Question

A hydrometer has a bulb of volume 10.5 cm³ and the area of the cross-section of its stem is 0.3 cm². The hydrometer weighs 13.5 gf. Calculate the density of the liquid if only 4 cm of the length of the stem gets immersed in the liquid.

Answer 1

Step 1: Given data

Volume of the bulb of the hydrometer, $V_b=10.5c{m}^3$

Area of the cross-section of the stem of the hydrometer, $A=0.3c{m}^2$

Weight of hydrometer, $W=13.5gf$

Weight of liquid displaced $=W_l$

Immersed length of the stem of the hydrometer, $L_i=4cm$

Submerged volume of the stem $=V_s$

Density of liquid, ${\rho }_l=?$

Step 2: Formula used

${\rho }_l=\frac{m_l}{v_l}$ ………………………………(a)

Where $m_l$ is the mass of the liquid displaced and $v_l$ is the volume of the liquid displaced

Step 3: Calculation of the density of the liquid

From the basic crux of the law of floatation, it follows that the volume of the liquid displaced will be equal to the summation of the volume of the bulb of the hydrometer and the submerged volume of the stem of the hydrometer.

$V_l=V_b+V_s$ ………………………(b)

The submerged volume of the stem of the hydrometer will be equal to the product of the area of the cross-section of the stem and the immersed length.

$$\begin{gathered} V_s=A_s\times L_i \\ {V}_s=0.3c{m}^2\times 4cm \end{gathered}$$

$V_s=1.2c{m}^3$ …………………….(c)

Substituting the value obtained from the equation (c) and the given values in equation (b), we get

$V_l=10.5c{m}^3+1.2c{m}^3$

$V_l=11.7c{m}^3$ ……………………(d)

From the basic crux of the law of floatation, it follows that the weight of the liquid displaced will be equal to the weight of the hydrometer.

$$\begin{gathered} W_l=W \\ {W}_l=13.5gf \end{gathered}$$

Converting the weight of the liquid displaced into the mass of the liquid displaced, we get

$m_l=\frac{W_l}{g}$

$m_l=13.5g$ ……………………….(e)

Substituting the values obtained from the equations (d) and (e) in equation (a), we get

$$\begin{gathered} {\rho }_l=\frac{13.5g}{11.7c{m}^3} \\ {\rho }_l=1.153g/c{m}^3 \end{gathered}$$

Hence, the density of the liquid is equal to 1.153 g/cm³.