A massive, uniform cylinder has a mass m and a radius R . It is accelerated by a

Question

A massive, uniform cylinder has a mass m and a radiusR. It is accelerated by a tension force T that isapplied through a rope wound around a light drum of radiusr that is attached to the cylinder. The coefficientof static friction is sufficient for the cylinder to roll withoutslipping. (a) Find the frictional force. (b) Find the acceleration a of the center of thecylinder. (c) Show that it is possible to choose r so thata is greater than T/m. (d) What is the direction of the frictional forcei nt hecircumstanfces of Part (c)?

Explanation / Answer

m a = T - F    where T is the tension and F isthe frictional force opposing the motion or    m R = T - F 1/2 m R2 = T r + FR     taking torques about the center ofmass 1/2 m R = T r / R + F 3 m R / 2 = T ( 1 + r /R)        adding the last to thefirst equation a = 2 T / (3 m) (1 + r /R)         for theacceleration of the center of mass if (1 + r / R) > 3 /2        a is greater than T /m     (r / R > 1/2) Then by our very first equation F must be in the direction ofthe motion At first glance it appears to be getting something fornothing, however if you consider the distance that T is moving then the work W =T d shows this is not the case.

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