Book edition 9th Edition

Author(s) Raymond A. Serway, John W. Jewett

Pages 1624 pages

ISBN 9781133947271

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Short Answer

A plastic sphere floats in water with $50 %$ of its volume submerged. This same sphere floats in glycerin with $40 %$ of its volume submerged. Determine the densities of (a) the glycerin and (b) the sphere.

(a) The density of glycerin is $1250 kg / m^{3}$ .

(b) The density of the sphere is $500 kg / m^{3}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Given data:

Percentage of volume of sphere submerged in water is $50 %$ .

Percentage of volume of sphere submerged in glycerin is $40 %$ .

Buoyancy and gravitational force:

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

$B = d g v$ ..…(I)

Here, $g$ is the acceleration due to gravity having value,

$g = 9.8 m / s^{2}$

The gravitational force on a body of volume $v$ and density $d$ is,

role="math" localid="1663748724511" $F = d v g$ ..... (II)

(a) Determining the density of glycerin:

Let the volume of the sphere be $V$ and its density be $d_{s}$ . Volume of water displaced is $\frac{V}{2}$ . Density of water is $1000 kg / m^{3}$ . Since the sphere is in equilibrium, equate equations (I) and (II) to get

Volume of displaced glycerin is .

Let the density of glycerin be .

Since the sphere is in equilibrium, equate equations (I) and (II) to get

$d_{s} V g = 1000 kg / m^{3} \times \frac{V}{2} \times g d_{s} = 500 kg / m^{3}$

Thus, the required density is,

$\frac{4}{10} V = 0.4 V$

Let the density of glycerin be $d_{g}$ .

Since the sphere is in equilibrium, equate equations (I) and (II) to get

$\begin{array}{rcl} d_{s} V g & = & d_{g} \times 0.4 V \times g \end{array}$

$\begin{array}{rcl} d_{g} & = & \frac{d_{s}}{0.4} \\ = & \frac{500}{0.4} kg / m^{3} \\ = & 1250 kg / m^{3} \end{array}$

Thus, the required density is $1250 kg / m^{3}$ .

(b) Determining the density of sphere:

This has already been calculated in the previous step. The required density is $500 kg / m^{3}$ .

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Physics For Scientists & Engineers

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Short Answer

A plastic sphere floats in water with $50 %$ of its volume submerged. This same sphere floats in glycerin with $40 %$ of its volume submerged. Determine the densities of (a) the glycerin and (b) the sphere.

Step by Step Solution

Given data:

Buoyancy and gravitational force:

(a) Determining the density of glycerin:

(b) Determining the density of sphere:

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Physics For Scientists & Engineers

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Short Answer

A plastic sphere floats in water with 50% of its volume submerged. This same sphere floats in glycerin with 40% of its volume submerged. Determine the densities of (a) the glycerin and (b) the sphere.

Step by Step Solution

Given data:

Buoyancy and gravitational force:

(a) Determining the density of glycerin:

(b) Determining the density of sphere:

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A plastic sphere floats in water with $50 %$ of its volume submerged. This same sphere floats in glycerin with $40 %$ of its volume submerged. Determine the densities of (a) the glycerin and (b) the sphere.