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 force T 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 acceleration (magnitude and direction) of the point on the cylinder which touches the horizontal surface?Make sure your answer is dimensionally correct. Now we

Explanation / Answer

a)

T = M*a

a = T/M

b)

angular acceleration = 0 (pure sliding, no rotation since no friction)

c)

a = T/M

Direction = along T

d)

Let friction force = F

Torque = (T-F)*R

Moment of inertia of cylinder about its axis I = 1/2*mR^2

Thus, (T - F)*R = [1/2*mR^2]*alpha................where alpha = ang.acceleration = a/R

Thus, (T - F)*R = [1/2*mR^2]*a/R

or, T - F = 1/2*ma

Also, T + F = ma

Adding both, 2T = 1.5 ma

a = 4T/(3m)

e)

For no slip, Us*mg = F

But F = ma - T = m*4T/(3m) - T = T/3

Us*mg = T/3

T= 3*Us*mg

a = 4T / (3m)

= 4*Us*g

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