Class Notes for Fall, 2000

Week of	Aug 23	Sep 6	Oct 2	Nov 6	Phy 205 page
	Aug 28	Sep 11	Oct 9	Nov 13
		Sep 18	Oct 16	Nov 20	Notes for Fall, 1999
		Sep 25	Oct 23	Nov 27
			Oct 30

Phy 205, Week 11

30 Oct Discuss some of the difficulties that some people have in examining the properties of algebraic expressions. You can't change the dimensions of something by making a simplifying approximation. An acceleration is not approximately a force.

Also, acceleration and velocity aren't the same thing. If an object is "going" forward, its acceleration can be either forward or backward.

Elastic collisions. In any collision, momentum and total energy are conserved. Energy comes in so many forms however that it's not always useful. It can happen that kinetic energy by itself is conserved. This is called a completely elastic collision, and is often a good approximation when things bounce. Some types of atomic collisions work this way, and in the demo with a pair of carts it's pretty well obeyed.

Work out the equations using conservation of momentum and of kinetic energy in one dimension. I did the special case where one of the two masses is at rest, as that shows most of the useful results. There's a fair amount of algebra, but I tried to spend most time on the interpretation of the results.

1 Nov Do an example from the homework problems, one involving a collision in two dimensions. Extend the question by asking what happens to the kinetic energy. Is it conserved? (no) What is the change in the kinetic energy? Here the main difficulty is in computing the kinetic energy after the collision. I set it up, but didn't finish it. You can compute the final speed squared needed for the kinetic energy either by a direct application of the Pythagorean theorem or by using the dot product.

Rotation. How to describe it by a single coordinate for rotation about a single axis. the coordinate here is an angle, q. In one dimensional motion I can describe properties of the system by the velocity v_x=dx/dt and by the acceleration a_x=dv_x/dt. In the same way I can use the derivatives of the angle to define the angular velocity and the angular acceleration

These are manipulated in pretty much the same way that you manipulated x, v_x, and a_x in one dimensional motion. Only here the interpretation will be in terms of angles instead of distances. The tachometer that appears on the dashboard of many cars measures w, thought not usually in radians per second.

There's a simple and very useful relation between angular motion and linear motion. When an object has angular velocity w, the linear speed of a point at a distance R from the center of rotation is v = wR. I went through the derivation of this, and you can see from the derivation that this works only when the angles are expressed in radians, not degrees or revolutions.

3 Nov Do one of the homework problems as an example.

Kinetic energy of a rotating, rigid body. Do this for two masses rotating about a common axis with angular speed w. The kinetic energy is the sum

The speeds of the two masses can be computed from their angular speed, using the relation v = rw.

The quantity in parentheses is the "moment of inertia" for the system, denoted I.

If you have more masses, you have more terms in the sum. If you have 10²³ masses, you have 10²³ terms. (Or you replace it with an integral.)

Demo: Various objects rolling downhill. A ring, a disk, a piece of chalk, a dumbbell. Why does one or the other of them win the race to the bottom consistently? It comes down to to conservation of energy. The initial potential energy becomes the final kinetic energy. I computed this and showed how the speed depends on the moment of inertia of the rolling object. For a given total mass and radius, the disk has a smaller moment of inertia because a lot of the mass is at a small distance from the center.

Examine the power (rate of change of energy) going into a cylinder when I wrap a string around it and pull. The power that I put in is F Dx/Dt in the limit as Dt approaches zero. This is Fv. (All one dimensional here.)

The rate of change of the kinetic energy of the rotating disk, assuming that it's rotation about a fixed axis is

d dt 1 2 I w2 = 1 2 I 2w d w dt = I wa.

Here I used the chain rule to do the derivative. This rate of change of the kinetic energy of the cylinder is the power input: Fv = FRw. Equate this to Iwa and the w's cancel, leaving FR = Ia.

The combination on the left is called the "torque" (at least for this case, where all the rotation is about a single axis). The equation plays a role for rotational motion much as F = ma does for what we've done before.