Reading: Taylor 4.6 (Thornton and Marion 2.6)

Giancoli 8.9

PH421: Oscillations – lecture 1

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Goals for the pendulum module:(1) CALCULATE the period of oscillation if we know the

potential energy (using exact and approximate forms); specific example is the pendulum

(2) MEASURE the period of oscillation as a function of oscillation amplitude

(3) COMPARE the measured period to models that make different assumptions about the potential

(4) PRESENT the data and a discussion of the models in a coherent form consistent with the norms in physics writing

(5) CALCULATE the (approximate) motion of a pendulum by solving Newton's F=ma equation

2

How do you calculate how long it takes to get from one point to another?

Separation of x and t variables!

But what if v is not constant?

3

Suppose total energy is CONSTANT (we have to know it, or be able to find out what it is)

The case of a conservative force

4

Classical turning points

xL xR

B

x0

Example: U(x) = ½ kx2 , the harmonic oscillator

5

xL xRx0

Symmetry - time to go there is the same as time to go back (no damping)

SHO - symmetry about x0

xL -> x0 same time as for x0 -> xR

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xL=-A xR=Ax0

SHO - do we get what we expect?

Another way to specify E is via the amplitude A

Independent of A!

x0=0

http://www.wellesley.edu/Physics/Yhu/Animations/sho.html

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xL=-A xR=Ax0

You have seen this before in intro PH, but you didn't derive it this way. 8

Period of SHO is INDEPENDENT OF AMPLITUDEWhy is this surprising or interesting?

As A increases, the distance and velocity change. How does this affect the period for ANY potential?

A increases -> further to travel -> distance increases -> period increases

A increases -> more energy -> velocity increases -> period decreases

Which one wins, or is it a tie?

9

Look at example of simple pendulum (point mass on massless string). This is still a 1-dimensional problem in the sense that the motion is specified by one variable,

Your lab example is a plane pendulum. You will have to generalize: what length does L represent in this case? 10

Plot these to compare to SHO to pendulum…… 11

Period increases because v(x) decreases (E decreases) for the same amplitude. Effect is magnified for larger amplitudes. 12

Equivalent angular version?

13

Integrate both sides

E and U() are known - put them in

Resulting integral• do approximately by hand using series expansion (pendulum period worksheet on web page)OR• do numerically with Mathematica (notebook on web page) 14

xL xR

B

x0

“Everything is a SHO!”

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