Practice for second exam!
The second exam will have two problems, from Chapters 8-10.
You are allowed one crib sheet of formulas or other material that you deem helpful for the exam.
You have had least practice on Chapter 10 material, so this page will concentrate on that chapter.
The two central ideas that go beyond elementary mechanics are
First, composition of angular velocities: Our favorite rigid body, a uniform circular disk of mass M and radius a, is spinning with constant angular velocity ω about its diameter as an axis.
A point-like but massive bug, also of mass M, is crawling with uniform speed around the periphery of the disk so it comes back to its starting point once per revolution of the disk. (For simplicity assume that there is no gravity, though the answers don't depend on that.)
- composition of angular velocities, and a total angular velocity that changes in direction as well as (possibly) in magnitude.
- ω and L not parallel, use of the moment of inertia tensor and principal axes
Note that the system of the above question is entirely constrained, so it makes no sense to ask about equations of motion, except to find constraint forces.
When the question is about an EOM, the Lagrangian approach may be useful. For example:
- What is the bug's angular velocity, in magnitude and direction (draw a diagram to specify the direction)? (Note that the angular velocity always means the total angular velocity with respect to an inertial frame)
- What is the total angular momentum vector L of the system disk + bug, about the disk's center, when the bug is on the disk's rotation axis?
Consider the disk + bug system but let the disk's angular velocity ω not be a driven, constant rate, but unconstrained; so you may describe the disk's configuration by a suitable variable φ.
The bug, however, is still crawling at a constant angular rate Ω with repsect to the disk. Find the EOM for φ and a first integral of that EOM (total energy is not conserved -- the bug does work while crawling -- but something else is.)
In the first question the bug and the disk have different angular velocities, so it is not surprising that the total angular momentum of the system is not parallel to either of these ω's. You may howevr have noticed that even for the bug separately, Lbug is not parallel to ωbug.
Here is an example where a similar thing happens:
Our favorite disk (M, a) is spinning with the usual, constant ω about an axle through its center and inclined at 45° with respect to the disk's symmetry axis.
- What is the disk's angular momentum vector L?
- What is the disk's kinetic energy?
- What is dL/dt? This equals the torque Γ that the bearings must exert in order to keep the disk spinning about that axis.
I can't offer any extremely relevant problem for chapters 8 and 9 but here is one at least "in the same ballpark":
Two particles move about each other in circular orbits under the influence of their mutual gravitational forces, with a period τ.
The motion is suddenly stopped at a given instant t = 0, and they are released and allowed to fall into each other. How long a time t elapses before they collide? (in terms of τ only!)
I will not post solutions, but you may ask questions about your solution by email or at my office at the following times:
- Tuesday 10 - 2, 3 - 4
- Wednesday 12 - 2, 3 - 6
- Thursday 10:30 - 2 but at the lunch in Toll room at 1 pm -- you can find me there