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

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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

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?

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?

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

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

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:

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...

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

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

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.

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

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.

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

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!

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

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?

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!

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

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?

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

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

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!

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.

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.