Oscillating block in liquid

The container shown contains a liquid of variable density which varies as d=d0(43yh0) kg/m3\displaystyle d = d_0 \left( 4 - \dfrac{3y}{h_0} \right) \text{ kg/m}^3 , where h0h_0 is total height of container, d0d_0 is constant and yy is measured from the bottom of the container. A solid block whose density is 52d0\dfrac{5}{2} d_0 and mass 'm' is released from bottom of the container. Given that block will execute SHM and is time period can be written as

T=2παh0βg\displaystyle T = 2\pi \sqrt{\dfrac{\alpha h_0}{\beta g}}

gcd(α,β)=1\text{gcd}(\alpha,\beta) = 1 and gg is acceleration due to gravity. Find α+β\alpha + \beta.

Details and Assumptions

  • Assume block to be cubical.
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