A small spool of radius *r*_{s} and a large Lucite disk of radius* r*_{d} 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*, *a** _{g}* and

*F*

_{T}.

(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, *F*_{T}, in the string (i.e. what equations relates m, *a*_{g} , *F*_{T} and a)? **Note: **In a system where

*F*_{T} – *ma*_{g} = *ma*,

if *a*<<*a*_{g} then* F*_{T} ≈ *ma*_{g}.

(c) What is the torque, τ, on the spool-axle-disk system as a result of the tension, *F*_{T}, 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, *r*_{s}, 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 *r*_{d} 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,* I*_{d}, of *a* disk of mass *M* and radius *r** _{d}* in terms of

*M*

_{d }and

*r*

_{d}?