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For the arrangement shown, a cylinder of...

For the arrangement shown, a cylinder of mass `m` with cross-sectional area `A`, initially in equilibrium poisition, is displaced slightly inside the liquid of density `p`. Prove that the motion is simple harmonic and also find its time-period.

Text Solution

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`T=kx_(0)`
and `T+Vrhog=mg`
`rArr kx_(0)+Vrhog=mg`
when `m` is displaced by `Deltax` downwards
`ma=T'-k(x_(0)+Deltax)`
`rArr ma=mg-(v+Adeltax)rhog-T'`
Adding `2ma=mg-k(x_(0)+Deltax)-(v+ADeltax)rhog`
`=-[k+Arhog]Deltax`, using equation (`3`)
`rArr a=-([k+Arhog])/(2m)Deltax`
`:. T=(2pi)/(omega)=2pisqrt((2m)/([k+Arhog]))`
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