A solid cylinder of mass 148.7 kg and diameter 0.72m has mounted to rotate aroun

Question

A solid cylinder of mass 148.7 kg and diameter 0.72m has mounted to rotate around its axis, with the axis parallel to the surface of the Earth. A very light rope is attached to the outside fo the cylidner, an spherical mass of 1.6kg was attached to the end of the rope. A time t=0, the cylinder has an initial angular velocity of wi, and the mass is 26m below the cylinder but it's being lifted by the cylinder taking up the rope. When the mass is 2m below the cylinder, it stops (because of the infuence of gravity on the spherical mass), and then starts descending again. Calculate wi.

Explanation / Answer

R = radius of cylinder = 0.72/2 = 0.36 m

g = 9.8 m/s^2

wi =

alpha = angular deacceleration

I = moment of inertia of cyleder = MR^2/2 = 148.7*(0.36)^2/2 = 9.63 kg*m^2

T = torque about the cylinder rotation axis due to weight of spherical mass

      = R*1.6*g

      = 5.64 N*m

alpha = T/I = 0.586 rad/s^2

d= ditance travelled by sphere in time t

= 26-2

= 24 m

theta = angular ditance moved by cylinder

     = d/R

         = 66.66 rad

used equation :: Wf^2 = Wi^2 - 2*(alpha)*(theta)

given Wf = final speed= 0 ,

hence    Wi^2 - 2*(alpha)*(theta) =0

             Wi   =[ 2*66.66*0.586]^0.5

             Wi =8.839 rad/s

hence   Wi =8.839 rad/s

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