A cylinder with a moment of inertia (about its axis of symmetry), mass , and rad

Question

A cylinder with a moment of inertia (about its axis of symmetry), mass , and radius has a massless string wrapped around it which is tied to the ceiling (Figure 1) . At time t=0 the cylinder is released from rest at height above the ground. Use g for the magnitude of the acceleration of gravity. Assume that the string does not slip on the cylinder. Let represent the instantaneous velocity of the center of mass of the cylinder, and let represent the instantaneous angular velocity of the cylinder about its center of mass. Note that there are no horizontal forces present, so for this problem v=-vy omega=-omegaz

Explanation / Answer

Apply Newton's second law ; That is F = M at            Fg - T = Mat Here weight of the cylinder W = Mg here at is translational acceleration Mg - T = M at           T= M(g-at)    ...... (1) -------------------------------------------------------------------- Now use = I                TR = I Where I is the moment of inertia I = (1/2) MR2 Acceleration () = at / R TR = [(1/2) MR2 ] (at / R ) TR = (1/2) MR(at ) T = (M /2) (at ) (at ) = 2T /M ...... (2) ------------------------------------------------------------------------- Substitute equation (2) in equation (1)         T = M ( g - 2T / M )           =Mg - 2T 3T = Mg
Tension (T) = Mg /3

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