just by counting the
number of crests that pass each second.
So, how fast is the wave moving?
Well, the wavelength gives me the
distance between crests. And the frequency
tells me how many pass per second, so
multiply the two gives me distance over time.
At least that has the right form!

25. v is the velocity of the wave
In fact, if you know the wavelength of a wave
and you know its frequency, you can find
its velocity. These three quantities are
related by the expression:
Where
v is the velocity of the wave
f is the frequency of the wave and
l is the wavelength of the wave.

26. Clearly, the medium through which the
waves travel affects the speed of the waves.
Let’s look at the speed at which waves
propagate through a tightly wound spring
and a loosely wound slinky.
The propagation speed is governed by
the mass per unit length (m) of the material
and the tension (FT) within the material.