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Question

One end of a long iron chain of linear mass density λ is fixed to a sphere of mass m and specific density 13 while the other end is free. The sphere along with the chain is immersed in a deep lake. If specific density of iron is 7. If the height h above the bed of the lake at which the sphere will float in equilibrium is xm3λ. then the value of x is (Assume that the part of the chain lying on the bottom of the lake exerts negligible force on the upper part of the chain)


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Solution

Weight of the sphere + chain = (m + λh)g
Volume of water displaced by sphere
= m1/3 = 3mg
Volume of water displaced by iron chain
= λh7g
Buoyant force = (3m +λh7)gFor equilibrium, weight = buoyant forcei.e m + λh = 3m +λh7h = 7m3λ

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