A thin uniform disk of mass M and radius R has a string wrapped around its edge and attached to the ceiling. The bottom of the disk is at a height 3R above the floor, as shown above. The disk is released from rest. The rotational inertia of a disk around its center I = 1 ⁄ 2 MR^2
A-On the circle below that represents the disk, draw and label the forces that act on the disk. Each force must be represented by a distinct arrow starting on, and pointing away from the disk, beginning at
the point where the force is exerted on the disk.
B-When released from rest, the disk falls and the string unwinds. The force the string exerts on the disk is FT, and the gravitational force exerted on the disk is Fg. Which of the following expressions correctly relates FT and Fg as the disk falls (see below)? justify your answer
C-Express all answers in terms of M, R, and physical constants, as appropriate.
•Derive an expression for the acceleration of the disk as it falls.
•Derive an expression for the time ∆t that it takes the disk to reach the ground.
•Derive an expression for the rotational kinetic energy Krot of the disk at the instant it reaches the ground.
D- A very narrow wedge is cut out of the thin uniform disk of mass M, as shown above. If r is the distance from the tip of the wedge, then the linear mass density of the wedge can be expressed as follows: (see below)
•Using integral calculus, derive an expression for the rotational inertia of the wedge around its tip.
• Derive an expression for the rotational inertia of the modified disk (i.e., the disk after the narrow wedge is
cut out) around its original center.
