Question

Describe briefly how you can show that the buoyant force acting on a solid body immersed in a liquid, does not depend on the nature of the solid body.

Answer 1

1. Let us consider a cylinder of height$h$, volume$V$, and area of cross-section$A$, is being held under the surface of a liquid.
2. Let pressure at the top of the cylinder be $P_1$and at the bottom of the cylinder be$P_2$.
3. Now, the force acting on the top of the cylinder $F_1=P_{1}A$
4. Similarly, the force acting on the bottom of the cylinder $F_2=P_{2}A$
5. Since the pressure is greater at greater depths, therefore, the upward force exceeds the downward force.
6. Consequently, the liquid applies to the cylinder a net upward force or buoyant force, whose magnitude $F_B$is given by:

$$\begin{gathered} F_B=F_2-F_1 \\ =P_{2}A-P_{1}A \\ =(P_2-P_1)A \end{gathered}$$

7. Now, the pressure difference between the top and bottom of the cylinder is given by, $P_2-P_1=hpg$ (where $p$ is the density of the fluid)

therefore,

$F_B=hpgA$

But $hA=V$is the volume of the cylinder or the volume of the fluid displaced,

and $V\times p=M$ mass of the fluid displaced.

Hence, buoyant force,$F_B=$$Mg$ = weight of the fluid displaced

Hence, the buoyant force acting on a body is equal to the weight of the displaced fluid and does not depend on the nature of the liquid.