A massive, uniform cylinder has a mass m and a radiusR. It is accelerated by a T

Question

A massive, uniform cylinder has a mass m and a radiusR. It is accelerated by a Tension force T that is appliedthrough a rope wound around a light drum of radius r thatis attached to the cylinder. The coefficient of staticfriction is sufficient for the cylinder to roll without slipping.A) Find the frictional force. B) Find the acceleration aof the center of the cylinder. C) Show that it is possible tochoose r so that a is greater than T/m.D) What is the direction of the frictional force in thecircumstances of part C?

(Figure of problem is in Figure 9-77) Thanks!

Explanation / Answer

   when tenstion is horizontal the cyllinder willroll forward and friction force will be in the direction ofT using newton 2nd lae Fx = T+f = ma -----(1) = Tr - fR = I ------(2)    subsitute I = MR2/2 and = q/Rplug in eq 2,    then Tr - fR = 1/2mRa ------(3) solve eq 3 for then f = Tr/R - 1/2ma -----4 subsitute eq 4 in eq 1 and solve for a    a = 2T/3m (1+r/R) -----5 substitue eq 5 in eq 4 we get f = T/3(2r/R -1) eq 4 gives aclleration of the cwnter of mass = a = 2T/3m(1+r/R) express the condition that a>T/m    2T/3m(1+r/R)>T/m >1 or r>1/2R if r>1/2R then f>0 means in thedirection of T

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