A heavy uniform solid cylinder of mass M and radius R is a pivoted frictionlessl

Question

A heavy uniform solid cylinder of mass M and radius R is a pivoted frictionlessly about its axis of symmetry. A light rope is wrapped many times around the cylinder, and a block of mass M/3 is suspended from the loose end of the rope. A) find the acceleration of the block? B) fin the tension in the string ? C) if the system is initially at rest, use the conversation of energy to find the velocity of the block after it had fallen through a vertical distance h ?
Please answer it and show all work A heavy uniform solid cylinder of mass M and radius R is a pivoted frictionlessly about its axis of symmetry. A light rope is wrapped many times around the cylinder, and a block of mass M/3 is suspended from the loose end of the rope. A) find the acceleration of the block? B) fin the tension in the string ? C) if the system is initially at rest, use the conversation of energy to find the velocity of the block after it had fallen through a vertical distance h ?
Please answer it and show all work A heavy uniform solid cylinder of mass M and radius R is a pivoted frictionlessly about its axis of symmetry. A light rope is wrapped many times around the cylinder, and a block of mass M/3 is suspended from the loose end of the rope. A) find the acceleration of the block? B) fin the tension in the string ? C) if the system is initially at rest, use the conversation of energy to find the velocity of the block after it had fallen through a vertical distance h ?
Please answer it and show all work

Explanation / Answer

If T is the tension in the rope

then equation of motion for M/3= M/3 x g - T= M/3 x a ..............eqn 1

T R= 1/2 MR^2 x a/R

so, T= 1/2 Ma eqn 2

so, Mg/3 - Ma/2= Ma/3

so, Mg/3 = M(a/2 +a/3)= 5Ma/6

or a= 2g/5

b. T = 1/2 M a = 1/2 x M x 2g/5= Mg/5

c. using conservation of energy

initial energy= potential energy = Mgh

work done by rope= T.h = Mg/5 x h= 0.2Mgh

so final kinetic energy= 1/2 x M x vf^2= Mgh-0.2Mgh=0.8Mgh

so vf = sqrt (0.8gh)

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