Question

Inner and outer radii of a spool are $r$ and $R$ respectively. A thread is wound over its inner surface and spool is placed over a rough horizontal surface. Thread is pulled by a force $F$ as shown in figure. In case of pure rolling :

• A. Thread unwinds, spool rotates anticlockwise and friction acts leftwards.
• B. Thread winds, spool rotates clockwise and friction acts leftwards.
• C. Thread winds, spool moves to the right and friction acts rightwards.
• D. Thread winds, spool moves to the right and friction does not come into existence.

Answer 1

Answer: (A), (C), (D)

The correct options are
A Thread unwinds, spool rotates anticlockwise and friction acts leftwards.
C Thread winds, spool moves to the right and friction acts rightwards.
D Thread winds, spool moves to the right and friction does not come into existence.

Since, the spool rolls over the horizontal surface, instantaneous axis of rotation passes through the point of contact of spool with the horizontal surface.
About the instantaneous axis of rotation, moment produced by F is clockwise. Therefore, the spool rotates clockwise. In that case, acceleration will be rightward and thread will wind.
If rotational motion of the spool is considered about its own, then the resultant moment on it must be clockwise. But moment produced by the force is anticlockwise and its magnitude is equal to $F_1/r$. Hence, moment produced by the friction (about its own axis) must be clockwise and its magnitude must be greater than Fr. It is possible only when friction acts leftwards. Therefore, option (b) is correct.
There are four forces acting on the spool. Frictional force $F_r$, Gravitational force $F_g$, Normal force N and the tension force F. Out of these only the tension force produces a moment about the instantaneous center of rotation. That moment would be clockwise as shown in the figure. Thus the spool would rotate clockwise and the thread would wind.