Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Problem 864

Two wires of copper having the length in the ratio \(4: 1\) and their radii are as \(1: 4\) are stretched by the same force. What will be the ratio of longitudinal strain in the two wires? (A) \(1: 16\) (B) \(16: 1\) (C) 1:64 (D) \(64: 1\)

Short Answer

Expert verified
The ratio of longitudinal strain in the two wires is \(16:1\).

Step by step solution

01

List the given information

- Length ratio: \(L_1:L_2 = 4:1\) - Radii ratio: \(r_1:r_2 = 1:4\) - Same force F is applied on both wires
02

Define longitudinal strain and Young's modulus (Y)

Longitudinal strain, \(\epsilon\), is the fractional change in length (\(\frac{\Delta L}{L}\)) when the force is applied. Young's modulus (Y) is a property of the material that relates the longitudinal stress and strain: \(Y = \frac{σ}{\epsilon}\) Where σ is the longitudinal stress, given by \(\frac{F}{A}\), and A is the cross-sectional area of the wire.
03

Find the expression for the longitudinal strain

For the two copper wires, the Young's Modulus is the same, as their material is identical. Thus, \(Y_1 = Y_2\) Using \(Y = \frac{σ}{\epsilon}\), \(\frac{σ_1}{\epsilon_1} = \frac{σ_2}{\epsilon_2}\) Then, stress can be written in terms of force and area. Since both wires experience the same force, F, \(\frac{F/A_1}{\epsilon_1} = \frac{F/A_2}{\epsilon_2}\) We simplify this expression to obtain the ratio of \(\epsilon\): \(\frac{\epsilon_1}{\epsilon_2} = \frac{A_2}{A_1}\)
04

Calculate the cross-sectional area ratio

The cross-sectional area of a wire is given by \(A = πr^2\). Using the given radius ratio, we can calculate the area ratio: \(\frac{A_1}{A_2} = \frac{πr_1^2}{πr_2^2} = \frac{(1)^2}{(4)^2} = \frac{1}{16}\)
05

Conclude the ratio of strain

Now, we can use the obtained area ratio to find the ratio of strains: \(\frac{\epsilon_1}{\epsilon_2} = \frac{A_2}{A_1} = \frac{16}{1}\) This corresponds to option (B) \(16:1\). So, the ratio of longitudinal strain in the two wires is \(16:1\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The compressibility of water \(4 \times 10^{-5}\) per unit atmospheric pressure. The decrease in volume of 100 cubic centimeter of water under a pressure of 100 atmosphere will be.......... (A) \(4 \times 10^{-5} \mathrm{CC}\) (B) \(4 \times 10^{-5} \mathrm{CC}\) (C) \(0.025 \mathrm{CC}\) (D) \(0.004 \mathrm{CC}\)

Read the assertion and reason carefully and mark the correct option given below. (a) If both assertion and reason are true and the reason is the correct explanation of the assertion. (b) If both assertion and reason are true but reason is not the correct explanation of the assertion. (c) If assertion is true but reason is false. (d) If the assertion and reason both are false. Assertion: The water rises higher in a capillary tube of small diametre than in the capillary tube of large diameter. Reason: Height through which liquid rises in a capillary tube is inversely proportional to the diameter of the capillary tube. (A) a (B) b (C) c (D) d

Surface tension of a liquid is found to be influenced by (A) It increases with the increase of temperature. (B) Nature of the liquid in contact. (C) Presence of soap that increase it. (D) Its variation with the concentration of the liquid.

Two solid spheres of same metal but of mass \(\mathrm{M}\) and \(8 \mathrm{M}\) full simultaneously on a viscous liquid and their terminal velocity are \(\mathrm{V}\) and ' \(\mathrm{nV}^{\prime}\) then value of 'n' is (A) 16 (B) 8 (C) 4 (D) 2

At what temperature the centigrade (celsius) and Fahrenheit readings at the same. \((\mathrm{A})-40^{\circ}\) (B) \(+40^{\circ} \mathrm{C}\) (C) \(36.6^{\circ}\) (D) \(-37^{\circ} \mathrm{C}\)
See all solutions

Recommended explanations on English Textbooks

Cues and Conventions

Read Explanation

Argumentative Essay

Read Explanation

Discourse

Read Explanation

English Grammar

Read Explanation

Email

Read Explanation

History of English Language

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free