Slipping and Rolling – A string is wrapped around a solid cylinder of mass M and

Question

Slipping and Rolling – A string is wrapped around a solid cylinder of mass M and radius R and pulls horizontally with force T from the top of the cylinder. The horizontal surface on which the cylinder rests is frictionless.

a. (1 pt) What is the acceleration of the center of mass of the cylinder?

b. (2 pts) What is the angular acceleration of the cylinder about its center of mass?

c. (2 pts) What is the acceleration (magnitude and direction) of the point on the cylinder which touches the horizontal surface? Make sure your answer is dimensionally correct.

Now we’re going to change the problem and give the surface on which the cylinder rests a nonzero coefficient of static friction. d. (3 pts) Assuming that the cylinder now rolls without slipping, what is the acceleration of its center of mass?

e. (2 pts) For a given coefficient of static friction, what is the largest acceleration of the center of mass which can be achieved without the cylinder slipping against the surface?

Explanation / Answer

a)   a = T / M      linear acceleration of C

b) T R = I * alpha = 1/2 M R^2 alpha

    alpha = 2 T / (M R)       angular acceleration

c) ap = T / M - R * (2 T / (M R))     acceleration of point on surface

ap = - T / (2 M )

d)   Now use the moment of inertia about the point of contact I = 3/2 M R^2

2 T R = I * alpha = I a / R    where a is acceleration of C.M.

2 T R = 3/2 M R a

a = 4 T / (3 M)

e) T = u M g   is the largest tension that can be applied without slipping

    3/4 M a = u M g

   a = 4/3 u g

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