A string is wrapped around a uniform cylinder of mass M and radius R. The cylind

Question

A string is wrapped around a uniform cylinder of mass M and radius R. The cylinder is released from rest with the string vertical and its top end tied to a fixed bar. (a) Show that the tension in the string is one-third the weight of the cylinder. (Do this on paper. Your instructor may ask you to turn in this work.) (b) Show that the magnitude of the acceleration of the center of gravity is 2g/3. (Do this on paper. Your instructor may ask you to turn in this work.) (c) Show that the speed of the center of gravity is (4gh/3)1/2 after the cylinder has descended through distance h. Verify your answer to (c) with the energy approach. (Do this on paper. Your instructor may ask you to turn in this work.)

Explanation / Answer

the moment of inertia of rigid cylinder is

I = ½ mr²

torque = T r

torque = I

T r = I a/r

T = (½ mr²)a/r²

T = ½ m a

F = m a

m g - T = m a

m g - ½ m a = m a

a = 2g/3


back to previous equation,

m g - T = m a

mg - T = 2mg/3

T = mg/3

T = w/3




(b).

a = 2g/3



(c).

E = 0

m g h - ½ m v² - ½ I ² = 0

m g h - ½ m v² - ½ (½ mr²)(v/r)² = 0

g h - ½ v² - ¼ v² = 0

gh = ¾ v²

v = (4gh/3)

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