Question

Referring to Figure$11.21$ , prove that the buoyant force on the cylinder is equal to the weight of the fluid displaced (Archimedes’ principle). You may assume that the buoyant force is$F_2-F_1$ and that the ends of the cylinder have equal areas A. Note that the volume of the cylinder (and that of the fluid it displaces) equals $(h_2-h_1)A$.

Short answer

The buoyant force acting on the body is equal to the weight of the object which is submerged into the liquid.

Step-by-step solution

Conceptual Introduction

Fluid statics, often known as hydrostatics, is a branch of fluid mechanics that investigates the state of balance of a floating and submerged body, as well as the pressure in a fluid, or imposed by a fluid, on an immersed body.

Given data

The ratio of the object's density to the density of the fluid in which it is immersed determines the fraction submerged in the fluid.

Sketch of the figure

Derivation of the formula of buoyant force

Here the $F_{by}$= buoyant force acting on the body

The two surfaces are named as A and B there the force is acted.

$F_{by}=F_b-F_T$

$$\begin{gathered} F_{by}=F_b-F_T \\ =P_b{A}_b-P_T{A}_T \\ =\left(h_2{\rho }_{w}g\right)A-\left(h_1{\rho }_{w}g\right)A \\ ={\rho }_w\times g\times A\left[h_2-h_1\right] \end{gathered}$$

Further simplifying, we get

$$\begin{gathered} F_{by}={\rho }_w\times g\times V_{wd} \\ =m_{wd}g \\ =W_{wd} \end{gathered}$$

Hence, the buoyant force acting on the body is equal to the weight of the object which is submerged into the liquid.