Mechanical Properties of Solids Test

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Mechanical Properties of Solids Test - 20

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Weekly Quiz Competition

Question 1
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The ratio of lengths of two wires made of same material is $$2 :3$$. The ratio of their respective longitudinal stress to produce same elongation is

A
$$4 : 9$$

B
$$9 : 4$$

C
$$2 : 3$$

D
$$3: 2$$

Solution

$$\dfrac{L_{1}}{L_{2}}=\dfrac{2}{3}$$

$$\Delta L_{1}=\Delta L_{2}$$ (given)

$$y_{1}=y_{2}$$ (given)

$$\dfrac{\sigma _{1}}{\epsilon _{1}}=\dfrac{\sigma _{2}}{\epsilon _{2}}$$ (as inferred from question)

$$\dfrac{\sigma _{1}}{\sigma _{2}}=\dfrac{\epsilon _{1}}{\epsilon _{2}}$$

$$\dfrac{\sigma _{1}}{\sigma _{2}}=\dfrac{\Delta L_{1}}{L_{1}}\times \dfrac{L_{2}}{\Delta L_{2}}$$$$=\dfrac{3}{2}\times 1$$
Question 2
1 / -0

Ratio of lengths of two brass wires is 3 : 4; their areas of cross section are in the ratio 2:3. When same force is applied on them, the elongations produced will be in the ratio:

A
9 : 8

B
8 : 9

C
2$$\sqrt{2}$$ :3

D
1 : 1

Solution

$$\dfrac{L_{1}}{L_{2}}=\dfrac{3}{4} , \dfrac{A_{1}}{A_{2}}=\dfrac{2}{3}$$

$$\dfrac{\Delta L_{1}}{\Delta L_{2}}=\dfrac{3}{4}\times \dfrac{3}{2}$$$$=\dfrac{9}{8}$$
Question 3
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The length of a wire of cross-sectional area $$1 \times $$ 10$$^{-6} m^{2}$$ is 10m. The young's modulus of the material of the wire is 25 G.pa. When the wire is subjected to a tensile force of $$100N$$, the elongation produced in $$mm$$ is:

A
$$0.04$$

B
$$0.4$$

C
$$4$$

D
$$40$$

Solution

$$\Delta L=\dfrac{100\times 10}{10^{-6}\times 25\times 10^{9}}$$$$=0.04 m$$$$=40 mm$$ (using standard result $$Y=\dfrac{(\dfrac{F}{A})}{(\dfrac{\text{change in length}}{\text{actual length}})}$$)
Question 4
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A steel wire of length 1 m has cross sectional area $$1cm$$$$^{2}$$. If young's modulus of steel is $$10^{11}N / m^{2}$$ ,then force required to increase the length of wire by 1 mm will be :

A
$$10^{11}N$$

B
$$10^{7}N$$

C
$$10^{4}N$$

D
$$10^{2}N$$

Solution

Given, $$L=1 m, A=10^{-4} m^{2}$$, $$\Delta l = 1mm$$, $$y= 10 ^{11} N/m^2$$

We know, $$\Delta L=\dfrac{FL}{Ay}$$

$$.001=\dfrac{F\times 1}{10^{-4}\times 10^{11}}$$

$$F=10^{4}N$$
Question 5
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Two steel wires have equal volumes. Their diameters are in the ratio 2 : 1. When same force is applied on them, the elongation produced will be in the ratio of:

A
$$1:8$$

B
$$8:1$$

C
$$1:16$$

D
$$16:1$$

Solution

Volume is equal.

i.e. $$\pi r_{1}^{2}L_{1} = \pi r_{2}^{2}L_{2}$$

given $$\dfrac{r_{1}}{r_{2}}=\dfrac{2}{1}$$

i.e. $$\dfrac{L_{1}}{L_{2}}=\dfrac{1}{4}$$

applying same force.

$$\dfrac{\Delta L_{1}}{\Delta L_{2}}=\dfrac{L_{1}}{L_{2}}\times \dfrac{A_{2}}{A_{1}}$$

$$=\dfrac{1}{4}\times \dfrac{1^{2}}{2^{2}}$$

$$=\dfrac{1}{16}$$
Question 6
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A $$3 cm$$ long copper wire is stretched to increase its length by $$0.3cm.$$ If poisson's ratio for copper is $$0.26$$, the lateral strain in the wire is

A
0.26

B
2.6

C
0.026

D
0.0026

Solution

$$\epsilon x=\dfrac{0.3}{3}$$$$=0.1$$ (standard result)

$$\sigma =0.26$$ (given)

$$0.26=\dfrac{-\epsilon y}{0.1}$$

$$-0.026=\epsilon y$$
Question 7
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The diameters of two steel wires are in the ratio 2: 3. Their lengths are equal. When same force is applied on them, the ratio of the elongation produced is

A
$$4 : 9$$

B
$$9 : 4$$

C
$$3 : 2$$

D
$$2 : 3$$

Solution

$$\dfrac{r_{1}}{r_{2}}=\dfrac{2}{3}$$
$$\dfrac{A_{1}}{A_{2}}=\dfrac{4}{9}$$
$$\dfrac{\Delta L_{1}}{\Delta L_{2}}=\dfrac{A_{2}}{A_{1}}$$$$=9/4$$
Question 8
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A copper wire and a steel wire of radii in the ratio $$1:2$$ lengths in the ratio $$2:1$$ are stretched by the same forces. If young's modulus of copper $$=1.1\times 10^{11}N / m^{2}$$, young's modulus of steel $$= 2\times 10^{11}N /m^{2}$$. Ratio of their extensions is :

A
$$160 : 11$$

B
$$16 : 110$$

C
$$1 : 61$$

D
$$6 : 11$$

Solution

Given, $$\dfrac{r_{c}}{r_{s}}=\dfrac{1}{2} , \quad \dfrac{L_{c}}{L_{s}}=\dfrac{2}{1} , \quad \dfrac{Area_{c}}{Area_{s}}=\dfrac{1^{2}}{2^{2}}=\dfrac{1}{4}$$

($$r_c$$,$$L_c$$ and $$Area_c$$ denote the corresponding radius,length and area of specimen used)

$$y_{c}=1.1\times 10^{11} N/m^{2} , \quad y_{s}=2\times 10^{11} N/m^{2}$$

$$\Delta L=\dfrac{FL}{Ay}$$ (F is same)

$$\therefore \dfrac{\Delta L_{C}}{\Delta L_{S}}=\dfrac{FL_{C}}{A_{C}y_{C}}\times \dfrac{A_{S}y_{S}}{FL_{S}}$$$$=\dfrac{2}{1}\times \dfrac{4}{1}\times \dfrac{2\times 10^{11}}{1.1\times 10^{11}}$$$$=\dfrac{16}{1.1}$$$$=\dfrac{160}{11}$$
Question 9
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The force that must be applied to a steel wire $$6m$$ long and diameter $$1.6mm$$ to produce an extension of 1mm [$$y=2.0 \times 10^{11}N.m^{-2}$$] is approximate.

A
$$100N$$

B
$$50N$$

C
$$67N$$

D
$$33.5N$$

Solution

$$L = 6m , d = 1.6\times 10^{-3} m$$
($$d$$,and $$L$$ denote the corresponding radius,and length of specimen used)
$$\Delta L=1\times 10^{-3} m , y=2\times 10^{11} N/m^{2}$$
$$F=\dfrac{\Delta L Ay}{L}$$
$$=\dfrac{1\times 10^{-3}\times \pi \times (1.6\times 10^{-3})^2\times 2\times 10^{11} }{6\times 4}$$
$$=67 N$$
Question 10
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A volume of $$10^{-3}m^{3}$$ is subjected to a pressure of 10 atmospheres. The change in volume is $$10^{-6}m^{3}$$. Bulk modulus of water is (Atmosphere pressure = 1x10$$^{5} N / m^{2}$$ ) :

A
1x10$$^{9}$$ N / m$$^{2}$$

B
1x10$$^{10}$$ N / m$$^{2}$$

C
1x10$$^{12}$$ N / m$$^{2}$$

D
1x10$$^{7}$$ N / m$$^{2}$$

Solution

Given,
$$\Delta V=-10^{-6}m^3$$
$$V=10^{-3}m^3$$
$$P=10\times 10^5 N/m^2$$
Bulk modulus,
$$B=-\dfrac{F/A}{\Delta V/V}$$
$$B=-\dfrac{P}{\Delta V/V}$$
$$B=-\dfrac{10\times 10^5}{-10^{6}/10^{-3}}=1\times 1-^9$$
$$B=1\times 10^9 N/m^2$$
The correct option is A.

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Re-Attempt Weekly Quiz Competition

Mechanical Properties of Solids Test - 20

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Mechanical Properties of Solids Test - 20

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SHARING IS CARING

Question 1

$$4 : 9$$

$$9 : 4$$

$$2 : 3$$

$$3: 2$$

Solution

Question 2

9 : 8

8 : 9

2$$\sqrt{2}$$ :3

1 : 1

Solution

Question 3

$$0.04$$

$$0.4$$

$$4$$

$$40$$

Solution

Question 4

$$10^{11}N$$

$$10^{7}N$$

$$10^{4}N$$

$$10^{2}N$$

Solution

Question 5

$$1:8$$

$$8:1$$

$$1:16$$

$$16:1$$

Solution

Question 6

0.26

2.6

0.026

0.0026

Solution

Question 7

$$4 : 9$$

$$9 : 4$$

$$3 : 2$$

$$2 : 3$$

Solution

Question 8

$$160 : 11$$

$$16 : 110$$

$$1 : 61$$

$$6 : 11$$

Solution

Question 9

$$100N$$

$$50N$$

$$67N$$

$$33.5N$$

Solution

Question 10

1x10$$^{9}$$ N / m$$^{2}$$

1x10$$^{10}$$ N / m$$^{2}$$

1x10$$^{12}$$ N / m$$^{2}$$

1x10$$^{7}$$ N / m$$^{2}$$

Solution

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