Tag: elastic energy

Questions Related to elastic energy

A uniform wire of length L and radius r is twisted by a angle $ \angle \alpha$. If modulus of rigidity of the wire is $ \eta  $, then the elastic potential energy stored in wire, is

elastic energy properties of material substances elasticity properties of matter physics

  1. $ \frac{\pi \eta r^{4}\alpha }{2L^{2}} $

  2. $ \frac{\pi \eta r^{4}\alpha^{2} }{4L} $

  3. $ \frac{\pi \eta r^{4}\alpha }{4L^{2}} $

  4. $ \frac{\pi \eta r^{4}\alpha^{2} }{2L} $

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Correct Option: B

The length of an elastic string is $x$ metre when the tension is $8\ N$. Its length is $y$ metre when the tension is $10\ N$. What will be its length, when the tension is $18\ N$?

elastic energy properties of material substances elasticity properties of matter physics

  1. $2x + y$

  2. $5y - 4x$

  3. $7y - 5x$

  4. $7y + 5x$

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Correct Option: B
Explanation:

Let, original length of the spring is L metre and, Y = $\dfrac { F.L }{ A.l } $

Now, when F = 8N, and l = (x - l)m then, $Y=\dfrac { 8.L }{ A.\left( x-L \right)  } \quad \longrightarrow (I)$
and when F=10N, and l = (y - l)m then, $y=\dfrac { 10.L }{ A.\left( y-L \right)  } m\quad \longrightarrow (II)$
From equation (I) and (II) we get,
$\dfrac { 8L }{ A\left( x-L \right)  } =\dfrac { 10L }{ A\left( y-L \right)  } $
or,  $8\left( y-L \right) =10\left( x-L \right) $
or,    $4y-4L=5x-5L$
or,                $L=5x-4y$
When, F=18N,
Let, length of the wire will be Z metre.
$\therefore \quad Y=\dfrac { 18.L }{ A.\left( Z-L \right)  } \quad \longrightarrow (III)$
From equation (I) and (III) we get,
$\dfrac { 8L }{ A\left( x-L \right)  } =\dfrac { 18L }{ A\left( Z-L \right)  } $
or,  $9\left( x-L \right) =4\left( Z-L \right) $
or,  $4Z=9x-9L+4L$
            $=9x-5L$
            $=9x-25x+20y$    [putting value of L]
or,  $Z=5y-4x$


Work done by restoring force in a string within elastic limit is $-10\ J$. The maximum amount of heat produced in the string is :

elastic energy properties of material substances elasticity properties of matter physics

  1. $10\ J$

  2. $20\ J$

  3. $5\ J$

  4. $15\ J$

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Correct Option: A

If work done in stretching a wire by 1 mm is 2J. Then the work necessary for stretching another wire of same material but with double the radius and half the length by 1 mm in joule is

elastic energy properties of material substances elasticity properties of matter physics

  1. 1/4

  2. 4

  3. 8

  4. 16

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Correct Option: D
Explanation:

The stretching force $F=\dfrac{YA\Delta l}{l}$

where $Y=$Young's modulus, $A=$Area of cross-section of wire, $l=$actual length of wire, $\Delta l=$increase in length.
$F=\dfrac{Y\pi r^{2}\Delta l}{l}$
As the material is same $Y$ does not change.
$\dfrac{F _1}{F _2}=\dfrac{\dfrac{r _1^{2}\Delta l _1}{l _1}}{\dfrac{r _2^{2}\Delta l _2}{l _2}}$
Here $\Delta l _1=1mm$
$\Delta l _2=1mm$
$l _2=\dfrac{1}{2}l _1$
$r _2=2r _1$
$\dfrac{F _1}{F _2}=\dfrac{\dfrac{r _1^{2}\times 1mm}{l _1}}{\dfrac{4r _1^{2}\times 1mm}{\dfrac{1}{2}l _1}}$
$\dfrac{F _1}{F _2}=\dfrac{1}{8}$
The work done in stretching wire by amount $\Delta l$ is $W=\dfrac{1}{2} F\Delta l$
Hence $\dfrac{W _1}{W _2}=\dfrac{F _1}{F _2}=\dfrac{1}{8}$
As $F _1=2$
$F _2=2\times 8=16$
Hence the correct option is (D).

When a body mass $M$ is attached to power end of a wire (of length $L$) whose upper end is fixed, then the elongation of the wire is $l$. In this situation mark out the correct statement(s).

elastic energy properties of material substances elasticity properties of matter physics

  1. Loss in gravitational potential energy of $M$ is $Mgl$.

  2. Elastic potential energy stored in the wire is $\dfrac {Mgl}{2}$

  3. Elastic potential energy stored in the wire is $Mgl$

  4. Elastic potential energy stored in the wire is $\dfrac {Mgl}{3}$

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Correct Option: A,B
Explanation:

Since it moves $l$ distance against gravity, so gravitational potential energy=Mgl
Elastic potential energy=$1/2\times Stress\times Strain\times Volume=1/2\times \dfrac{Mg}{A} \times \dfrac{l}{L}\times AL=\dfrac{Mgl}{2}$

State whether true or false :
The hollow shaft is much stronger than a solid shaft of same mass, same length and same material.

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

  1. True

  2. False

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Correct Option: A
Explanation:

Modulus of elasticity = $\dfrac { stress }{ strain } $     and, stress = $\dfrac { Force }{ Area } $

$\therefore $  Elasticity $\alpha \dfrac { 1 }{ Area } $
As follow shaft, so, elasticity of the follow shaft is more than a solid one. As elasticity measures rigidity. So, hollow shaft is stronger.

In which year Robert Hooke presented his law of elasticity?

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

  1. 1672

  2. 1674

  3. 1676

  4. 1678

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Correct Option: C
Explanation:

In 1676, Robert Hooke presented his law of elasticity, now called Hooke's law.

 A cable that can support a load of 1000 N is cut into equal parts. the maximum load that can be supported by the either part is:-
physics mechanical properties of solids applications of elasticity elastic energy properties of matter

  1. 1000 N

  2. 2000 N

  3. 500 N

  4. 250 N

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Correct Option: A
Explanation:

Maximum load supported by the cable is directly proportional to the breaking stress.

Since,

Breaking stress = F / A

where F is the force and A is the cross-sectional area.

As we see that the breaking stress is independent of the length of the cable. So, if the cable is cut in two equal parts, the maximum load that can be supported by either parts of the cable remain the same as before.

State whether true or false :
The metal used in construction of a bridge should have high Young's modulus.

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

  1. True

  2. False

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Correct Option: A
Explanation:

Young's modulus measures modulus of elasticity of a material. 

Elasticity defines rigidity of the material. 
So, for construction, we need metals which have high elasticity i.e. high young's modulus.

A silver wire of length $10 $ metre and cross-sectional area $10^{-8} m^{2}$ is suspended vertically and a weight of $10 N$ is attached to it. Young's modulus of silver and its resistivity are $7 \times 10^{10} N/m^{2}$ and $1.59 \times 10^{8} N/m^{2}$ \Omega - m$ respectively. The increase in its resistance is equal to:-

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

  1. $0.0455 \Omega$

  2. $0.455 \Omega$

  3. $0.91 \Omega$

  4. $0.091 \Omega$

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Correct Option: C