Question

A cylindrical block of density d stays fully immersed in a beaker filled with two immiscible liquids of different densities $d_1$ and $d_2$.The block is in equilibrium with half of it in liquid 1 and the other half in liquid 2. If the block is given a displacement downward and released ,then (neglecting fricitional losses).

• A. It executes SHM
• B. It executes periodic motion but not SHM
• C. The frequency of oscillation is independent of the size of cylinder
• D. None of these

Answer 1

The correct option is A It executes SHM

Let $A$ and $a$ be the cross-sectional area and the height of the block respectively.

$⟹$ Volume of the block $V=Aa$

Initially, the block is in equilibrium with half of it in liquid 1 and the other half in liquid 2.

Using Archimede's principle, $Mg=$ Weight of liquid displaced

$\therefore \left(Vd\right)g=\left(\frac{V}{2}{d}_1\right)g+\left(\frac{V}{2}{d}_2\right)g$ ...............(1)

Now the block is displaced by $x$ units downward.

Thus the net change in force $F=\left(Vd\right)g-(A(\frac{a}{2}+x)d_{2}g+A(\frac{a}{2}-x)d_{1}g)$

where $V=Aa$, $d_2>d_1$ and using (1),

$\therefore F=-(d_2-d_1)Agx⟹F\propto -x$

Hence it executes $SHM.$