Question

A solid sphere of volume $V$ and density $\rho$ floats at the interface of two immiscible liquids of densities $\rho_1$ and $\rho_2$ respectively. If $\rho_1$ < $\rho_2$ < $\rho_3$, then the ratio of volume of the parts of the sphere in upper and lower liquid is

• A. $\frac{?-?_{1}}{?_{2}-?}$
• B. $\frac{?_{2}-?}{?-?_{1}}$
• C. $\frac{? + ?_{1}}{?+?_{2}}$
• D. $\frac{?+?_{2}}{?+?_{1}}$

Answer 1

The Correct Option is B

V = Volume of solid sphere. Let V = Volume of the part of the sphere immersed in a liquid of density $\rho_{1}$ and V = Volume of the part of the sphere immersed in liquid of density $\rho_{2}$. According to law of floatation, $V\rho g=V_{1}\rho_{1}g+V_{2}\rho_{2}g\quad\quad\quad\quad\quad\quad... \left(i\right)$ and $V=V_{1}+V_{2}\quad\quad \quad \quad \quad \quad \quad \,\,\,\,\, ... \left(ii\right)$ Hence from eqns (i) and (ii), $V_{1}\rho g+V_{2}\rho g = V_{1}\rho_{1}g+V_{2}\rho_{2}g$ or$\quad V_{1}\left(\rho-\rho_{1}\right)g=V_{2}\left(\rho_{2}-\rho\right)g$ or$\quad \frac{V_{1}}{V_{2}}=\frac{\rho_{2}-\rho}{\rho-\rho_{1}}$

Answer 2

Ans. Archimedes Principle explains the relationship between the apparent weight of the object immersed in water and the weight of the water that is displaced by it. It includes factors like Buoyant Force which helps establish the relationship. Archimedes’ Principle, or the physical law of buoyancy, was discovered by Greek mathematician Archimedes. The formula of Archimedes’ Principle is: Fb = ρ x g x V.

Here,

• Fb = buoyant force
• ρ = density of the fluid
• V = submerged volume
• g = acceleration due to gravity

Archimedes’ principle states that a body at rest, completely or partially submerged in fluid, is acted upon by a buoyant force, the magnitude of which is equivalent to the weight of displaced fluid. This is considered the first condition of equilibrium. The displaced portion of the fluid's weight is equal to the magnitude of the buoyant force. One example of where its application is used are Hydrometers, which is based on the principle of Archimedes.

The thrust force, also known as the buoyant force, is responsible for enabling objects to float. Thus, this equation can also be called the “law of buoyancy”. Mathematically, the formula for Archimedes' Principle can be given as:

F_b = ρ x g x V

• Fb = buoyant force
• g = acceleration due to gravity
• p = density 
• V = volume

When an object is immersed in water, it either floats or sinks based on the density of the object and the liquid. Let us consider W1 as the actual weight of the object and W2 as the buoyant force.

• When W₁>W₂, the density of the object is more than that of the liquid causing it to sink.
• When W₁=W₂, the density of the object and the liquid are equal and the object will float in the water at any depth and remain completely submerged in the liquid.
• When W₁< W₂, the density of the object is less than that of the liquid causing it to float(partially submerged).