A 1.25 ft diameter drum has a weight of 90 lbs., raius of gyration of 0.60 ft. A

Question

A 1.25 ft diameter drum has a weight of 90 lbs., raius of gyration of 0.60 ft. A 46 ft long chain having a weight of 3.25 lb/ft is wrapped about the drum diameter so that a chain length of s=3 feetis suspendedas shown. If the drum is originally at rest, determine its angular velocity after the chain end has decended s=22 ft. Neglect the thickness of the chain and the imbalance due to "partially" wrapped chain.

w(angular velocity)=_______________________

Dynamics In class Assignment Solve this problem in class in two steps: Set up: Kinematic diagram Newton's Equation of motion Express for mass moment of inertia Kinematic expression to find angular velocity After checking and verification by instructor' solve problem Numerical result with units HINT: Integration You are free to collaborate with your classmates.

Explanation / Answer

as the chin descend, the drum will rotate and in the final configuration, drum will have rotational KE and chain will have KE.

These KE will come from the PE of the descended chain.

Now, as the chain descend, for every x, drum will rotate ar where a is angle. Simillarly, at end configuration, v = w*r

Initial PE (assuming center of drum to be zero) = -3/2*3.25*g (COM at s/2)

Final PE = -22/2 * 3.25*g

Decrease in PE = -9.5*3.25*g

Increase in KE = m*v*v/2 + I*w*w/2 = 22*3.25*v*v/2 + (90*0.6*0.6)*v*1.25*v*1.25/2

Sum of decrease in PE + INcrease in KE = 0

solve, v = 4.022

So, w = v/r = 3.22 rad/s

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