Mechanical Properties of Solids Test

Result

Mechanical Properties of Solids Test - 79

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
1 / -0

A $$2\ m$$ long light metal rod $$AB$$ is suspended from the ceiling horizontally by means of two vertical wires of equal length, tied to its ends. One wire is of brass and has cross-section of $$0.2 \times 10 ^ { - 4 }\ m ^ { 2 }$$ and the other is of steel with $$0.1 \times 10 ^ { 4 }\ m^2$$ cross-section. In order to have equal stresses in the two wires, a weight should be hung from the rod from end $$A$$ a distance of

A
$$66.6\ cm$$

B
$$133\ cm$$

C
$$44.4\ cm$$

D
$$155.6\ cm$$

Solution

$$\begin{aligned}&\text { Since stress is equal }\rightarrow\\&\begin{aligned}\dfrac{T_{1}}{0.2\times10^{-4}}&=\dfrac{T_{2}}{0.1\times10^{-4}}\\\Rightarrow \quad T_{1}&=2T_{2}\longrightarrow(1)\end{aligned}\end{aligned}$$

$$\begin{array}{l}\text { Torque about } c \text { should be } 0 . \sum \tau=0 \\\qquad T_{1} x=T_{2}(2-x)\\\Rightarrow\quad T_{1} x=2 T_{2}-x T_{2} \\\Rightarrow x\left(T_{1}+T_{2}\right)=2 T_{2} \\\Rightarrow x=\dfrac{2 T_{2}}{3 T_{2}}=\dfrac{2}{3} m=66.6 \mathrm{~cm}\end{array}$$
Question 2
1 / -0

A particle of mass $$m$$ is moving with constant speed $$v$$ in a circular path on a smooth horizontal plane by a spring as shown. If the natural length of the spring is $$l _ 0$$ and stiffness of the spring is $$k$$, the elongation of the spring is :

A
$$\dfrac { { k }^{ 2 }{ l }_{ 0 }^{ 2 }+4{ mv }^{ 2 }{ k }-{ kl }_{ 0 } }{ 2k } $$

B
$$\dfrac { \sqrt { { k }^{ 2 }{ l }_{ 0 }^{ 2 }+{ 4mv }^{ 2 }k } -{ kl }_{ 0 } }{ 2k } $$

C
$$\dfrac { \sqrt { { k }^{ 2 }{ l }_{ 0 }^{ 2 }+{ 4mv }^{ 2 }k-{ kl }_{ 0 } } }{ 2k } $$

D
$$\dfrac { \sqrt { { k }^{ 2 }{ l }_{ 0 }^{ 2 }+{ 4mv }^{ 2 }k } +{ kl }_{ 0 } }{ 2k } $$

Solution
Question 3
1 / -0

Two wires of the same material (young's modules Y) and same length L but radii $$R$$ and $$2R$$ respectively are joined end to end and a weight $$W$$ is suspended from the combination as shown in the figure. the elastic potential energy in the system in equilibrium is

A
$$\dfrac { 3{ W }^{ 2 }L }{ 4\pi { R }^{ 2 }Y }$$

B
$$\dfrac { 3{ W }^{ 2 }L }{ 8\pi { R }^{ 2 }Y }$$

C
$$\dfrac { 5{ W }^{ 2 }L }{ 8\pi { R }^{ 2 }Y }$$

D
$$\dfrac { { W }^{ 2 }L }{ \pi { R }^{ 2 }Y } $$

Solution

$$W = {\dfrac{1}{2}}\times{F}\times{e} = {\dfrac{w^2L}{2AY}}$$

$$W = {\dfrac{w^2L}{2(\pi{R^2})Y}} + {\dfrac{w^2L}{2{\pi}(2R)^2Y}}$$

= $${\dfrac{5w^2L}{8{\pi}R^2Y}}$$
Question 4
1 / -0

The intensity at the maximum in a Young's double slit experiment is $${ I }_{ 0 }$$. Distance between two slits is $$d=5\lambda $$, where $$\lambda $$ is the wavelength of light used in the experiment. What will be the intensity in front of one of the slits on the screen at a distance $$D = 10 d$$ ?

A
$${ I }_{ 0 }$$

B
$$\cfrac { { I }_{ 0 } }{ 4 } $$

C
$$\cfrac { 3 }{ 4 } { I }_{ 0 }$$

D
$$\cfrac { { I }_{ 0 } }{ 2 } $$

Solution

$$\begin{array}{l}\text { Here } D=5 \lambda, D=10 d,y=\frac{d}{2} \\\text { We have to find resultant intersily at } y=\frac{d}{2}\end{array}$$

$$\begin{aligned}\text { Path difference, } \Delta x=d\tan\theta=\frac{d y}{D}=\dfrac{d^{2}}{2 D} &=\frac{d^{2}}{2 \times 10 d} \\&=\frac{d}{20} \\&=\dfrac{\lambda}{4}\end{aligned}$$

$$\begin{array}{l}\phi=\dfrac{2 \pi}{\lambda},\Delta x=\dfrac{\pi}{2} \\I_{y}=I_{0} \cos ^{2}(\phi / 2)=I_{0} \cos ^{2}\left(\dfrac{\pi}{4}\right)=\dfrac{I_{0}}{2}\end{array}$$
Question 5
1 / -0

Find the stress in $$CD.$$
Area of $$CD = 2\ m^2$$

A
$$15\ N/m^2$$

B
$$5\ N/m^2$$

C
$$20\ N/m^2$$

D
$$None \ of \ these$$

Solution

$$\begin{array}{c}\text { Stress in } C D=\dfrac{30}{2}=15\mathrm{~N} / \mathrm{m}^{2} \\\text { Ans : } A \text { . }\end{array}$$
Question 6
1 / -0

A uniform steel wire hangs from the ceiling and elongates due to its own weight. The ratio of elongation of the lower half of wire is

A
4:1

B
3:1

C
3:2

D
1:1

Solution
Question 7
1 / -0

A wire is stretched by 0.01 m by a certain force 'F' another wire of same material whose diameter and lengths are double to original wire is stretched b the same force then its elongation will be-

A
0.005 m

B
0.01m

C
0.02 m

D
0.04 m

Solution

$$\begin{array}{l}\text { Given, Force applied = F }\\\text { Let, length }=L \\\text { Area of cross section }=A=\pi r^{2} \\\text { Young's modulus }=Y\end{array}$$

$$\begin{array}{l}\text { Case1) Given elongation= }0.01 \mathrm{~m} \\\text { elongation formula, }e=\frac{F L}{A y} \\\Rightarrow e=\frac{F L}{\pi r^{2} y}=0.01\end{array}$$

$$\begin{array}{l}\text { Case 2)} \text { Length and diameter are doubled. } \\\text {That means radius } P \text { s also doubled. } \\\text { Now elongation } e^{\prime}=\frac{F(2 L)}{\pi(2 r)^{2} y}\end{array}$$

$$\begin{aligned}e^{\prime} &=\frac{2 F L}{4 \pi r^{2} Y}=\frac{e}{2} \\e^{\prime} &=\frac{0.01}{2}\\\Rightarrow&e^{\prime}=0.005\mathrm{~m}\end{aligned}$$
Question 8
1 / -0

A wire of length $$1\, m$$ and its area of cross-section is $$1\, cm^3$$. The Young's modulus of the wire is $$10^{11}\, Nm^{-2}$$. Two forces each equal to $$F$$ are applied on its two ends in the opposite directions. If the change in length be $$1\, mm$$, what is the value of $$F$$?

A
$$0.5 \times 10^4 \, N$$

B
$$10^4 \, N$$

C
$$2 \times 10^4 \, N$$

D
None of these

Solution
Question 9
1 / -0

One end of a uniform wire of length Land of weight Wis attached rigidly to a point in the roof and a weight $$W_1$$, is suspended from its lower end. If $$A$$ is the area of cross section of the wire, the stress in the wire at a height $$\cfrac{6L}{8}$$ from its lower end is

A
$$\cfrac{W_1}{A}$$

B
$$\cfrac{W_1+(W/4)}{A}$$

C
$$\cfrac{W_1+(3W/4)}{A}$$

D
$$\cfrac{W_1+W}{A}$$

Solution
Question 10
1 / -0

A massless and thin string is wrapped several times around a disc kept on a rough horizontal surface. A boy standing at a distance 'd' for the cylinder holds free end of the string pulls the cylinder towards him. If there is no slipping, length of the string passed through the hand of the boy while the cylinder reaches his hands is

A
$$d$$

B
$$2d$$

C
$$3d$$

D
$$4d$$

Solution