Mechanical Properties of Solids Test

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Mechanical Properties of Solids Test - 56

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Weekly Quiz Competition

Question 1
1 / -0

Two copper wires having the length in ratio $$4:1$$ and their radii ratio as $$1:4$$ are stretched by the same force. Then the ratio of the longitudinal strain in the two will be

A
$$1:16$$

B
$$16:1$$

C
$$1:64$$

D
$$64:1$$

Solution

Stress$$ \propto$$ Strain $$\propto \frac{F}{A}$$
Ratio of strain $$=\frac{A_2}{A_1}=\left(\frac{r_2}{r_1}\right)^2=\left(\frac{4}{1}\right)^2=\left(\frac{16}{1}\right)$$
Question 2
1 / -0

A block of mass of $$2kg$$ is attached to one end of a wire of cross sectional area $$1mm \,\,\,2 $$ and is. Find the elongation of the wire when the block is at top of the circle $$(Y=2\times 10^{11}Nm^{-2})$$

A
$$0.2892 mm$$

B
$$0.054 mm$$

C
$$01446 mm$$

D
$$0.0017 mm$$
Question 3
1 / -0

Three wires P,Q and R the same materials and length have radii $$0.12cm,0.2cm$$ and $$0.3cm$$ respectively. Which wire has the highest value of Young's modules of elasticity ?

A
P

B
Q

C
R

D
All have the same value

Solution

We know Young's Modulus of elasticity $$E=\dfrac{stress}{strain}$$------(A)
Also we know stress $$\sigma=\dfrac{F}{A}$$where F and A are the force and area of cross section.
And strain $$\epsilon=\dfrac{{l}_{0}}{\Delta l}$$ where $${l}_{0}$$ and $$\Delta L$$ are the original length and elongation length.
Reducing the above found values of stress and strain in A we get,
$$E=\dfrac{F{l}_{0}}{A\Delta l}$$
Considering force applied in each wires to be same and given length of wires P,Q and R are same.
$${E}_{P}=\dfrac{F{l}_{0}}{0.12\Delta {l}_{P}}$$-------(B)

$${E}_{Q}=\dfrac{F{l}_{0}}{0.2\Delta {l}_{Q}}$$-------(C)

$${E}_{R}=\dfrac{F{l}_{0}}{0.3\Delta {l}_{R}}$$ -------(D)
Now comparing equation B,C and D we can say
$$E\alpha \dfrac{1}{A\Delta l}$$
Thus we can say that wire P will have the highest Young's modulus of elasticity.
Question 4
1 / -0

A wire of density $$9\times{10^3}kg/{m^3}$$ is stretched between two clamps 1 m apart and is stretched to an extension of $$4.9\times{10^{ - 4}}metre.$$ Young's modulus of material is $$9\times{10^{10}}N/{m^2}$$. Then

A
The lowest frequency of standing wave is 35 Hz

B
The frequency of 1st overtone is 70 Hz

C
The frequency of 1st overtone is 105 Hz

D
The stress in the wire is $$4.41\times{10^7}N/{m^2}$$

Solution

For wire if M = mass, r = density, A = Area of cross section V = volume, l = length, Dl = change
in length Then mass per unit length $$ m = \dfrac{M}{l} = \dfrac{Al\rho }{l} = A\rho $$ And Young? s modules of elasticity
$$ y = \dfrac{T/A}{\Delta l/l}Þ T = \dfrac{Y\Delta lA}{l}. $$ Hence lowest frequency of vibration $$ n = \dfrac{1}{2l}\sqrt{\dfrac{T}{m}} $$
$$ \dfrac{1}{2l} \sqrt{\dfrac{y\left(\dfrac{\Delta l}{l}\right)A}{A\rho }}= \dfrac{1}{2l}\sqrt{\dfrac{y\Delta l}{l\rho}}Þ\, n = \dfrac{1}{2\times 1}\sqrt{\dfrac{9\times 10^{10}\times 4.9\times 10^{-4}}{1\times 9\times 10^{3}}} = 35Hz $$
Question 5
1 / -0

A two meter long rod is suspended with the help of two wires of equal length. One wire is of steel and its cross-section is 0.1 $$cm^{2} $$ and another wire is of brass and its cross-section area is 0.2 $$cm^{2} $$. If a load W is suspended from the rod and introduced in both the wires is same then the ratio of tensions in them will be

A
(a) will depend on the position of W

B
(b) $$T_{1}/T_{2}=2$$

C
(c) $$T_{1}/T_{2}=1$$

D
(d) $$T_{1}/T_{2}=0.5$$

Solution

Given,$$A_S=0.1 cm^2$$
$$A_B=0.2 cm^2$$
$$T_1x=T_2.(l-x)$$
Here 'x' is not given
So it will depend on the position of W.
Question 6
1 / -0

A spherical ball contracts in volume by $$0.01$$% when subjected to a normal uniform pressure of $$100$$ atmospheres. The bulk modulus of its material in dyne $${cm}^{-2}$$ is:

A
$$10\times {10}^{11}$$

B
$$100\times {10}^{2}$$

C
$$1\times {10}^{11}$$

D
$$2\times {10}^{11}$$

Solution

$$\dfrac{\Delta V}{V}=\dfrac{0.01}{100}=10^{-4}$$
$$p=100atm$$
$$B=\dfrac{p}{\Delta V/V}=\dfrac{100 \times 10^5}{10^{-4}}=10^{11} N/m^2$$
$$=10^{12} dyne/cm^2$$
Question 7
1 / -0

The length of of a metal wire $$l$$ when the tension in is $$'F'$$ and $$'xl'$$ when the tension is $$'yF'$$. Then the natural length of the wire is

A
$$\dfrac { \left( x-y \right) l }{ x-1 } $$

B
$$\dfrac { \left( y-x \right) l }{ y-1 } $$

C
$$\dfrac { \left( x-y \right) l }{ x+1 } $$

D
$$\dfrac { \left( y-x \right) l }{ y+1 } $$

Solution

Let, Natural length $${{L}_{o}}$$

Stress = Young modulus x strain

$$ \dfrac{F}{A}=Y\times \left( \dfrac{L-{{L}_{o}}}{{{L}_{o}}} \right)\ ......\ (1) $$

$$ \dfrac{yF}{A}=Y\times \left( \dfrac{xL-{{L}_{o}}}{{{L}_{o}}} \right)\ ......\ (2) $$

Divide equation (2) by (1)

$$ y=\dfrac{xL-{{L}_{o}}}{L-{{L}_{o}}} $$

$$ \Rightarrow y\left( L-{{L}_{o}} \right)=\left( xL-{{L}_{o}} \right) $$

$$ \Rightarrow {{L}_{o}}=\dfrac{\left( y-x \right)L}{y-1} $$

Natural length of wire is $$\dfrac{\left( y-x \right)L}{y-1}$$
Question 8
1 / -0

A tungsten wire of length 20 cm is stretched by 0.1 cm. Find the strain on the wire.

A
0.002

B
0.005

C
0.001

D
0.004

Solution

Given,

Length of wire $$=20\,cm$$

Elongation $$=0.1\,cm$$

We have the equation,

$$Strain=\dfrac{Elongation}{Length}=\dfrac{0.1}{20}=0.005$$
Question 9
1 / -0

When temperature of a gas is $$20^oC$$ and pressure is changed from $${ P }_{ 1 }=1.0\times { 10 }^{ 5 } Pa$$ to $${ P }_{ 2 }=1.65\times { 10 }^{ 5 } Pa$$ and the volume is changed by 10%. The bulk modulus is :

A
$$1.55\times { 10 }^{ 5 }Pa$$

B
$$1.15\times { 10 }^{ 5 }Pa$$

C
$$1.4\times { 10 }^{ 5 }Pa$$

D
$$1.01\times { 10 }^{ 5 }Pa$$

Solution

Given,
$$T=20^0C$$
$$P_1=1.01\times 10^5 Pa$$
$$P_2=1.165\times 10^5 Pa$$
$$\Delta P=P_2-P_1=1.165\times 10^5-1.01\times 10^5 =0.155\times 10^5 Pa$$
$$\dfrac{\Delta V}{V}=10$$% $$=0.1$$
Bulk modulus, $$B=\dfrac{\Delta P}{\Delta V/V}$$
$$B=\dfrac{0.65\times 10^5}{0.1}=1.55\times 10^5 Pa$$
The correct option is A.
Question 10
1 / -0

A solid cube is subjected to a pressure of $$\left( 5\times { 10 }^{ 5 }\quad N/{ m }^{ 2 } \right) $$. Each side of the cube is shortened by then volumetric strain and Bulk modulus of the cube are

A
$$0.03,5\times { 10 }^{ 5 }\quad N/{ m }^{ 2 }$$

B
$$0.03,1.67\times { 10 }^{ 7 }\quad N/{ m }^{ 2 }$$

C
$$3,1.67\times { 10 }^{ -7 }\quad N/{ m }^{ 2 }$$

D
$$0.01,1.67\times { 10 }^{ 7 }\quad N/{ m }^{ 2 }$$

Solution

$$P=5\times 10^5N/m^2$$, let initial side=$$a$$
$$V=a^3$$
Shortened side=$$0.99a$$
Shortened volume=$$(0.99)^3a^3$$
$$B=\frac { -V\triangle P }{ \triangle v } \Rightarrow \dfrac { -V\times 5{ \times 10 }^{ 5 } }{ -{ a }^{ 3 }\left( 1-{ (0.99) }^{ 3 } \right) } \\ \Rightarrow \dfrac { 5{ \times 10 }^{ 5 } }{ 0.030 } \Rightarrow 166.66{ \times 10 }^{ 5 }\\ \Rightarrow 1.67{ \times 10 }^{ 7 }N/{ m }^{ 3 }$$
$$\dfrac { \triangle v }{ V } =0.03\rightarrow $$ Volumetric strain.

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Re-Attempt Weekly Quiz Competition

Mechanical Properties of Solids Test - 56

Result

Mechanical Properties of Solids Test - 56

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SHARING IS CARING

Question 1

$$1:16$$

$$16:1$$

$$1:64$$

$$64:1$$

Solution

Question 2

$$0.2892 mm$$

$$0.054 mm$$

$$01446 mm$$

$$0.0017 mm$$

Question 3

P

Q

R

All have the same value

Solution

Question 4

The lowest frequency of standing wave is 35 Hz

The frequency of 1st overtone is 70 Hz

The frequency of 1st overtone is 105 Hz

The stress in the wire is $$4.41\times{10^7}N/{m^2}$$

Solution

Question 5

(a) will depend on the position of W

(b) $$T_{1}/T_{2}=2$$

(c) $$T_{1}/T_{2}=1$$

(d) $$T_{1}/T_{2}=0.5$$

Solution

Question 6

$$10\times {10}^{11}$$

$$100\times {10}^{2}$$

$$1\times {10}^{11}$$

$$2\times {10}^{11}$$

Solution

Question 7

$$\dfrac { \left( x-y \right) l }{ x-1 } $$

$$\dfrac { \left( y-x \right) l }{ y-1 } $$

$$\dfrac { \left( x-y \right) l }{ x+1 } $$

$$\dfrac { \left( y-x \right) l }{ y+1 } $$

Solution

Question 8

0.002

0.005

0.001

0.004

Solution

Question 9

$$1.55\times { 10 }^{ 5 }Pa$$

$$1.15\times { 10 }^{ 5 }Pa$$

$$1.4\times { 10 }^{ 5 }Pa$$

$$1.01\times { 10 }^{ 5 }Pa$$

Solution

Question 10

$$0.03,5\times { 10 }^{ 5 }\quad N/{ m }^{ 2 }$$

$$0.03,1.67\times { 10 }^{ 7 }\quad N/{ m }^{ 2 }$$

$$3,1.67\times { 10 }^{ -7 }\quad N/{ m }^{ 2 }$$

$$0.01,1.67\times { 10 }^{ 7 }\quad N/{ m }^{ 2 }$$

Solution

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