Tag: elastic energy

Questions Related to elastic energy

When a body undergoes a linear tensile strain if experience a lateral contraction also. The ratio of lateral contraction to longitudinal strain is known as

physics properties of material substances poisson ratio poisson's ratio elastic energy

  1. Young's modulus

  2. Bulk modulus

  3. Poisson's law

  4. Hooke's law

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Correct Option: C
Explanation:

Poisson's ratio is the ratio of transverse contraction strain to longitudinal extension strain in the direction of stretching force

A compressive force is applied to a uniform rod of rectangular cross-section so that its length decreases by $1\%$. If the Poisson’s ratio for the material of the rod be $0.2$, which of the following statements is correct ? The volume approximately .....”

physics properties of material substances poisson ratio poisson's ratio elastic energy

  1. decreases by $1\%$

  2. decreases by $0.8\%$

  3. decreases by $0.6\%$

  4. increases by $0.2\%$

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Correct Option: C
Explanation:

$V=Al=abl; \dfrac{\triangle a}{a}=\dfrac{\triangle b}{b}\left[\because \sigma=\dfrac{\dfrac{-\triangle a}{a}}{\dfrac{\triangle l}{l}}=\dfrac{\dfrac{\triangle b}{b}}{\dfrac{\triangle l}{l}}\right]$
$\Rightarrow \dfrac{\triangle V}{V}=2\dfrac{\triangle a}{a}+\dfrac{\triangle I}{l}=-2\sigma \dfrac{\triangle I}{I}+\dfrac{\triangle I}{I}\Rightarrow \dfrac{\triangle V}{V}=\dfrac{\triangle I}{I}(1-2\sigma)-1(1-2\times 0.2)=-1(1-0.4)=-0.6$
$\because$ The volume approximately decreases by $0.6\%$.

When a rubber cord is stretched, the change in volume is negligible compared to the change in its linear dimension. Then poisson's ratio for rubber is

physics properties of material substances poisson ratio poisson's ratio elastic energy

  1. infinite

  2. zero

  3. 0.5

  4. -1

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Correct Option: C
Explanation:

By Lame's relation, $\ \nu = \dfrac { 1 }{ 2 } -\dfrac { E  }{ 6B} ,$ where  $B$ is bulk modulus.
Given, volume change is negligible, thus B tends to infinity. $(B=-V\dfrac { dP }{ dV } )$
 Thus, $\nu=\dfrac { 1 }{ 2 } $

The Poisson's ratio $\sigma$ should satisfy the relation :

physics properties of material substances poisson ratio poisson's ratio elastic energy

  1. -1< $\sigma $ < 0.5

  2. -0.5 < $\sigma $ < 1.0

  3. 0.5 < $\sigma $ < 1.0

  4. -1.0 < $\sigma $ < -0.5

Show answer
Correct Option: A
Explanation:

Poisson's ratio is the ratio of transverse contradiction strain to longitudinal extension strain in the direction of stretching force.

The Poisson's ratio $\sigma$ should satisfy the relation,
$-1<\sigma <0.5$

A metallic wire of young's modulus Y and poisson's ratio $\sigma$, length L and area of cross section A is stretched by a load of W kg. The increase in volume of the wire is:

physics properties of material substances poisson ratio poisson's ratio elastic energy

  1. $\sigma (W^2 L/2AY^2)$

  2. $\sigma (W^2 L/AY^2)$

  3. $\sigma (W^2 L/4AY^2)$

  4. $\sigma (2W^2 L/AY^2)$

Show answer
Correct Option: B
Explanation:

We know that $\sigma =(\Delta A/A) / (\Delta L/L)=(\Delta V/V)/(\Delta L/L)^2 \implies \Delta V=\sigma (\Delta L/L)^2V$

We also know $Y=(W/A)/(\Delta L/L) \implies (\Delta L/L)=W/AY$

Substituting this value in the previous expression, we get, $\Delta V=\sigma (W/AY)^2V=\sigma (W^2 L/AY^2)$

The correct option is (b)

Poisson' ratio is defined as the ratio of 

physics properties of material substances poisson ratio poisson's ratio elastic energy

  1. longitudinal stress and longitudinal strain

  2. longitudinal stress and lateral stress

  3. lateral stress and longitudinal stress

  4. lateral stress and lateral strain

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Correct Option: C
Explanation:

Poisson' ratio is defined as the ratio of lateral stress and longitudinal stress

The correct option is (c)

For which material the poisson's ratio is greater than 1

physics properties of material substances poisson ratio poisson's ratio elastic energy

  1. Steel

  2. Copper

  3. Aluminium

  4. None of the above

Show answer
Correct Option: D
Explanation:

Poisson's ratio can lie only between 0 to 1. It cannot be greater than 1 for any material

The correct option is (d)

A metal wire of length L is loaded and an elongation of $\Delta L$ is produced. If the area of cross section of the wire is A, then the change in volume of the wire, when elongated is . Take Poisson's ratio as 0.25

physics properties of material substances poisson ratio poisson's ratio elastic energy

  1. $\Delta V=(\Delta L)^2A/L$

  2. $\Delta V=(\Delta L)^2A/4L$

  3. $\Delta V=(\Delta L)^2A/2L$

  4. $\Delta V=(\Delta L)^2A/3L$

Show answer
Correct Option: B
Explanation:

We know that $\sigma =(\Delta A/A) / (\Delta L/L)=(\Delta V/V)/(\Delta L/L)^2 \implies \Delta V=\sigma (\Delta L/L)^2V=\sigma (\Delta L/L)^2(LA)=\sigma (\Delta L)^2A/L$

Substituting $\sigma=0.25$, we get, $\Delta V=(\Delta L)^2A/4L$

The correct option is (b)

The change in unit volume of a material under tension with increase in its poisson's ratio will be

physics properties of material substances poisson ratio poisson's ratio elastic energy

  1. Increase

  2. Decrease

  3. Remains same

  4. Initially increases and then decreases

Show answer
Correct Option: B
Explanation:

The poisson's ratio is related to modulus of elasticity as $Y = 3B(1-2 \sigma)$. Since stress is same for Y and B, we get, $dL/L=dV/3V(1-2 \sigma) \implies dV=3V (dL/L)(1-2 \sigma)$
As $\sigma$ is increased, $dV$ decreases. 

The correct option is (b)

The formula relating youngs modulus (Y), rigidity modulus (n) and Poisson's ratio ($\sigma$) is 

physics properties of material substances poisson ratio poisson's ratio elastic energy

  1. $Y=2n(1- \sigma)$

  2. $Y=2n(1+\sigma)$

  3. $Y=n(1- 2\sigma)$

  4. $Y=n(1+2 \sigma)$

Show answer
Correct Option: B
Explanation:

Young's modulus and rigidity modulus can be related to poisson's ratio as $Y=2n(1+\sigma)$

The correct option is (b)