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 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
- [JEE MAIN 2013]
- 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
$2\pi {\left[ {\frac{M}{{\left( {k + A\sigma g} \right)}}} \right]^{1/2}}$
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A mass m is suspended from a spring of length l and force constant $K$. The frequency of vibration of the mass is ${f_1}$. The spring is cut into two equal parts and the same mass is suspended from one of the parts. The new frequency of vibration of mass is ${f_2}$. Which of the following relations between the frequencies is correct
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A spring having with a spring constant $1200\; N m ^{-1}$ is mounted on a hortzontal table as shown in Figure A mass of $3 \;kg$ is attached to the free end of the spring. The mass is then pulled sideways to a distance of $2.0 \;cm$ and released
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Two bodies of masses $1\, kg$ and $4\, kg$ are connected to a vertical spring, as shown in the figure. The smaller mass executes simple harmonic motion of angular frequency $25\, rad/s$, and amplitude $1.6\, cm$ while the bigger mass remains stationary on the ground. The maximum force exerted by the system on the floor is ..... $N$ ( take $g = 10\, ms^{-2}$)
- [JEE MAIN 2014]
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- [AIPMT 2004]