A spool of inner radius R and outer radius 3R has moment of inertia I=MR² about its centre. A thread wound on the inner surface is pulled horizontally by a constant force Mg. Find the acceleration (in m/s²) of the pulled point of the thread, assuming pure rolling.

Question

G1 Physics · Accepted Answer

Let the centre accelerate rightward with a and take anticlockwise angular acceleration α as positive. Pure rolling at outer radius 3R gives a=-3Rα. Translation gives Mg+f=Ma. Torque about the centre gives -MgR+3Rf=MR²α=-MaR/3. Solving gives a=6g/5=12 m/s² for g=10 m/s². The point where the thread leaves the inner hub has additional tangential acceleration a/3 in the pulling direction, so the thread-point acceleration is a+a/3=16 m/s². Hence the integer answer is 16.