Class Notes, Fall 2001

Week of	Aug 22	Sep 5	Oct 1	Nov 5	Phy 205 page
	Aug 27	Sep 10	Oct 8	Nov 12
		Sep 17	Oct 15	Nov 19
		Sep 24	Oct 22	Nov 26
			Oct 29		Notes for Fall, 2000

Phy 205, Week 11

29 Oct What is an elastic collision? Under what circumstances do they occur? The essential idea is that a collision is elastic (or very nearly so) if no energy (or very little energy) can go into forms other than kinetic. The most common ways for energy to dissipate is as heat or as bending of the material.

In the extreme opposite case, where everything sticks together, kinetic energy is certainly lost, going into other forms of energy. I explicitly calculated the change of kinetic energy for the case that one moving mass collides with a stationary one. Conservation of momentum gives the single equation in one dimension.

The change in the kinetic energy is

Use the computed value for v' and substitute into this expression to get

Spend some time analyzing this in the cases that one or the other mass is a lot larger than the other.

Density. What is it? Precisely?

31 Oct Restate the various ways to write relationships about energy. The original form of the work-energy theorem,

Wtotal = ó õ xf xi  Fx( x ) dx = Δ K = m vf2 / 2 - m vi2 / 2

When the force is a function of x alone, you can find the anti-derivative of F_x and evaluate the integral.

and the theorem can be rearranged as

where these expressions are short for the initial and final kinetic and potential energies.

Not every force has a potential energy associated with it. The common case of friction doesn't because when something is sliding, the value of the force depends not only on where it is but on which direction it is moving.

Do problem 6-71 in detail.

Various kinds of density. The first is "average volume-mass density" and that is the ratio of a mass to its volume. Just as with velocity, you have to take a limit to define this at a point. Just as Δx / Δt become dx / dt, this ΔM / ΔV becomes dM / dV.

You can have other densities. Population density involves people per area instead of mass per volume. You can have mass per area instead of mass per volume; sometimes that's more convenient.

One that we will use right away is "linear mass density" and that is a mass per length instead of a mass per volume. dM / dx.

2 Nov Use linear mass density to see how to evaluate the center of mass of a non-uniform (but straight) object. I worked out the approach in detail, starting from the definition for two point masses and building up from there. i did in detail the CM for an object with a linear density given by

where the numerical value of A = 0.01 kg/m³. In the end the numerical value of A didn't matter for the center of mass; it canceled out. In order to manipulate angular variables it's important to understand the meaning of the radian as a unit of angle. It comes down to one simple drawing: an angle at the center of a circle. The arc on the circumference is proportional to that angle and to the radius of the circle.

The only question is the value of the proportionality constant, c. As with many previous problems, it requires some other information. The circumference of a circle is 2 π R. If you use degrees for the angle you get the equation

so c = π / 180. This is very awkward to use, so the radian was invented as that unit of angle so that c comes out to be one. That implies that there have to be 2 π radians in a full circle. One radian is a bit less than 60 degrees. In calculus you learn how to differentiate the trigonometric functions, such as d(sin θ)/d θ = cos θ. These formula have to be in radians to work! If you are in degrees there are factors such as π/180 floating around.