Question

Refer to Problem 36 and Figure P14.36. A hydrometer is to be constructed with a cylindrical floating rod. Nine fiduciary marks are to be placed along the rod to indicate densities of $\mathbf{0}\mathbf{.}\mathbf{98}\mathbf{}\mathit{g}\mathbf{/}\mathit{c}{\mathit{m}}^{\mathbf{3}}$, $\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{}\mathit{g}\mathbf{/}\mathit{c}{\mathit{m}}^{\mathbf{3}}$,$\mathbf{1}\mathbf{.}\mathbf{02}\mathbf{}\mathit{g}\mathbf{/}\mathit{c}{\mathit{m}}^{\mathbf{3}}$ , $\mathbf{1}\mathbf{.}\mathbf{04}\mathbf{}\mathit{g}\mathbf{/}\mathit{c}{\mathit{m}}^{\mathbf{3}}$, . . . ,$\mathbf{1}\mathbf{.}\mathbf{14}\mathbf{}\mathit{g}\mathbf{/}\mathit{c}{\mathit{m}}^{\mathbf{3}}$ . The row of marks is to start$\mathbf{0}\mathbf{.}\mathbf{2}\mathbf{}\mathit{c}\mathit{m}$ from the top end of the rod and end$\mathbf{1}\mathbf{.}\mathbf{8}\mathbf{}\mathit{c}\mathit{m}$ from the top end.

(a) What is the required length of the rod?

(b) What must be its average density?

(c) Should the marks be equally spaced? Explain your answer.

Short answer

(a) The required length of the rod is $\mathbf{11}\mathbf{.}\mathbf{6}\mathbf{}\mathit{c}\mathit{m}$.

(b) The average density of the rod must be$\mathbf{0}\mathbf{.}\mathbf{963}\mathbf{}\mathit{g}\mathbf{/}\mathit{c}{\mathit{m}}^{\mathbf{3}}$ .

(c) The marks on the cylinder should not be equispaced since the rate of change of height of cylinder not immersed in liquid with respect to liquid density is not constant.

Step-by-step solution

Given data

Lowest density to be measured by the Hydrometer

${\rho }_l=0.98{\text{g/cm}}^{\text{3}}$

Highest density to be measured by the Hydrometer

${\rho }_h=1.14{\text{g/cm}}^{\text{3}}$

Length of red not immersed for the lowest density

$h_l=0.2\text{cm}$

Length of red not immersed for the highest density

$h_d=1.8\text{cm}$

Hydrometer relation

The density $\rho$of the liquid measured in the Hydrometer is a function of the average density ${\rho }_0$ of the measuring cylinder, the length $L$ of the cylinder and the height $h$ of the cylinder not immersed in the liquid is as follows

$\mathit{\rho }\mathbf{=}\frac{{\mathbf{\rho }}_{\mathbf{0}}\mathbf{L}}{\mathbf{L}\mathbf{-}\mathbf{h}}$ .....(I)

Determining the length of the cylinder

Equation (I) for the lowest density liquid to be measured is as follows

${\rho }_l=\frac{{\rho }_{0}L}{L-h_l}$

Substitute the values to get

$0.98{\text{g/cm}}^{\text{3}}=\frac{{\rho }_{0}L}{L-0.2\text{cm}}$ .....(II)

Equation (I) for the highest density liquid to be measured is as follows

${\rho }_h=\frac{{\rho }_{0}L}{L-h_h}$

Substitute the values to get

$1.14{\text{g/cm}}^{\text{3}}=\frac{{\rho }_{0}L}{L-1.8\text{cm}}$ .....(III)

Divide equation (III) by equation (II) to get

$\begin{array}{rcl}\frac{1.14{\text{g/cm}}^{\text{3}}}{0.98{\text{g/cm}}^{\text{3}}}& =& \frac{L-0.2\text{cm}}{L-1.8\text{cm}}\\ 1.14L-2.052\text{cm}& =& 0.98L-0.196\text{cm}\\ 0.16L& =& 1.856\text{cm}\\ L& =& 11.6\text{cm}\\ & & \end{array}$

Thus, the required length is $11.6\text{cm}$.

Determining the average density of the cylinder

Substitute the value of the length of the cylinder in equation (II) to get

$\begin{array}{rcl}0.98{\text{g/cm}}^{\text{3}}& =& \frac{{\rho }_0\times 11.6\text{cm}}{11.6\text{cm}-0.2\text{cm}}\\ & =& 1.0175{\rho }_0\\ {\rho }_0& =& \frac{0.98{\text{g/cm}}^{\text{3}}}{1.0175}\\ & =& 0.963{\text{g/cm}}^{\text{3}}\end{array}$

Thus, the average density is$0.963{\text{g/cm}}^{\text{3}}$ .

Determining the spacing of the marks

Equation (I) is written as

$h=L\left(1-\frac{{\rho }_0}{\rho }\right)$

The rate of change of height with liquid density is

$\frac{dh}{d\rho }=\frac{L{\rho }_0}{{\rho }^2}$

This is not constant. Hence the change in height decreases with increase in density. Thus the marks can't be equispaced.