Question

A rectangular block is pushed face-down into three liquids, in turn. The apparent weight ${\mathit{W}}_{\mathbf{a}\mathbf{p}\mathbf{p}}$of the block versus depth hin the three liquids is plotted in Fig. 14-26. Rank the liquids according to their weight per unit volume, greatest first.

Short answer

The rank of the liquids according to their weight per unit volume, greatest first is${\gamma }_c>{\gamma }_b>{\gamma }_a$

Step-by-step solution

The given data 

(a) Graph of versus $W_{app}$is plotted for all the three liquids.

(b) The rectangular block is faced down.

Understanding the concept of the apparent weight

Using the concept of the apparent weight of a block and the buoyant force, we can get the value of the required ratio which is the weight per unit volume. The value of weight per unit volume of fluid increases with a decrease in the value of the apparent weight.

Formula:

The buoyant force applied on a body by the fluid due to its weight,$F_b=\rho gV$ (i)

The apparent weight of a body due to displaced volume, $W_{app}=mg-F_B$ (ii)

Calculation of the rank of the liquids according to their weight per unit volume

From equation (i), the force or the weight per unit volume of a body can be given as:

Weight per unit volume$\left(\gamma \right)=\rho g.................\left(a\right)$

Substituting the value of buoyant force from equation (i) in equation (ii), we can get the apparent weight of the block as follows:

$$\begin{gathered} W_{app}=mg-\rho gV \\ =mg-\gamma v \\ \gamma =\frac{mg-W_{app}}{V}........................\left(b\right) \end{gathered}$$

From equation (b), we can say that as $W_{app}$ decreases when weight per unit volume increases.

Therefore, the rank of weight per unit volume is${\gamma }_c>{\gamma }_b>{\gamma }_a$.