Mechanical Properties of Solids Test

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Mechanical Properties of Solids Test - 30

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Question 1
1 / -0

Two identical wires of different materials have Young's moduli of elasticity as $$22\times10^{10}N/m^2$$ and $$11\times10^{10}N/m^2$$ respectively. If these are stretched by equal loads then the ratio of extensions produced in them will be:

A
2 : 1

B
1 : 2

C
4 : 1

D
1 : 4

Solution

We know that $$\Delta l=\dfrac{FL}{AY}$$. Thus, for same load and same length and area (because similar wires), extension is inversely proportional to Young's modulus.
Thus, for Y in ratio $$2:1$$, extension will be in ratio $$1:2$$
Question 2
1 / -0

A spherical ball is compressed by $$0.01$$% when a pressure of $$100 $$ atmosphere is applied on it. Its bulk modulus of elasticity in $$dyne/cm^{2}$$ will be approximately

A
$$10^{12}$$

B
$$10^{14}$$

C
$$10^{6}$$

D
$$10^{24}$$

Solution

The pressure is 100atm i.e. $$100\times 10^5 Pa=10^7 N/m^2$$
Now as $$1 N/m^2=10 dyne/cm^2$$
we get
pressure as $$10^8 dyne/cm^2$$
Now as $$\Delta V/V$$ is given as $$\dfrac{0.01}{100}=10^{-4}$$
Thus we get Bulk modulus as $$\dfrac{10^8}{10^{-4}}=10^{12}$$
Question 3
1 / -0

Two identical wires are suspended from a roof, but one is of copper and other is of iron. Young's modulus of iron is thrice that of copper. The weights to be added on copper and iron wires so that the ends are on the same level must be in the ratio of

A
1 : 3

B
2 : 1

C
3 : 1

D
4 : 1

Solution

We know that $$\Delta l=\dfrac{FL}{AY}$$. Thus, for same length and same length, extension is inversely proportional to Y and directly proportional to force, or $$\dfrac{F}{Y}$$
Thus, for $$Y$$ in ratio $$3:1$$ for iron and copper respectively, extension will be in ratio $$1:1$$ only if Forces are in ratio $$1:3.$$
Question 4
1 / -0

For a material $$\sigma=-0.25$$ under an external stress, the longitudinal strain is $$10^{-2}$$. The percentage change in the diameter of the wire is

A
$$+1$$%

B
$$-1$$%

C
$$+0.25$$%

D
$$-0.25$$%

Solution

$$=\dfrac { \Delta d }{ d } \times 100$$
$$=-\sigma \dfrac { \Delta l }{ l } \times 100\quad $$
$$=-(-0.25)\times 1{ 0 }^{ -2 }\times 100$$
$$= +0.25$$%
(Negative sign is there because increase in length causes decrease in diameter)
Question 5
1 / -0

A wire of density $$9\times10\space kg\space m^{-3}$$ stretched between two clamps $$1\space m$$ apart is subjected to an extension of $$4.9\times10^{-4}\space m$$. If Young's modulus of the wire is $$9\times10^{-4}\space m$$. The lowest frequency of transverse vibrations in the wire is

A
$$35\space Hz$$

B
$$70\space Hz$$

C
$$105\space Hz$$

D
$$140\space Hz$$

Solution

Fundamental frequency $$\nu=\dfrac{1}{2l}\sqrt{\dfrac{T}{m}}$$.
Now, Young's modulus $$Y=\dfrac{T/A}{\Delta l/l}=\dfrac{Tl}{A\Delta l}\Rightarrow T= \dfrac{YA\Delta l}{l}$$.
So, $$\nu=\dfrac{1}{2l}\sqrt{\dfrac{\dfrac{YA\Delta l}{l}}{A\rho}}=\dfrac{1}{2l}\sqrt{\dfrac{YA\Delta l}{lA\rho}}=\dfrac{1}{2}\sqrt{\dfrac{9\times10^{10}\times 4.9\times 10^{-4}}{9\times10^3}}=35$$ Hz.
Ans: A
Question 6
1 / -0

A steel rod of length l, area of cross section A, Young's modulus E and linear coefficient of expansion a is heated through $$t^{\circ}C$$. The work that can be performed by the rod when heated is

A
$$(EAat)\times(lat)$$

B
$$\displaystyle \frac{1}{2}(EAat)\times(lat)$$

C
$$\displaystyle \frac{1}{2}(EAat)\times\frac{1}{2}(lat)$$

D
$$2(EAat)(lat)$$

Solution

Work done = Energy stored = $$1/2\times Stress\times Strain \times Volume=1/2\times (Eat)\times (at) \times Al$$, which is given is option B.
Question 7
1 / -0

A wire made of the material of Young's modulus $$Y$$ has an stress $$S$$ applied to it. If Poisson's ratio of the wire is $$\sigma$$, the lateral strain is:

A
$$\displaystyle \sigma\frac{S}{Y}$$

B
$$\displaystyle \sigma\frac{Y}{S}$$

C
$$\displaystyle \sigma Y\times S$$

D
$$\displaystyle \frac{S}{\sigma Y}$$

Solution

Longitudinal strain = $$S/Y$$ ;
Poisson's ratio is given as $$\dfrac{Lateral \ strain}{Longitudinal\ Strain}$$
$$Lateral \ Strain = Poisson's\ ratio \times Longitudinal\ Strain=\sigma \dfrac{S}{Y}$$
Question 8
1 / -0

Which of the following pairs is not correct?

A
strain-dimensionless

B
stress-$$N/m^{2}$$

C
modulus of elasticity-$$N/m^{2}$$

D
poisson's ratio-$$N/m^{2}$$

Solution

stress is $$\dfrac{F}{A}$$ hence unit $$N/m^2$$
strain is $$\dfrac{\Delta l}{L}$$ so unit $$m/m$$ therefore dimensionless
modulus of elasticity is $$ \dfrac{stress}{strain}$$ hence same unit as stress as the denominator is dimensionless
poisson's ratio $$\dfrac{-\epsilon_t}{\epsilon_l} $$ so its also going to be dimensionless
Question 9
1 / -0

If a wire is stretched by applying force at one of its ends, then the elastic potential energy density in terms of Young's modulus Y and linear strain $$\alpha$$ will be

A
$$\displaystyle \frac{\alpha Y^2}{2}$$

B
$$\displaystyle \frac{Y\alpha}{2}$$

C
$$\displaystyle \frac{\alpha^2Y}{2}$$

D
$$2\alpha^2Y$$

Solution

$$Potential\quad Energy\quad Density=\dfrac { 1 }{ 2 } \times Stress\times Strain=\dfrac { 1 }{ 2 } \times \dfrac { xY }{ l } \times \dfrac { x }{ l } \quad (\because Stress=Y\times Strain)\\ or,\quad \dfrac { { x }^{ 2 }Y }{ 2{ L }^{ 2 } } \\ Since\quad { (\dfrac { x }{ L } ) }^{ 2 }=Strai{ n }^{ 2 }\quad =\quad { \alpha }^{ 2 }\\ Thus\quad giving,\quad \dfrac { { \alpha }^{ 2 }Y }{ 2 } $$
Question 10
1 / -0

A steel girder can bear a load of $$20 tons$$. If the thickness of girder is double, then for the same depression it can bear a load of :

A
40 ton

B
80 ton

C
160 ton

D
5 ton

Solution

Let the load on the girder be $$W_{ 1 }$$ and the thickness be $$t_{1}$$.
Now, when the thickness is doubled, i.e. $$t_{2}=2t_{1}$$, the load on the girder is $$W_{2}$$.
Depression $$\delta =\dfrac { W{ l }^{ 3 } }{ 4b{ t }^{ 3 }Y } $$
Since depression is same in both cases,
$$\delta_{1}=\delta_{2}$$
$$\dfrac { { W }_{ 1 } }{ { t }_{ 1 }^{ 3 } } =\dfrac { { W }_{ 2 } }{ { t }_{ 2 }^{ 3 } } $$
Hence,
$$W_{2}= 160 N$$

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Re-Attempt Weekly Quiz Competition

Mechanical Properties of Solids Test - 30

Result

Mechanical Properties of Solids Test - 30

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SHARING IS CARING

Question 1

2 : 1

1 : 2

4 : 1

1 : 4

Solution

Question 2

$$10^{12}$$

$$10^{14}$$

$$10^{6}$$

$$10^{24}$$

Solution

Question 3

1 : 3

2 : 1

3 : 1

4 : 1

Solution

Question 4

$$+1$$%

$$-1$$%

$$+0.25$$%

$$-0.25$$%

Solution

Question 5

$$35\space Hz$$

$$70\space Hz$$

$$105\space Hz$$

$$140\space Hz$$

Solution

Question 6

$$(EAat)\times(lat)$$

$$\displaystyle \frac{1}{2}(EAat)\times(lat)$$

$$\displaystyle \frac{1}{2}(EAat)\times\frac{1}{2}(lat)$$

$$2(EAat)(lat)$$

Solution

Question 7

$$\displaystyle \sigma\frac{S}{Y}$$

$$\displaystyle \sigma\frac{Y}{S}$$

$$\displaystyle \sigma Y\times S$$

$$\displaystyle \frac{S}{\sigma Y}$$

Solution

Question 8

strain-dimensionless

stress-$$N/m^{2}$$

modulus of elasticity-$$N/m^{2}$$

poisson's ratio-$$N/m^{2}$$

Solution

Question 9

$$\displaystyle \frac{\alpha Y^2}{2}$$

$$\displaystyle \frac{Y\alpha}{2}$$

$$\displaystyle \frac{\alpha^2Y}{2}$$

$$2\alpha^2Y$$

Solution

Question 10

40 ton

80 ton

160 ton

5 ton

Solution

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