Question 415: Properties of Matter

Question

A solid sphere of radius R and density $$\rho $$ is attached to one end of a mass-less spring of force constant k.
The other end of the spring is connected to another solid sphere of radius R and density 3$$\rho $$. The complete
arrangement is placed in a liquid of density 2$$\rho $$ and is allowed to reach equilibrium. The correct
statement(s) is (are)

Accepted Answer

Considering FBDs of spheres A and B, we get
Here, $x$ is the extension produced in the spring
At equilibrium $\Sigma \mathrm{F}_{\mathrm{A}}=0$ and $\Sigma \mathrm{F}_{\mathrm{B}}=0$
$$ \begin{aligned} \frac{4}{3} \pi R^3(2 ho) g & =k x+\frac{4}{3} \pi R^3 ho g ...........(i)\\ k x+\frac{4}{3} \pi R^3(2 ho) g & =\frac{4}{3} \pi R^3(3 ho) g .........(ii)\end{aligned} $$
Subtract (ii) from (i), we get
$$ \begin{aligned} -k x & =k x-\frac{4}{3} \pi \mathrm{R}^3(2 ho) g \\ x & =\frac{4 \pi \mathrm{R}^3 ho g}{3 k} \end{aligned} $$
(Option A is correct)
If we consider both the spheres as a system, then at equilibrium,
$$ \begin{aligned} 	ext { total weight } & =	ext { buoyant force } \ 	ext { Total weight }(\mathrm{A}+\mathrm{B}) & =\frac{4}{3} \pi \mathrm{R}^3(4 ho) g \ 	ext { Total buoyant force } & =\frac{4}{3} \pi \mathrm{R}^3(4 ho) g \end{aligned} $$
Hence, sphere A (light sphere) is completely submerged (option D is correct)

Answer

The net elongation of the spring is $${{4\pi {R^3}\rho g} \over {3k}}$$

Answer

The net elongation of the spring is $${{8\pi {R^3}\rho g} \over {3k}}$$

Answer

The light sphere is partially submerged

Answer

The light sphere is completely submerged

Properties of Matter

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