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ritwik06
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Homework Statement
Refer to diagram for this question:
A uniform flexible chain of length 1.50 m rests on a fixed smooth sphere of radius [tex]R=\frac{2}{pi}[/tex]such that one end A of the chain is at the top of the sphere while the other end B is hanging freely. Chain is held stationary by a horizontal thread PA as shown in the figure. Calculate the acceleration when the thread is burnt.
The Attempt at a Solution
I am attempting the whole solution from my side. I have even got the final answer. But I am not sure of it. Please point out errors if any.
First I calculate the length of the cirumference of the sphere to know the part of the chain which is freely hanging. I assume that mass is uniformly distributed about the chain u kg/m
So the force due to the hanging part= 0.5*u*g
consider a very small sections of the chain with length dx. The dx*u*[tex]sin\theta[/tex] component is balanced by the Normal reaction.
Only the tangential force of each particle supports the motion along the boundary of the sphere. Therefore the total force supporting the motion:
[I am at present not considring the part which is hanging.]
[tex]ug\int^{\frac{\pi}{2}}_{0}dx cos \theta[/tex]
dx=r* d[tex]\theta[/tex] [I have doubt in this step, is it right?]
[tex]\frac{2ug}{\pi}\int^{\frac{\pi}{2}}_{0}d\theta cos \theta[/tex]
=[tex]\frac{2ug}{\pi}[/tex]
Net force=[tex]\frac{2ug}{\pi}+\frac{ug}{2}[/tex]
Net acceleration=Net Force/Mass of Chain
a=[tex]\frac{g(4+\pi)}{3\pi}[/tex]
Is this process correct?