Describe angular momentum about a chosen axis through inertia, angular velocity and impact geometry, then test whether proposed relationships are plausible.
(12 points, suggested time 25 minutes)
The left end of a rod of length d and rotational inertia I is attached to a frictionless horizontal surface by a frictionless pivot, as shown above. Point C marks the center (midpoint) of the rod. The rod is initially motionless but is free to rotate around the pivot. A student will slide a disk of mass mdisk toward the rod with velocity v0 perpendicular to the rod, and the disk will stick to the rod a distance x from the pivot. The student wants the rod
disk system to end up with as much angular speed as possible.
Suppose the rod is much more massive than the disk. To give the rod as much angular speed as possible, should the student make the disk hit the rod to the left of point C, at point C, or to the right of point C ? To the left of C At C To the right of C
Briefly explain your reasoning without manipulating equations.
1 point
Correct answer: "To the right of C "
Reasoning cannot earn credit if the incorrect selection is made.
For an explanation that the torque exerted by the disk or the angular momentum of the disk is greater when farther from the pivot
Example 1: The disk exerts a greater torque on the rod when it pushes the rod farther from the pivot.
Example 2: The disk has greater angular momentum when it's farther from the pivot. The disk loses almost all its speed during the collision and hence gives the rod almost all its angular momentum. So the rod ends up with more angular momentum when the disk hits it farther from the pivot.
On the Internet, a student finds the following equation for the postcollision angular speed ω of the rod in this situation: ω=Imdisk xv0. Regardless of whether this equation for angular speed is correct, does it agree with your qualitative reasoning in part (a) ? In other words, does this equation for ω have the expected dependence as reasoned in part (a) ? Yes No
Briefly explain your reasoning without deriving an equation for ω.
2 points
Correct answer: "Yes"
If "No" is selected, the explanation may still earn full credit if an incorrect selection was made in part (a).
For a selection consistent with the selection from part (a)
For indicating that the equation shows that ω increases with increasing x
Example: According to the equation, ω increases with x;
a bigger x produces a bigger angular speed. This agrees with my reasoning from part (a), where I said a bigger x creates a bigger angular speed after the collision.
Another student deriving an equation for the postcollision angular speed ω of the rod makes a mistake and comes up with ω=mdisk d4Ixv0. Without deriving the correct equation, how can you tell that this equation is not plausible-in other words, that it does not make physical sense? Briefly explain your reasoning.
For parts (d) and (e), do NOT assume that the rod is much more massive than the disk.
3 points
For focusing on functional dependence (instead of, for example, considering units/dimensions)
For addressing mdisk ,I, or both
For correctly concluding that the equation is wrong because of the dependence on mdisk ,I, or both
Example: If mdisk is large, then more angular momentum will be transferred during the collision. But the equation shows the angular speed decreasing with increasing mdisk , because it is in the denominator.
Distribution