Torque and Rotational Dynamics unit test
A test on this unit alone, marked as a percentage and a letter grade — for the test your class is actually sitting, rather than for May. Answer everything, then submit once: seeing the answer to question 3 before attempting question 4 makes the final percentage meaningless.
Moments of inertia add
Falling spool or yo-yo
Moment of inertia by integration
Standard moments of inertia
Why the axis matters
Equilibrium of a leaning ladder
Angular kinematics with calculus
Moment arm
Pulley with mass
Rotational equilibrium
Static equilibrium conditions
Slipping vs rolling
Short answer 1. Define or explain: Parallel axis theorem
3 ptsShort answer 2. Define or explain: Torque as a cross product
3 ptsShort answer 3. Define or explain: Spherical shell
3 ptsShort answer 4. Define or explain: Solid sphere
3 ptsFree response
8 ptsQUALITATIVE/QUANTITATIVE TRANSLATION (Question 4, 8 points). A uniform disk and a ring, each of mass M and radius R with outer edges of the same material, roll without slipping along a horizontal surface, each with center-of-mass speed v. They transition smoothly onto a ramp inclined at angle θ and continue rolling without slipping up the ramp. The ring travels farther along the ramp than the disk before momentarily stopping.
A. While the disk and ring roll up the ramp without slipping, the static friction magnitudes on them are fD and fR. Indicate whether fD is greater than, less than, or equal to fR, and justify using qualitative reasoning beyond referencing equations.
B. A cylinder of mass M, radius R, and rotational inertia I about its central axis rolls without slipping up a ramp inclined at angle θ. Derive an expression for the magnitude of the static frictional force f on the cylinder, in terms of M, R, I, θ, and physical constants, as appropriate.
C. In a different scenario, the disk and ring have the same initial speed v but the new ramp causes both to slip as they roll up it. Indicate whether the kinetic friction magnitude on the disk is greater than, less than, or equal to that on the ring while both slip, and briefly justify.