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Question

A uniform cylinder of length L and mass M having cross sectional area A is suspended, with its length vertical, form a fixed point by a massless spring, such that it is half submerged in a liquid of density σ at equilibrium position. When the cylinder is given a downward push and released, it starts oscillating vertically with a small amplitude. The time period of the oscillations of the cylinder will be:

A
Smaller than 2π[M(k+Aρg)]1/2
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B
2πMk
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C
arger than 2π[M(k+Aρg)]1/2
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D
2π[M(k+Aρg)]1/2
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Solution

The correct option is A Smaller than 2π[M(k+Aρg)]1/2
Solution:
When cylinder is displaced by an amount x from its mean position, spring force and upthrust both will increase.
Hence,
Net restoring force = extra spring force + extra upthrust
or F=(kx+Axρg)
or a=(k+ρAgM)x
Now, f=12πax
=12πk+ρAgM
Now, f=1T=2πMk+ρAg


Hence A is the correct option

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