| 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 |
6 Nov
Repeat where the idea of moment of inertia comes from in terms of
kinetic energy of a rotating rigid body. What is it for a continuous
system such as a meter stick? Do it by a sequence of successive
approximations, leading to an integral:
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For the most common use of the word, you are referring to volumes and masses. Pick a volume DV and it will contain a mass DM. The quotient, DM/DV is the average volume mass density. It's like saying that you went a distance 120 miles in 2 hours, so you have an average speed of 60 mph. The instantaneous value of the quantity requires you to take a limit as Dt approaches zero. Similarly, the density requires you to take a limit as DV goes to zero: dM/dV.
Sometimes this type of density is not the one that you want. If you are looking at something long and thin, such as a wire, the "linear mass density" is more appropriate. Instead of a volume, DV, use a length, Dx. The limit is most concisely denoted as dM/dx.
Back to the moment of inertia of the meter stick. Here the linear mass density is pretty nearly constant, dm/dx = M/L. This means that the factor dm is
Put this back into the integral of x2dm to get
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Back to torque. Repeat the derivation of the torque = Ia equation from energy. Demonstrate what torque means and what this moment arm, R, means. Apply this to the example of a mass hung over a pulley. Set up the F=ma equation for the hanging mass and the torque equation for the pulley.
Note: In each case I still have to go through the exercise of seeing what things act on each object. For the hanging mass, it's the usual old material. For the pulley I have to figure out the torque cause by each thing acting, and in this case to determine that the only significant torque comes from the pull of the string.
8 Nov One of the homework problems involved vector products. Do an example just like it.
Finish the problem of the mass hanging from the pulley and analyze the results to see if they make sense and to see if intuition about the results matches the computed answer.
Demo of angular momentum. Use the weighted bicycle wheel and the rotating stand. Also see what happens if I stand on the platform and pull in my arms as I rotate.
10 Nov Do one of the homework problems in detail, including a demo.
Go over the definition of vector product again. Now apply it to F = dp/dt and do all the manipulations needed to find the relationship between torque and (d/dt of) angular momentum. Along the way demonstrate again the conservation of angular momentum using the rotating platform.
