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A wire having a length L and cross-sectional area A is suspend

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A wire having a length $L$ and cross-sectional area $A$ is suspended at one of its ends from a ceiling. Density and Young's modulus of material of the wire are $ρ$ and $Y$ , respectively. Its strain energy due to its own weight is $\frac{ρ ^{2} g ^{2} A L ^{3}}{α Y}$ . Find the value of $α$ .

Name: Video solution 1: A wire having a length L and cross-sectional area A is suspended at one of its ends from a ceiling. Density and Young's modulus of material of the wire are ρ and Y, respectively. Its strain energy due to its own weight is αYρ2g2AL3. Find the value of α.
Uploaded: 2022-06-15T04:26:57.000Z
Duration: 4 min
Description: Video solution 1: A wire having a length L and cross-sectional area A is suspended at one of its ends from a ceiling. Density and Young's modulus of material of the wire are ρ and Y, respectively. Its strain energy due to its own weight is αYρ2g2AL3. Find the value of α.

Views: 5,810 students

Updated on: Jun 15, 2022

Book:

Mechanics Volume 2 (DC Pandey)

Found 4 tutors discussing this question

Amelia Discussed

A wire having a length

L

and cross-sectional area

A

is suspended at one of its ends from a ceiling. Density and Young's modulus of material of the wire are

ρ

and

Y

, respectively. Its strain energy due to its own weight is

\frac{ρ ^{2} g ^{2} A L ^{3}}{α Y}

. Find the value of

α

5 mins ago

Discuss this question LIVE

5 mins ago

Text solutionVerified

Consider an element as shown in the figure.
Solution Image

Stress in the element

= \frac{Force}{Area} = \frac{x A ρ g}{A} = x ρ g

Now, elastic potential energy stored in the wire is

d U d U = \frac{1}{2} (Stress) (Strain) (Volume) = \frac{1}{2} \cdot \frac{( Stress ) ^{2}}{Y} (Volume) = \frac{1}{2} \cdot \frac{( x ρ g ) ^{2}}{Y} A d x = \frac{1}{2} \cdot \frac{ρ ^{2} g ^{2} A}{Y} x^{2} d x

Total elastic potential energy

= \frac{1}{2} \cdot \frac{ρ ^{2} g ^{2} A}{Y} \int_{0}^{L} x^{2} d x

= \frac{ρ ^{2} g ^{2} A L ^{3}}{6 Y}

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Question Text	A wire having a length $L$ and cross-sectional area $A$ is suspended at one of its ends from a ceiling. Density and Young's modulus of material of the wire are $ρ$ and $Y$ , respectively. Its strain energy due to its own weight is $\frac{ρ ^{2} g ^{2} A L ^{3}}{α Y}$ . Find the value of $α$ .
Updated On	Jun 15, 2022
Topic	Mechanical Properties of Solids
Subject	Physics
Class	Class 11
Answer Type	Text solution:1 Video solution: 1
Upvotes	239
Avg. Video Duration	4 min

A wire having a length $L$ and cross-sectional area $A$ is suspended at one of its ends from a ceiling. Density and Young's modulus of material of the wire are $ρ$ and $Y$ , respectively. Its strain energy due to its own weight is $\frac{ρ ^{2} g ^{2} A L ^{3}}{α Y}$ . Find the value of $α$ .

Text solutionVerified

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A wire having a length L and cross-sectional area A is suspended at one of its ends from a ceiling. Density and Young's modulus of material of the wire are ρ and Y, respectively. Its strain energy due to its own weight is αYρ2g2AL3​. Find the value of α.

Text solutionVerified

Filo tutor solutions (1)

Practice questions from Mechanics Volume 2 (DC Pandey)

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A wire having a length $L$ and cross-sectional area $A$ is suspended at one of its ends from a ceiling. Density and Young's modulus of material of the wire are $ρ$ and $Y$ , respectively. Its strain energy due to its own weight is $\frac{ρ ^{2} g ^{2} A L ^{3}}{α Y}$ . Find the value of $α$ .