Tag: elastic energy

Questions Related to elastic energy

The depression produced at the end of a $50 cm$ long cantilever on applying a load is $15 mm$. The depression produced at a distance of $30 cm $ from the rigid end will be

physics mechanical properties of solids applications of elasticity elastic energy properties of matter

  1. 3.24 mm

  2. 1.62 mm

  3. 6.48 mm

  4. 12.96 mm

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Correct Option: C
Explanation:

Length of cantilever, $L= 50cm=0.5m$
Deflection at the end $w(L)= 15mm$
Find: Deflection $w(x)$ at 30cm from rigid end.
Depression at a point in a cantilever beam with load at one end is given by,$ w(x)=\dfrac { P{ x }^{ 2 }(3L-x) }{ 6EI } $
We have,
$\dfrac { w(x) }{ w(L) } =\dfrac { \dfrac { P{ x }^{ 2 }(3L-x) }{ 6EI }  }{ \dfrac { PL^{ 3 } }{ 3EI }  } $

$\dfrac { w(0.3m) }{ w(0.5m) } =\dfrac { \dfrac { P{ (0.3) }^{ 2 }(3(0.5)-0.3) }{ 6EI }  }{ \dfrac { P(0.5)^{ 3 } }{ 3EI }  } $

This gives, $w(0.3m)=6.48mm$

A solid cylindrical rod of radius $3 mm$ gets depressed under the influence of a load through $8 mm$. The depression produced in an identical hollow rod with outer and inner radii of $4 mm$ and $2 mm$ respectively, will be

physics mechanical properties of solids applications of elasticity elastic energy properties of matter

  1. 2.7mm

  2. 1.9mm

  3. 3.2mm

  4. 7.7mm

Show answer
Correct Option: A
Explanation:

Depression in Solid cylinder $\delta _{1}= 8mm$
Its radius $r _{1}= 3mm$
Outer radius of Hollow cylinder $R _{2}= 4mm$
Inner radius of Hollow cylinder $r _{2}= 2mm$
Let depression in this cylinder be $\delta _{2}$.
Depression $\delta =\dfrac { W{ l }^{ 3 } }{ 12\pi{r }^{ 4 }Y } $
From the above equation we know that $\delta$ is proportional to $\dfrac{1}{r^{4}}$
Hence, we have
$\dfrac{\delta _{1}}{\delta _{2}}= \dfrac{{R _{2}}^{4}-{r _{2}}^{4}}{r _{1}^{4}}$
Substituting the values in above equation, we get
$\delta _{2}=2.7mm$

A beam of cross section area A is made of a material of Young modulus Y. The beam is bent into the arc of a circle of radius R. The bending moment is proportional to

physics mechanical properties of solids applications of elasticity elastic energy properties of matter

  1. $\displaystyle \frac{Y}{R}$

  2. $\displaystyle \frac{Y}{RA}$

  3. $\displaystyle \frac{R}{Y}$

  4. $YR$

Show answer
Correct Option: A
Explanation:

Bending moment $C=\dfrac{Y{I} _{G}}{R}$
Therefore, $C$ is proportional to $\dfrac{Y}{R}$

A steel wire of length $L$ and area of cross-section A shrinks by $\Delta l$ during night. Find the tension developed at night if Young's modulus is $Y$ and wire is clamped at both ends

physics mechanical properties of solids applications of elasticity elastic energy properties of matter

  1. $\displaystyle \frac{AYL}{\Delta l}$

  2. $AYL$

  3. $AY\Delta l$

  4. $\displaystyle \frac{AY\Delta l}{L}$

Show answer
Correct Option: D
Explanation:

$\displaystyle \frac{\Delta l}{L}=\frac{F}{AY}:or:F=\frac{AY\Delta l}{L}$ (using standard equation)

A wire of radius $1 mm$ is bent in the form of a circle of radius $10 cm$. The bending moment will be $(Y = 2\times10^{11}N/m^{2})$

physics mechanical properties of solids applications of elasticity elastic energy properties of matter

  1. 3.14 N/m

  2. 6.28 N/m

  3. 1.57 N/m

  4. 15.7 N/m

Show answer
Correct Option: C
Explanation:

Bending Moment $C=\dfrac { Y{ I } _{ G } }{ R } $
where $Y= 2\times { 10 }^{ 11 }N/{m}^{2}$
$R= 10cm= 0.1m$
${ I } _{ G }=\dfrac{\pi{R}^{4}}{4} = \dfrac{\pi{0.1}^{4}}{4} =7.85\times{10}^{-5} m^{4}$
Put these in the expression for $C$,
$C=1.57N/m$

A body of mass 3.14 kg is suspended from one end of a wire of length 10 m. The radius of cross-section of the wire is changing uniformly from $5 \times 10^{-4}$ m at the top (i.e. point of suspension) to $9.8 \times 10^{-4}$ m at the bottom. Young's modulus of elasticity is $2 \times 10^{11} \ N/m^2$. The change in length of the wire is

physics mechanical properties of solids applications of elasticity elastic energy properties of matter

  1. $4 \times 10^{-3}$ m

  2. $3 \times 10^{-3}$ m

  3. $ 10^{-3}$ m

  4. $2 \times 10^{-3}$ m

Show answer
Correct Option: C

A wire of cross section $A$ is stretched horizontally between two clamps located $2lm$ apart. A weight $Wkg$ is suspended from the mid-point of the wire.If the Young's modulus of the material is $Y$, the value of extension $x$ is

physics mechanical properties of solids applications of elasticity elastic energy properties of matter

  1. $ { \left( \cfrac { Wl }{ YA } \right) }^{ 1/3 }$

  2. $ { \left( \cfrac { YA }{ WI } \right) }^{ 1/3 }$

  3. $\cfrac { 1 }{ l } { \left( \cfrac { Wl }{ YA } \right) }^{ 2/3 }$

  4. $ l{ \left( \cfrac { W }{ YA } \right) }^{ 2/3 }$

Show answer
Correct Option: D
Explanation:
Let $M$, $L$ and $T$ represent the dimensions of mass, length and time respectively. Then:
${ [(\dfrac { Wl }{ YA } ) }^{ \dfrac { 1 }{ 3 }  }]=\dfrac { [{ M }^{ \dfrac { 1 }{ 3 }  }{ L }^{ \dfrac { 2 }{ 3 }  }{ T }^{ -\dfrac { 2 }{ 3 }  }] }{ [{ M }^{ \dfrac { 1 }{ 3 }  }{ L }^{ \dfrac { 2 }{ 3 }  }{ T }^{ -\dfrac { 2 }{ 3 }  }] } =[{ M }^{ 0 }{ L }^{ 0 }{ T }^{ 0 }]$
Since extension has dimensions of length, the only option which fits the requirement is the last one where an additional length term is multiplied.

Relation among elastic contents $Y, G, B, \sigma $

physics mechanical properties of solids applications of elasticity elastic energy properties of matter

  1. $\dfrac{9}{Y} = \dfrac{1}{B} + \dfrac{3}{G}$

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

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

  4. $\sigma = \dfrac{3B - 2G}{2(G + 3B)}$

Show answer
Correct Option: A

You are given three wires $  \mathrm{A}, \mathrm{B}  $ and $ \mathrm{C}  $ of the same length and cross section. They are each stretched by applying the same force to the ends. The wire A is stretched least comes back to its original length when the stretching force is removed. The wire $  B  $ is stretched more than $  A  $ and also comes back to its original length when the stretching force is removed. The wire C is stretched most and remains stretched even when the stretching force is removed. The greatest Young's modulus of elasticity is possessed by the material of a wire

physics mechanical properties of solids applications of elasticity elastic energy properties of matter

  1. A

  2. B

  3. C

  4. All have the same elasticity

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Correct Option: A

In designing, a beam for its use to support a load. The depression at center is proportional to (where, $Y$ is Young's modulus).

physics mechanical properties of solids applications of elasticity elastic energy properties of matter

  1. $Y^2$

  2. $Y$

  3. $\dfrac{1}{Y}$

  4. $\dfrac{1}{Y^2}$

Show answer
Correct Option: C