Question

We have three containers with different liquids. The gauge pressure p_gversus depth $\mathit{h}$is plotted in Fig. 14-28 for the liquids. In each container, we will fully submerge a rigid plastic bead. Rank the plots according to the magnitude of the buoyant force on the bead, greatest first.

Short answer

The rank of the plots according to the magnitude of the buoyant force on the bead, greatest first is$F_a>F_b>F_c$ .

Step-by-step solution

The given data 

Graph: Gauge pressure$\left(P_g\right)$ versus depth$\left(d\right)$ is plotted for the liquids.

Understanding the concept of the buoyant force

The buoyant force is proportional to the density of fluid and acceleration due to gravity if the volume is constant. The slope of gauge pressure versus depth graph gives the value of the product of the density and gravitational acceleration of the body.

Formulae:

The buoyant force applied on a block by the fluid, $F_B=\rho Vg$ (i)

The gauge pressure acting on a body, $P_g=\rho gh$ (ii)

Calculation of the rank of the plots according to the buoyant force

Here, the volume of a submerged bead is constant.

Thus, from equation (i), we can get that

$F_B\alpha \rho g....................\left(a\right)$

From equation (ii), the slope of gauge pressure versus depth gives the value of

$\rho g\frac{P_g}{h}\rho g$

As slope is greatest for $P_g$versus $h$graph, the buoyant force would be greatest considering equation (a).

Therefore, rank of buoyant force is $F_a>F_b>F_c$.