Oscillating block in liquid

Classical Mechanics Level 3

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

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

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

Details and Assumptions

  • Assume block to be cubical.

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