Question

A hydrometeris an instrument used to determine liquid density. A simple one is sketched in Figure P14.36. The bulb of a syringe is squeezed and released to let the atmosphere lift a sample of the liquid of interest into a tube containing a calibrated rod of known density. The rod, of length $\mathit{L}$and average density ${\mathit{\rho }}_{\mathbf{0}}$, floats partially immersed in the liquid of density $\mathit{\rho }$. A length $\mathit{h}$ of the rod protrudes above the surface of the liquid. Show that the density of the liquid is given byrole="math" localid="1663764227560" $\mathit{\rho }\mathbf{=}\frac{{\mathbf{\rho }}_{\mathbf{0}}\mathbf{L}}{\mathbf{L}\mathbf{-}\mathbf{h}}$.

Short answer

It is proved that the density of the liquid is $\rho =\frac{{\rho }_{0}L}{L-h}$.

Step-by-step solution

Given data:

The length of the rod is $L$.

The length of the rod not immersed in the liquid is $h$.

The average density of the rod is ${\rho }_0$.

The density of the liquid is $\rho$.

Buoyancy and gravitational force:

The buoyant force from a liquid of density$\mathit{d}$when volume of the liquid displaced$\mathit{v}$is,

$\mathit{B}\mathbf{=}\mathit{g}\mathit{d}\mathit{v}$ ….. (1)

Here,$\mathit{g}$ is the acceleration due to gravity.

The gravitational force on a body of mass$\mathit{m}$is,

$\mathit{F}\mathbf{=}\mathit{m}\mathit{g}$ ….. (2)

Show that the density of the liquid is ρ=ρ0LL−h:

Length of the rod immersed in the liquid is $L-h$.

Let $A$be the cross sectional area of the rod. Volume of the rod immersed in the liquid is $A\left(L-h\right)$.

This is also the volume of the liquid displaced. From equation (1), the buoyancy force on the rod directed upward is,

$B=\rho gA(L-h)$

The total volume of the rod is $AL$and thus its mass is ${\rho }_{0}AL$. Thus from equation (2), the gravitational force acting on the rod directed downward is,

$F={\rho }_{0}ALg$

At equilibrium, these two forces cancel each other. Thus

$$\begin{gathered} \rho gA(L-h)={\rho }_{0}ALg \\ \rho =\frac{{\rho }_{0}L}{L-h} \end{gathered}$$

Hence it is proved.