A small spool of radius rs and a large Lucite disk of radius rd are connected by an axle that is free to rotate in an almost frictionless manner inside of a bearing as shown in the diagram below.
A string is wrapped around the spool and a mass m, which is attached to the string, is allowed to fall.
(a) Draw a free body diagram showing the forces on the falling mass, m, in terms of m, ag and FT.
(b) If the magnitude of the linear acceleration of the mass, m, is measured to be a, what is the equation that should be used the calculate the tension, FT, in the string (i.e. what equations relates m, ag , FT and a)? Note: In a system where
FT – mag = ma,
if a<<ag then FT ≈ mag.
(c) What is the torque, τ, on the spool-axle-disk system as a result of the tension, FT, in the string acting on the spool?
(d) What is the magnitude of the angular acceleration, α, of the rotating system as a function of the linear acceleration, a, of the falling mass and the radius, rs, of the spool?
(e) If the rotational inertia of the axle and the spool are neglected, what is the rotational inertia, I, of the large disk of radius rd as a function of the torque on the system,τ, and the magnitude of the angular acceleration, α?
(f) What is the theoretical value of the rotational inertia, Id, of a disk of mass M and radius rd in terms of Md and rd?