# 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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