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Chapter 10: Q32P (page 260)

Archimedes' principle can be used to determine the specific gravity of a solid using a known liquid(Example 10鈥�8). The reverse can be done as well. (a) Asan example, a 3.80-kg aluminum ball has an apparent mass of 2.10 kg when submerged in a particular liquid: calculate the density of the liquid. (b) Determine a formula for finding the density of a liquid using this procedure.

Short Answer

Expert verified

The density of the liquid is \(1210\;{{{\rm{kg}}} \mathord{\left/{\vphantom {{{\rm{kg}}} {{{\rm{m}}^3}}}} \right.} {{{\rm{m}}^3}}}\).

The formula for the density of the liquid is \({\rho _{{\rm{liquid}}}} = \left( {\frac{{{m_{{\rm{object}}}} - {m_{{\rm{apparent}}}}}}{{{m_{{\rm{object}}}}}}} \right){\rho _{{\rm{object}}}}\).

Step by step solution

01

Step1:Understanding the density of a liquid

The density of a liquid is the ratio of the mass of liquid and its volume. The density of a liquid decreases with an increase in temperature. It is expressed in kilograms per cubic meter.

02

Step2:Identification of the given data

Mass of the aluminum ballis \({m_{{\rm{ball}}}} = 3.80\;{\rm{kg}}\).

The apparent mass of aluminum ball submerged in a liquidis \({m_{{\rm{apparent}}}} = 2.10\;{\rm{kg}}\).

03

Step3:Part (a) - Determination of the density of liquid

The mass of the liquid displaced by the ball is equal to the difference between the actual mass and the apparent mass of the aluminum ball.

\(\Delta m = {m_{{\rm{actual}}}} - {m_{{\rm{apparent}}}}\)

The mass of the liquid displaced is the product of the ball's volume and the liquid's density.

\(\begin{aligned}{\rho _{{\rm{liquid}}}}{V_{{\rm{ball}}}} &= \Delta m\\{\rho _{{\rm{liquid}}}}\frac{{{m_{{\rm{ball}}}}}}{{{\rho _{{\rm{Al}}}}}} &= \Delta m\\{\rho _{{\rm{liquid}}}} &= \frac{{\Delta m}}{{{m_{{\rm{ball}}}}}}{\rho _{Al}}\\{\rho _{{\rm{liquid}}}} &= \frac{{{m_{{\rm{actual}}}} - {m_{{\rm{apparent}}}}}}{{{m_{{\rm{ball}}}}}}{\rho _{{\rm{Al}}}}\end{aligned}\)

Substitute the values in above formula.

\(\begin{aligned}{\rho _{{\rm{liquid}}}} &= \frac{{\left( {3.80\;{\rm{kg}} - 2.10\;{\rm{kg}}} \right)}}{{3.80\;{\rm{kg}}}}\left( {2.70 \times {{10}^3}\;{{{\rm{kg}}} \mathord{\left/{\vphantom {{{\rm{kg}}} {{{\rm{m}}^3}}}} \right.} {{{\rm{m}}^3}}}} \right)\\{\rho _{{\rm{liquid}}}} &= 1210\;{{{\rm{kg}}} \mathord{\left/{\vphantom {{{\rm{kg}}} {{{\rm{m}}^3}}}} \right.} {{{\rm{m}}^3}}}\end{aligned}\)

Hence, the density of the liquid is \(1210\;{{{\rm{kg}}} \mathord{\left/{\vphantom {{{\rm{kg}}} {{{\rm{m}}^3}}}} \right.} {{{\rm{m}}^3}}}\).

04

Part (b) :Determination of the formula of the density of liquid

Generalizing the relation from above, we have

\({\rho _{{\rm{liquid}}}} = \left( {\frac{{{m_{{\rm{object}}}} - {m_{{\rm{apparent}}}}}}{{{m_{{\rm{object}}}}}}} \right){\rho _{{\rm{object}}}}\).

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Most popular questions from this chapter

(II) A two component model used to determine percent body fat in a human body assumes that a fraction \(f\left( { < 1} \right)\) of the body鈥檚 total mass m is composed of fat with a density of \({\rm{0}}{\rm{.90 g/c}}{{\rm{m}}^3}\), and that the remaining mass of the body is composed of fat-free tissue with a density of \({\rm{1}}{\rm{.10 g/c}}{{\rm{m}}^3}\). If the specific gravity of the entire body鈥檚 density is \({\rm{X}}\), show that the percent body fat \(\left( { = f \times 100} \right)\)is given by

\(\% {\rm{Body fat}} = \frac{{{\rm{495}}}}{X} - 450\).

Intravenous transfusions are often made under gravity, as shown in Fig. 10鈥�55. Assuming the fluid has a density of \({\bf{1}}{\bf{.00\;g/c}}{{\bf{m}}^{\bf{3}}}\)at what height h should the bottle be placed so the liquid pressure is (a) 52 mm-Hg, and (b) \({\bf{680\;mm - }}{{\bf{H}}_{\bf{2}}}{\bf{O?}}\)(c) If the blood pressure is 75 mm-Hg above atmospheric pressure, how high should the bottle be placed so that the fluid just barely enters the vein?

What diameter must a \({\bf{15}}{\bf{.5}}\;{\bf{m}}\)-long air duct have if the ventilation and heating system is to replenish the air in a \({\bf{8}}{\bf{.0}}\;{\bf{m \times 14}}{\bf{.0}}\;{\bf{m \times 4}}{\bf{.0}}\;{\bf{m}}\) room every \({\bf{15}}{\bf{.0}}\;{\bf{min}}\)? Assume the pump can exert a gauge pressure of \({\bf{0}}{\bf{.710 \times 1}}{{\bf{0}}^{\bf{3}}}\;{\bf{atm}}\).

(II) The maximum gauge pressure in a hydraulic lift is 17.0 atm. What is the largest-size vehicle (kg) it can lift if the diameter of the output line is 25.5 cm?

An ice cube floats in a glass of water filled to the brim. What can you say about the density of ice? As the ice melts, will the water overflow? Explain
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