Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

Find the change in volume of an aluminum sphere with an initial radius of 10 cm when the sphere is heated from $0 . 0^{o} C to 100 . 0^{o} C$ .

The change in volume of an aluminum sphere is $29 {cm}^{3}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: The given data

Initial radius of sphere $r_{1} = 10 cm$ at $T_{1} = 0^{o} C$
Final temperature at which sphere expands $T_{2} = 100^{o} C$
Linear expansion coefficient for aluminum is $α = 23 \times 10^{- 6} / {}^{o}C$

Step 2: Understanding the concept of linear expansion

When an object's temperature changes, it expands and grows larger, a process known as thermal expansion. Thermal expansion can occur due to an increase in temperature. For the given problem, we have to use the formula for volume expansion. Change in volume expansion depends on the original volume, the temperature change, and the volume expansion coefficient.

Formula:

The linear expansion of a body, $∆ L = Lα ∆ T$ …(i)

Where is the coefficient of linear expansion of body.

The volume change in expansion of a body, $∆ V = Vβ ∆ T$ …(ii)

Where is the coefficient of volume expansion of the rod

The volume of the sphere, $V = \frac{4}{3} {πr}^{3}$ …(iii)

Step 3: Calculation of the volume change

So, the coefficient of volume expansion of the rod is given as:

$\begin{array}{rcl} β & = & 3 α \\ = & 3 \times 23 \times 10^{- 6} / {}^{o}C \\ = & 69 \times 10^{- 6} / {}^{o}C \end{array}$

Therefore, we can calculate the initial volume of the sphere by using the formula of equation (iii) as given:

$\begin{array}{rcl} V & = & \frac{4}{3} \times 3.14 \times {(10 cm)}^{3} \\ = & 4186.66 {cm}^{3} \end{array}$

We can now find the change in volume using equation (ii) as:

$\begin{array}{rcl} ∆ V & = & 4186.66 {cm}^{3} \times 69 \times 10^{- 6} / {}^{o}C \times {(100 - 0)}^{o} C \\ = & 28.88 {cm}^{3} \\ \approx & 29 {cm}^{3} \end{array}$

Hence, the change in volume is $29 {cm}^{3}$

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Fundamentals Of Physics

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Short Answer

Find the change in volume of an aluminum sphere with an initial radius of 10 cm when the sphere is heated from $0 . 0^{o} C to 100 . 0^{o} C$ .

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of linear expansion

Step 3: Calculation of the volume change

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Fundamentals Of Physics

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Short Answer

Find the change in volume of an aluminum sphere with an initial radius of 10 cm when the sphere is heated from 0.0oC to 100.0oC.

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of linear expansion

Step 3: Calculation of the volume change

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