## Classical Mechanics: Lecture 30 Notes

**Overview**

Today is a 50 minute lecture on simple harmonic motion, a topic that has applications for nearly every area of engineering. There is a fantastic lecture 33 coming up relating to the ideal gas law. Something to look forward to.

**Details**

For a spring,

F = ma = -kx

T = Ia = -r x F

Tp = mgl sin ) = – Ip O

sin 0= 0 if theta is in radians by the small angle approximation

0 + (lmg/Ip) 0 = 0

0= 0max cos(wt + p)

w = sqrt (bmg/Ip)

T = 2 pi/w

Yet again, he does a few demonstrations. He derives the period of a rod using the parallel axis theorem.

He does a very interesting derivation at 21 minutes in with a tube full of liquid. The water will oscillate in the tube.

Mass water = Volume x length x density

Lift the right side a distance y. The velocity anywhere inside the tube is the same. Assume no energy loss, no heat, no friction.

1/2 M v^2 = Area x height x row x g x height

or KE = PE

Damping makes the period longer.

He does a torsional pendulum. These are like a 1D spring where you twist the rope to make it rotate. How far can you go and get away with it? If you twist the spring too far, then you deform it. Then the period does become a function of the extension. Usually the period is independent of theta max.

**Review**

P = mv

U = mgh