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JEE Mains · Physics · STD 11 - 13. oscillations
A uniform cylinder of length \(L\) and mass \(M\) having cross-sectional area \(A\) is suspended, with its length vertical, from a fixed point by a massless spring, such that it is half submerged in a liquid of density \(\sigma \) at equilibrium position. When the cylinder is given a downward push and released, it starts oscillating vertically with a small amplitude. The time period \(T\) of the oscillations of the cylinder will be
- A Smaller than \(2\pi {\left[ {\frac{M}{{\left( {k + A\sigma g} \right)}}} \right]^{1/2}}\)
- B \(2\pi \sqrt {\frac{M}{k}} \)
- C Larger than \(2\pi {\left[ {\frac{M}{{\left( {k + A\sigma g} \right)}}} \right]^{1/2}}\)
- D \(2\pi {\left[ {\frac{M}{{\left( {k + A\sigma g} \right)}}} \right]^{1/2}}\)
Answer & Solution
Correct Answer
(A) Smaller than \(2\pi {\left[ {\frac{M}{{\left( {k + A\sigma g} \right)}}} \right]^{1/2}}\)
Step-by-step Solution
Detailed explanation
If \(x\) is the displacement then, \(M{\omega ^2}x = \left[ {\rho Ag + k} \right]x\) \( \Rightarrow \omega = {\left[ {\frac{{\rho Ag + k}}{m}} \right]^{1/2}}\) \( \Rightarrow T=2\pi {\left[ {\frac{M}{{\left( {k + A\sigma g} \right)}}} \right]^{1/2}}\) It Should be less than…
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