Figure shows a spool with thread wound on it placed on a smooth plane inclined at angle from horizontal. The spool has mass m, edge radius `R`, and is wound up to a radius r. its moment of inertia about its own axis is `I`. The free end of the thread is attached as shown in the figure. So that the thread is parallel to the inclined plane. `T` is the tension in the thread. Which of the following is correct?

Options

A. The linear acceleration of the spool axis down the slope mg is `(mgsintheta-T)/m`
B. Angular acceleration is `(Tr)/(2I)`
C. The linear acceleration of the spool axis down the plane is `(Tr^(2))/I`
D. The acceleration of the spool axis down the slope is `(gsintheta)/(1+I/(mr^(2)))`

Answer

A, C, D

Explanation

Linear acceleration `=(mgsintheta=T)/m`
Angular acceleration `=("Torque" (Tr))/("Momentum of inertia" (I))`
Linear acceleration `=r[(Tr)/I]=(Tr^(2))/I`
Since the two linear acceleration must match,
`:. (Tr^(2))/I=(mgsintheta-T)/m` or `(Tmr^(2))/I+T=mgsintheta`
or `T[1+(mr^(2))/I]=mgsintheta` or `T=(mgsintheta)/(1+(mr^(2))/I)`