Figure shown a container having liquid of variable density. The density of liquid veriesas `rho=rho_(0)(4-(3h)/(h_(0)))`. Here, `h_(0)` and `rho_(0)` are constants and h is measured from bottom of the container. A solid block of small dimensions whose density is `(5)/(2) rho_(0)` and mass m is released from bottom of the tank. Prove that the block will execute simple harmonic motion. Find the frequency of oscillation.
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Explanation

Net force on the block at a height h from the bottom is

`F_("net")=`upthrust-weight (upwards)
`=((m)/((5)/(2)rho_(0))) rho_(0)(4-(3h)/(h_0))g-mg`
`F_("net")=0 at h=(h_0)/(2)`
so, `h=(h_0)/(2)` is the equilibrium positive of the block.
For `hgt(h_0)/(2)` weightgtuupthrust
i.e. net force is downwards and for `hlt(h_0)/(2)`
weight lt upthrust
i.e. net force is upwards
For upward displacement x from mean position, net downward force is
`F=-[((m)/(5/2 rho_0))rho_(0){4-(3(h-x))/(h_0)}g-mg](h=(h_0)/(2))`
`:. F=-(6mg)/(5h_(0))x` ....(i)
(because at `h=(h_0)/(2)` upthrust and weight are equal)
since, `F prop -x`
Oscillations are simple harmonic in nature Rewriting Eq. (i)
`ma=-(6mgx)/(5h_(0))`
or, `a=-(6g)/(5h_(0))x`
`:. f=(1)/(2pi)sqrt(|(a)/(x)|)`
`f=(1)/(2 pi) sqrt((6g)/(5h_(0)))`.