Question

A solid cube and a solid sphere both made of same material are completely submerged in water but to different depths. The sphere and the cube have same surface area. The buoyant force is

• A. Greater for the cube than the sphere
• B. Greater for the sphere than the cube
• C. Same for the sphere and the cube
• D. Greater for the object that is submerged deeper

Answer 1

The correct option is B Greater for the sphere than the cube

Let the surface area of the Sphere and Cube be $A$.

$\Rightarrow$ Side of the cube is $a=\sqrt{\frac{A}{6}}$

Volume of the cube is $V_{\text{cube}}=a^3=6^{-\frac{3}{2}}{A}^{\frac{3}{2}}$

Similarly, Radius of the sphere is $r=\sqrt{\frac{A}{4\pi }}$

Volume of the sphere is $V_{\text{sphere}}=\frac{4}{3}\pi r^3=\frac{4\pi }{3}(4\pi {)}^{-\frac{3}{2}}{A}^{\frac{3}{2}}$

$\therefore V_{\text{cube}}\approx 0.06804A^{1.5}$ and $V_{\text{sphere}}\approx 0.09403A^{1.5}$

Clearly, $V_{\text{sphere}}>V_{\text{cube}}$.

We know that, Buoyant force is proportional to the Volume of the immersed body.

Thus, it is greater for Sphere than that for Cube.