Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

Answers without the blur.

Just sign up for free and you're in.

Short Answer

A steel rod is 3.00 cm in diameter at $25 . 00^{o} C$ . A brass ring has an interior diameter of 2.992 cm at $25 . 00^{o} C$ . At what common temperature will the ring just slide onto the rod?

The temperature at which ring just slides into the rod is $360 . 0^{o} C$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: The given data

Diameter of the steel rod at 25^oC is $D_{s} = 3.000 cm$
Diameter of the brass ring at 25^oC is $D_{b} = 2.992 cm$
Expansion coefficient for steel is $α_{s} = 11 \times 10^{- 6} / {}^{o}C$
Expansion coefficient for brass is $α_{b} = 19 \times 10^{- 6} / {}^{o}C$

Step 2: Understanding the concept of thermal expansion

When an object's temperature changes, it expands and grows larger, a process known as thermal expansion. Thermal expansion can be occurring due to an increase in temperature. For the given problem, we have to use the formula for linear expansion to calculate the final temperature.

Formula:

The linear expansion of diameter of a body, $∆ D = Dα ∆ T$ …(i)

Where $α$ is the coefficient of linear expansion of body, $∆ T$ is the temperature difference and D is the diameter

Step 3: Calculation of the temperature

The total length after expansion is given as:

$\begin{array}{rcl} D & = & D_{s} + ∆ D_{s} \\ = & D_{s} + D_{s} α_{s} ∆ T . . . . . . . . . . . . . . (a) (from equation (i), the change of expansion) \end{array}$

The total length after expansion is given as:

$\begin{array}{rcl} D & = & D_{b} + ∆ D_{b} \\ = & D_{b} + D_{b} α_{b} ∆ T . . . . . . . . . . . . . . (b) (from equation (i), the change of expansion) \end{array}$

The rod will fit through the ring only if the diameter of the ring and the rod is the same after the expansion so that, from equations (a) and (b), it is given as:

$\begin{array}{rcl} D_{s} + D_{s} α_{s} ∆ T & = & D_{b} + D_{b} α_{b} ∆ T \\ D_{s} - D_{b} & = & D_{b} α_{b} ∆ T - D_{s} α_{s} ∆ T \\ D_{s} - D_{b} & = & (D_{b} α_{b} - D_{s} α_{s}) ∆ T \\ ∆ T & = & \frac{D_{s} - D_{b}}{D_{b} α_{b} - D_{s} α_{s}} \end{array}$

By substituting the given values in the above equation, we get the temperature change as:

$\begin{array}{rcl} ∆ T & = & \frac{3.000 cm - 2.992 cm}{(2.992 cm \times 19 \times 10^{- 6} / {}^{o}C - 3.000 cm \times 11 \times 10^{- 6} / {}^{o}C)} \\ = & \frac{0.008 cm}{(56.848 - 33) \times 10^{- 6} cm / {}^{o}C} \\ = & \frac{0 . 008^{o} C}{(23.848) \times 10^{- 6}} \\ = & 335 . 0^{o} C \end{array}$

Using this value, we can calculate the total temperature is given as:

$\begin{array}{rcl} T & = & T_{i} + ∆ T \\ = & 25 . 00^{o} C + 335 . 0^{o} C \\ = & 360 . 0^{o} C \end{array}$

Hence, the value of the temperature is $360 . 0^{o} C$

Find Study Materials for

Create Study Materials

Select your language

Fundamentals Of Physics

Answers without the blur.

Short Answer

A steel rod is 3.00 cm in diameter at $25 . 00^{o} C$ . A brass ring has an interior diameter of 2.992 cm at $25 . 00^{o} C$ . At what common temperature will the ring just slide onto the rod?

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of thermal expansion

Step 3: Calculation of the temperature

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Select your language

Fundamentals Of Physics

Answers without the blur.

Short Answer

A steel rod is 3.00 cm in diameter at 25.00oC. A brass ring has an interior diameter of 2.992 cm at 25.00oC. At what common temperature will the ring just slide onto the rod?

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of thermal expansion

Step 3: Calculation of the temperature

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Work Energy and Power

Radiation

Astrophysics

Physics of Motion

Scientific Method Physics

Fluids

Torque and Rotational Motion

Nuclear Physics

Fields in Physics

Measurements

Space Physics

Electricity

Physical Quantities and Units

Medical Physics

Electrostatics

Further Mechanics and Thermal Physics

Mechanics and Materials

Engineering Physics

Energy Physics

Force

Particle Model of Matter

Waves Physics

Magnetism

Modern Physics

Atoms and Radioactivity

Dynamics

Electricity and Magnetism

Conservation of Energy and Momentum

Fundamentals of Physics

Kinematics Physics

Circular Motion and Gravitation

Linear Momentum

Rotational Dynamics

Oscillations

Translational Dynamics

Geometrical and Physical Optics

Thermodynamics

Magnetism and Electromagnetic Induction

Electromagnetism

Electric Charge Field and Potential

Turning Points in Physics

A steel rod is 3.00 cm in diameter at $25 . 00^{o} C$ . A brass ring has an interior diameter of 2.992 cm at $25 . 00^{o} C$ . At what common temperature will the ring just slide onto the rod?