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Calculate the period of small oscillatio...

Calculate the period of small oscillations of a floting box as shown in figure, which was pushed down inverical direction. The mass of box is `m`, area of its base is `A` and the density of liquid is `rho`. The resistance of the liquid is assumed to be negligible.

Text Solution

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Initially when box is floating in liquid, if its h depth is sumerged in liquid then buoyancy force on it is
`F_(B) =` weight of liquid displaced `= Ahrhog`

As the box is in equilibrium, we have `Ahrhog = mg`
Now if box is further pushed down by a distance x, net restorinn gorce ono it in upward (toward mean position) direction is
`F_(B) = - [A(h+x)rhog-mg] = -Axrhog` [ as `mg = Ahrhog`]
It a is the accleration of box in upward direction we have `a = -((Arhog)/(m))x`
Equation shows that the box executes SHM with angular frequency `omega` given as `omega = sqrt((Arhog)/(m))`
Thus time peiod of its equilibrium can be given as `T = (2pi)/(omega) = 2pisqrt((m)/(Arhog))`
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