Mechanical Properties of Solids Test

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Mechanical Properties of Solids Test - 36

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Weekly Quiz Competition

Question 1
1 / -0

Which of the following affects the elasticity of a substance?

A
Hammering and annealing

B
Change in temperature

C
Impurity in substance

D
All of the above

Solution

Elasticity depends on temperature.
Also, elasticity changes due to impurities because they increase binding of crystal grains., hence enhancing elasticity.
Hammering also changes the structure of material as crystal grains break up into smaller units which plays a large role in determining elasticity (increases elasticity). Also, annealing tends to form a uniform orientation of crystal grains, hence producing larger crystal and changing elasticity.
Question 2
1 / -0

A wire is stretched by a force $$F$$. If $$s$$ is the strain developed and $$Y$$ is Young's modulus of material of wire, then work done per unit volume is

A
$$\displaystyle\frac{Ys^2}{2}$$

B
$$\displaystyle\frac{s^2}{2Y}$$

C
$$\displaystyle\frac{1}{2}Fs$$

D
$$\displaystyle\frac{Y}{2s^2}$$

Solution

Work done per unit volume is same as energy per unit volume which is given by:
$$\dfrac{1}{2}\times Stress \times Strain=1/2\times (Y\times Strain)\times Strain=\dfrac{1}{2}\times Y\times {(Strain)}^{2}=\dfrac{Y{s}^{2}}{2}$$
Question 3
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Two wires of same material and length but cross-sections in the ratio $$1:2$$ are used to suspend the same loads. The extensions in them will be in the ratio

A
$$1:2$$

B
$$2:1$$

C
$$4:1$$

D
$$1:4$$

Solution

Let the cross section of wire be A , 2A
$$\Delta L_1=\dfrac{Fl}{AY}$$
$$\Delta L_2=\dfrac{Fl}{2AY}$$
so, $$\dfrac{\Delta L_1}{\Delta _2}=2/1$$
Question 4
1 / -0

A spherical ball contracts in volume by $$0.02\%$$ when subjected to a pressure of $$100$$ atmosphere. Assuming one atmosphere $$=10^8\ Nm^{-2}$$, the bulk modulus of the material of the ball is

A
$$0.02\times 10^5\ N/m^2$$

B
$$0.02\times 10^7\ N/m^2$$

C
$$50\times 10^7\ N/m^2$$

D
$$50\times 10^9\ N/m^2$$

Solution

$$B=-V\dfrac{dP}{dV}$$
Given $$dP=100atm, \dfrac{dV}{V}=-0.0002$$
Thus, $$B=-\dfrac{100}{-0.0002}=5\times{10}^{5}$$ in atm units.
$$1\ atm = {10}^{5}Pa$$, thus in SI units, $$B=5\times {10}^{10}=50\times{10}^{9}\ N/{m}^{2}$$
Question 5
1 / -0

The bulk modulus of a perfectly rigid body is

A
infinity

B
zero

C
some finite value

D
non-zero constant

Solution

$$B=-V\dfrac{dP}{dV}$$
Perfectly rigid body means there can be no change in volume, or $$dV$$ tends to zero. Thus, B tends to infinity.
Question 6
1 / -0

The amount of work done in increasing the length of a wire through $$1cm$$ will be

A
$$\displaystyle\frac{Y A}{2L}$$

B
$$\displaystyle\frac{Y L}{2A}$$

C
$$\displaystyle\frac{Y L^2}{2A}$$

D
None of these

Solution

Work done = change in elastic potential energy=$$1/2\times Stress\times Strain\times Volume=1/2\times Y\times (Strain)^2\times AL$$
=$$1/2\times Y\times (1cm/L)^2\times AL=\dfrac{AY}{2L}$$
Question 7
1 / -0

On stretching a wire, the elastic energy stored per unit volume is

A
$$Fl/2AL$$

B
$$FA/2L$$

C
$$FL/2A$$

D
$$FL/2$$

Solution

Energy stored per unit volume is given as $$\dfrac{1}{2}Stress\times Strain$$=$$\dfrac { 1 }{ 2 } \left[ \dfrac { F }{ A } \right] \left[ \dfrac { l }{ L } \right] =\dfrac { Fl }{ 2AL } $$
Question 8
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A $$5$$ metre long wire is fixed to the ceiling. A weight of $$10kg$$ is hung at the lower end and is $$1$$metre above the floor. The wire was elongated by $$1mm$$. The energy stored in the wire due to stretched is

A
zero

B
$$0.05$$joule

C
$$100$$ joule

D
$$500$$ joule

Solution

$$W=\displaystyle\frac{1}{2}\times F\times l=\frac{1}{2}mgl$$
$$=\displaystyle\frac{1}{2}\times 10\times 10\times 1\times 10^{-3}=0.05J$$
Question 9
1 / -0

A steel ring of radius $$r$$ and cross sectional area $$A$$ is fitted onto a wooden disc of radius $$R(R>r)$$. If the Young's modulus of steel is $$Y$$, then the force with which the steel ring is expanded is

A
$$AY(R/r)$$

B
$$AY(R-r)/r$$

C
$$(Y/A)[(R-r)/r]$$

D
$$Yr/AR$$

Solution

Let $$T$$ be the tension in the ring, then
$$Y=\displaystyle\frac{T.2\pi r}{A.2\pi(R-r)}=\frac{Tr}{A(R-r)}$$
$$\therefore T=\displaystyle\frac{YA(R-r)}{r}$$
Question 10
1 / -0

A metallic rod of length $$l$$ and cross-sectional area $$A$$ is made of a material of Young modules $$Y$$. If the rod is elongated by an amount $$y$$, then the work done is proportional to

A
$$y$$

B
$$\dfrac{1}{y}$$

C
$$y^2$$

D
$$\dfrac{1}{y^2}$$

Solution

$$ Volume =A\times L $$ or $$V=Al$$
$$strain=\displaystyle \frac{Elongation}{\text{Original length}}=\frac{Y}{l}$$
Young's modulus $$=\displaystyle \frac{stress}{strain}$$
Work done, $$W=\displaystyle \frac{1}{2}\times stress \times strain \times volume$$
$$W=\displaystyle \frac{1}{2}\times Y\times (strain)^2\times Al$$
$$=\displaystyle \frac{1}{2}\times Y\left[\frac{y}{l}\right]^2\times Al=\displaystyle\frac{1}{2}\left[\frac{YA}{l}\right]y^2\Rightarrow W\propto y^2$$

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Re-Attempt Weekly Quiz Competition

Mechanical Properties of Solids Test - 36

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Mechanical Properties of Solids Test - 36

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Question 1

Hammering and annealing

Change in temperature

Impurity in substance

All of the above

Solution

Question 2

$$\displaystyle\frac{Ys^2}{2}$$

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

$$\displaystyle\frac{1}{2}Fs$$

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

Solution

Question 3

$$1:2$$

$$2:1$$

$$4:1$$

$$1:4$$

Solution

Question 4

$$0.02\times 10^5\ N/m^2$$

$$0.02\times 10^7\ N/m^2$$

$$50\times 10^7\ N/m^2$$

$$50\times 10^9\ N/m^2$$

Solution

Question 5

infinity

zero

some finite value

non-zero constant

Solution

Question 6

$$\displaystyle\frac{Y A}{2L}$$

$$\displaystyle\frac{Y L}{2A}$$

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

None of these

Solution

Question 7

$$Fl/2AL$$

$$FA/2L$$

$$FL/2A$$

$$FL/2$$

Solution

Question 8

zero

$$0.05$$joule

$$100$$ joule

$$500$$ joule

Solution

Question 9

$$AY(R/r)$$

$$AY(R-r)/r$$

$$(Y/A)[(R-r)/r]$$

$$Yr/AR$$

Solution

Question 10

$$y$$

$$\dfrac{1}{y}$$

$$y^2$$

$$\dfrac{1}{y^2}$$

Solution

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