A cloth tape is wound around the outside of a uniform solid cylinder (mass M, radius R) and fastened to the ceiling as shown in the diagram below. They cylinder is held with the tape vertical and then released from rest. as the cylinder descends, it unwinds from the tape without slipping. The moment of inertia of a uniform solid cylinder about its center is 1/2MR^2.
In terms of g, what is the downward acceleration of the center of the cylinder as it unrolls from the tape?
In terms of g, what is the downward acceleration of the center of the cylinder as it unrolls from the tape?
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mg – T = m a
and
Tr = I α
Where T is the tension and ω is the angular velocity and α is the angular acceleration
Linear acceleration a = r. α
Eliminating T
mg – (I /r) α = m a
mg – (a/ r²) = m a
a = g / {1 + I / m r²)
since I / m r² = ½
a = 2g /3.
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and
Tr = I α
Where T is the tension and ω is the angular velocity and α is the angular acceleration
Linear acceleration a = r. α
Eliminating T
mg – (I /r) α = m a
mg – (a/ r²) = m a
a = g / {1 + I / m r²)
since I / m r² = ½
a = 2g /3.
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