A uniform, hollow, cylindrical spool has inside radius R/2,outside radius R, and

Question

A uniform, hollow, cylindrical spool has inside radius R/2,outside radius R, and mass M. It is mounted so that it rotates on afixed, horizontal axel. A counterweight of mass m is connectedto the end of a string wound around the spoo. The clounterweightfalls from rest at t=0 to a position y at time t. Show thatthe torque due to friction forces between spool and axel is f = R[m(g- (2y/t^2))-M(5y/4t^2)] A uniform, hollow, cylindrical spool has inside radius R/2,outside radius R, and mass M. It is mounted so that it rotates on afixed, horizontal axel. A counterweight of mass m is connectedto the end of a string wound around the spoo. The clounterweightfalls from rest at t=0 to a position y at time t. Show thatthe torque due to friction forces between spool and axel is f = R[m(g- (2y/t^2))-M(5y/4t^2)]

Explanation / Answer

   in the given problem we can see that    f will oppose the torquedue to the hanging object     = I          = T R -f    f = T R - I ................. (1)    now we find T, I and and substitutein (1)     Fy = T - m g            =- m a    T = m (g - a) ........... (2)    and also    y = vi t + (a t2/ 2)    a = 2 y / t2...............(3)     = a / R       = 2 y / Rt2...............(4)    I = (1 / 2) M [R2 + (R / 2)2]      = (5 / 8) MR2.............(5)        on substituting (2), (3), (4) and (5) in (1)we get    f = m [g - (2 y /t2)] R - (5 / 8) [M R2 (2 y) / Rt2]       

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