Chapter 10, Problem 31P

(a)

To determine

To Calculate: The density of the liquid.

Answer to Problem 31P

  $1031.9{\text{kg/m}}^{\text{3}}$

Explanation of Solution

Given:

Apparent mass, $M_{\text{app}}=2.10\text{kg}$

Actual mass, $M_{\text{sample}}=3.40\text{kg}$

Formula used:

The volume is given as,

  $V=\frac{m}{\rho }$

Where $m$ is the mass of the substance, $\rho$ is the density and $V$ is the volume.

Calculation:

The volume of the aluminum,

  $V_{Al}=\frac{m}{\rho }=\frac{3.40}{2699}$

When the block is immersed in the liquid, this volume is equal to the volume of the displaced, The loss in weight is given as,

  $=3.40-2.10=1.3\text{kg}$

The density of the liquid is

  $\begin{array}{c}\rho =\frac{m}{V}\\ =\frac{1.3\text{kg}}{\frac{3.40}{2699{\text{m}}^3}}\\ =\frac{1.3\times 2699}{3.4}\\ =1031.9{\text{kg/m}}^{\text{3}}\end{array}$

Conclusion:

The density of the liquid is $1031.9{\text{kg/m}}^{\text{3}}$ .

(b)

To determine

To Derive:The formula for density of a liquid.

Answer to Problem 31P

  ${\rho }_{\text{liquid}}=\frac{m_{\text{solid}}-m_{\text{apparent}}}{m_{\text{solid}}}\times {\rho }_{\text{solid}}$

Explanation of Solution

Formula used:

The volume is given as,

  $V=\frac{m}{\rho }$

Where $m$ is the mass of the substance, $\rho$ is the density and $V$ is the volume.

Calculation:

Consider a solid and a liquid sample whose apparent mass is represented by the $m_{\text{apparent}}$ , $m_{\text{solid}}$ is the mass of the solid, $g$ is the acceleration due to gravity and $F_B$ is the buoyant force.

  $\begin{array}{l}{F}_B=m_{\text{solid}}g-m_{\text{apparent}}g\\ m_{\text{solid}}g=m_{\text{apparent}}g-F_B\\ =m_{\text{apparent}}g-\left(-V_{\text{solid}}\times {\rho }_{\text{liquid}}\times g\right)\\ m_{\text{solid}}-m_{\text{apparent}}=V_{\text{solid}}\times {\rho }_{\text{liquid}}\\ m_{\text{solid}}-m_{\text{apparent}}=\frac{m_{\text{solid}}}{{\rho }_{\text{solid}}}\times {\rho }_{\text{liquid}}\\ {\rho }_{\text{solid}}=\frac{m_{\text{solid}}}{m_{\text{solid}}-m_{\text{apparent}}}\times {\rho }_{\text{liquid}}\\ \text{or}\\ {\rho }_{\text{liquid}}=\frac{m_{\text{solid}}-m_{\text{apparent}}}{m_{\text{solid}}}\times {\rho }_{\text{solid}}\end{array}$

Conclusion:

The density formula is ${\rho }_{\text{liquid}}=\frac{m_{\text{solid}}-m_{\text{apparent}}}{m_{\text{solid}}}\times {\rho }_{\text{solid}}$ .