Be Careful with Liquid

A rectangular tank having base \( 15\text{ cm} \times 20\text{ cm}\) is filled with water (density \(\rho = 1000 \text{ kg/m}^3\)) up to \(20\text{ cm}\) height. One end of an ideal spring of natural length \({h_0 = 20\text{ cm}\) and force constant \(K = 280\text{ N/m}\) is fixed to the bottom of the tank so that the spring remains vertical. This system is in an elevator moving downwards with acceleration \( a = 2\text{ m/s}^2\). A cubical block of side \( l = 10\text{ cm}\) and mass \( m = 2\text{ kg}\) is gently placed over the spring and released gradually.

  1. Find the compression \(a\) (in \(\text{cm}\)) in the spring in the equilibrium position.
  2. If the block is slightly pushed down from the equilibrium position and released, find the frequency of oscillation about the equilibrium position \(f = \dfrac {b\sqrt c}\pi\) (in \(\text{Hz}\)), where \(b\) and \(c\) are positive integers with \(c\) being square-free.

Enter your answer as \( abc\).

This is a part of my set Aniket's Mechanics Challenges.

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