A string is wrapped around a solid cylinder of mass M and radius R and pulls hor

Question

A string is wrapped around a solid cylinder of mass M and radius R and pulls horizontally with forceT from the top of the cylinder. The horizontal surface on which the cylinder rests is frictionless. a) What is the acceleration of the center of mass of the cylinder? b)What is the angular acceleration of the cylinder about its center of mass? c)What is the velocity (magnitude and direction) of the point on the cylinder which touches the horizontal surface? Make sure your answer is dimensionally correct. Now were going to change the problem and give the surface on which the cylinder rests a non zero coefficent of static friction, uS d) Assuming that the cylinder now rolls without slipping, what is the acceleration of its center of mass e) For a given coefficient of static friction, uS, what is the largest acceleration of the center of mass which can be achieved without the cylinder slipping against the surface?

Explanation / Answer

Force on the cylinder = T

Moment of Inertia I = MR2 /2

if a is the acceleration of the center of mass then

Ma = T   as here is no friction

a = T/M

angular acceleration =

For rotational motion

I = T

= T/I = 2T/MR2

Velocity of the point that touches the surface is 0, we assume the cyclinder is rolling without slipping.

let f be the frictional force, as the point of contact has a tendency to slip backward, friction will act forward in the direction of T

f = sMg

linear motion is given by

T+f = Ma

a= (T+ sMg)/M

for the cylinder to roll without slipping

= a/R

I = T +f

= (T+f)/I

     = 2(T + sMg)/MR2

    =a/R

maximum acceleration without slipping

a = 2(T + sMg)/MR

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