Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

An aluminum-alloy rod has a length of 10.00 cm at 20.00⁰C and a length of 10.015 cm at the boiling point of water.
(a) What is the length of the rod at the freezing point of water?
(b) What is the temperature if the length of the rod is 10.009 cm?

The length of the rod at the freezing point of water is 9.996 cm
The temperature value is $68^{o} C$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: The given data

Aluminum-alloy rod length is $L_{1} = 10.00 cm$ at $T_{1} = 20^{o} C$
Aluminum-alloy rod length is $L_{2} = 10.015 cm$ at $T_{2} = 100^{o} C$

Step 2: Understanding the concept of linear expansion

Thermal expansion can occur due to an increase in temperature. It has three types’ i.e. linear expansion, surface expansion, and volume expansion. For the given problem, we have to use the formula for linear expansion to find the length of the rod and the temperature of the rod.

Formula:

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

Step 3:(a) Calculation of length of the rod at the freezing point

Using equation (i), the coefficient of linear expansion for aluminum-alloy is given as:

$\begin{array}{rcl} α & = & \frac{∆ L}{L ∆ T} \\ = & \frac{10.015 cm - 10.00 cm}{10.0 cm \times {(100 - 20)}^{o} C} \\ = & \frac{0.015 cm}{10.0 cm \times 80^{o} C} \\ = & 1.88 \times 10^{- 5} / {}^{o}C \end{array}$

We have to calculate the change in length of the rod at freezing point of the water so that the change in temperature is given as:

$\begin{array}{rcl} ∆ T & = & 0^{o} C - 100^{o} C \\ = & - 100^{o} C \end{array}$

Length of rod at $100^{o} C$ is $L_{2} = 10.015 cm$

So, the change in linear expansion of the rod is given as:

$\begin{array}{rcl} ∆ L & = & 10.015 cm \times 1.88 \times 10^{- 5} / {}^{o}C \times (- 100^{o} C) \\ = & - 1.88 \times 10^{- 2} cm \end{array}$

Total length at 0⁰C is given as the sum of original length and the expanded length, that is:

$\begin{array}{rcl} L^{'} & = & L + ∆ L \\ = & 10.015 cm - 1.88 \times 10^{- 2} cm \\ = & 10.015 cm - 0.0188 cm \\ = & 9.996 cm \end{array}$

Hence, the value of the length of the rod is 9.996 cm

Step 4: (b) Calculation of the temperature

The length of the rod is given as $L = 10.009 cm$

But we know that $∆ L = 10.009 cm - 10.00 = 0.009 cm$

From equation (i), the change in the temperature due to expansion is given as:

$\begin{array}{rcl} 0.009 cm & = & 10.00 cm \times 1.88 \times 10^{- 5} / {}^{o}C \times ∆ T \\ ∆ T & = & \frac{0.009 cm}{10.00 cm \times 1.88 \times 10^{- 5} / {}^{o}C} \\ = & 47 . 87^{o} C \\ \approx & 48^{o} C \\ = \end{array}$

But, $∆ T = T_{x} - 20^{o} C$ ,

where, T_x is the temperature at which the length of the rod is $L = 10.009 cm$

$\begin{array}{rcl} T_{x} - 20^{o} C & = & 48^{o} C \\ T_{x} & = & 48^{o} C + 20^{o} C \\ T_{x} & = & 68^{o} C \end{array}$

Hence, the value of the required temperature is $68^{o} C$ .

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Fundamentals Of Physics

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Short Answer

An aluminum-alloy rod has a length of 10.00 cm at 20.00⁰C and a length of 10.015 cm at the boiling point of water.
(a) What is the length of the rod at the freezing point of water?
(b) What is the temperature if the length of the rod is 10.009 cm?

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of linear expansion

Step 3:(a) Calculation of length of the rod at the freezing point

Step 4: (b) Calculation of the temperature

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Fundamentals Of Physics

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Short Answer

An aluminum-alloy rod has a length of 10.00 cm at 20.000C and a length of 10.015 cm at the boiling point of water. (a) What is the length of the rod at the freezing point of water? (b) What is the temperature if the length of the rod is 10.009 cm?

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of linear expansion

Step 3:(a) Calculation of length of the rod at the freezing point

Step 4: (b) Calculation of the temperature

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An aluminum-alloy rod has a length of 10.00 cm at 20.00⁰C and a length of 10.015 cm at the boiling point of water.
(a) What is the length of the rod at the freezing point of water?
(b) What is the temperature if the length of the rod is 10.009 cm?