Mechanical Properties of Solids

The strain-stress plot for materials $A, B, C$ and $D$ is shown in the figure. Which material has the largest Young's modulus?

A uniform wire of length $l$ of weight $w$ is suspended from the roof with a weight of $W$ at the other end. The stress in the wire at $\frac{l}{3}$ distance from the top is $\left(\frac{W}{A} + \frac{2}{\gamma}\frac{w}{A}\right)$, where $A$ is the cross sectional area of the wire. The value of $\gamma$ is ________.

With rise in temperature, the Young's modulus of elasticity:

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A): The property of body, by virtue of which it tends to regain its original shape when the external force is removed, is Elasticity.

Reason (R): The restoring force depends upon the bonded inter atomic and inter molecular force of solid.

In the light of the above statements, choose the correct answer from the options given below:

Match List I with List II:

Choose the correct answer from the options given below:

One end of a metal wire is fixed to a ceiling and a load of $2\text{ kg}$ hangs from the other end. A similar wire is attached to the bottom of the load and another load of $1\text{ kg}$ hangs from this lower wire. Then the ratio of longitudinal strain of upper wire to that of the lower wire will be ________. [Area of cross section of wire $= 0.005\text{ cm}^2, Y = 2 \times 10^{11}\text{ N m}^{-2}$ and $g = 10\text{ m s}^{-2}$ ]

Two blocks of mass $2\text{ kg}$ and $4\text{ kg}$ are connected by a metal wire going over a smooth pulley as shown in figure. The radius of wire is $4.0 \times 10^{-5}\text{ m}$ and Young's modulus of the metal is $2.0 \times 10^{11}\text{ N/m}^2$. The longitudinal strain developed in the wire is $\frac{1}{\alpha\pi}$. The value of $\alpha$ is _________. [Use $g = 10\text{ m/s}^2$]

Each of three blocks $P, Q$ and$R$ shown in figure has a mass of $3\text{ kg}$. Each of the wires $A$ and $B$ has cross-sectional area $0.005\text{ cm}^2$ and Young's modulus $2 \times 10^{11}\text{ N m}^{-2}$. Neglecting friction, the longitudinal strain on wire $B$ is ________ $\times 10^{-4}$. (Take $g = 10\text{ m/s}^2$)

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R

Assertion A: Steel is used in the construction of buildings and bridges.

Reason R: Steel is more elastic and its elastic limit is high.

In the light of above statements, choose the most appropriate answer from the options given below

Let a wire be suspended from the ceiling (rigid support) and stretched by a weight $W$ attached at its free end. The longitudinal stress at any point of cross-sectional area $A$ of the wire is:

A metal block of mass $m$ is suspended from a rigid support through a metal wire of diameter $14\text{ mm}$. The tensile stress developed in the wire under equilibrium state is $7 \times 10^{5}\text{ N m}^{-2}$. The value of mass $m$ is _________ $\text{kg}$. (Take, $g = 9.8\text{ m s}^{-2}$ and $\pi = \frac{22}{7}$)

A steel rod has a radius of $20\text{ mm}$ and a length of $2.0\text{ m}$. A force of $62.8\text{ kN}$ stretches it along its length. Young's modulus of steel is $2.0 \times 10^{11}\text{ N/m}^2$. The longitudinal strain produced in the wire is _____________ $\times 10^{-5}$

Given below are two statements: One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A): The stretching of a spring is determined by the shear modulus of the material of the spring.

Reason (R): A coil spring of copper has more tensile strength than a steel spring of same dimensions.

In the light of the above statements, choose the most appropriate answer from the options given below:

A block of mass M is attached to a wire. The upper part of wire is made of steel and lower part of wire of copper ( as shown in figure ). Both parts have the same area of cross section. Neglecting the mass of wires, the ratio of longitudinal strain developed in steel and copper wires is: ( Given, Young's modulus of steel $= 2 \times 10^{11}\text{ N/m}^2$ and Young's modulus of copper $= 1.5 \times 10^{11}\text{ N/m}^2$)

A string of area of cross-section $4\text{ mm}^2$ and length $0.5\text{ m}$ is connected with a rigid body of mass $2\text{ kg}$. The body is rotated in a vertical circular path of radius $0.5\text{ m}$. The body acquires a speed of $5\text{ m/s}$ at the bottom of the circular path. Strain produced in the string when the body is at the bottom of the circle is _________ $\times 10^{-5}$.

(use Young's modulus $10^{11}\text{ N/m}^2$ and $g = 10\text{ m/s}^2$)

Four identical hollow cylindrical columns of mild steel support a big structure of mass 50 $\times$ 103 kg. The inner and outer radii of each column are 50 cm and 100 cm respectively. Assuming uniform local distribution, calculate the compression strain of each column. [Use Y = 2.0 $\times$ 1011 Pa, g = 9.8 m/s2]

Two wires $A$ and $B$ are stretched by the same load. The radius of wire $A$ is double the radius of wire $B$. The stress on the wire $B$ as compared to the stress on the wire $A$ is

Two wires of different materials have same length $L$ and same diameter $d$. The second wire is connected at the end of the first wire and forms one single wire of double the length. This wire is subjected to stretching force $F$ to produce the elongation I. The two wires have

Two identical wires of substances ' $P$ ' and ' $Q$ ' are subjected to equal stretching force along the length. If the elongation of ' $Q$ ' is more than that of ' $P$ ', then

A steel wire having a radius of 2.0 mm, carrying a load of 4 kg, is hanging from a ceiling. Given that g = 3.1$\pi$ ms–2, what will be the tensile stress that would be developed in the wire?

The stress-strain curves are drawn for two different materials X and Y. It is observed that the ultimate strength point and the fracture point are close to each other for material X but are far apart for material Y.

We can say that materials X and Y are likely to be (respectively),

A man grows into a giant such that his linear dimensions increase by a factor of 9. Assuming that his density remains same, the stress in the leg will change by a factor of:

The Young's modulus of steel wire of radius $r$ and length $L$ is $Y$.

If the radius $r$ and length $L$ of the wire are doubled then the value of $Y$

The two wires $A$ and $B$ of equal cross-section but of different materials are joined together. The ratio of Young's modulus of wire $A$ and wire $B$ is 20/11. When the joined wire is kept under certain tension the elongations in the wires $A$ and $B$ are equal. If the length of wire $A$ is 2.2 m, then the length of wire $B$ is

______ m.

Two wires $A$ and $B$ made of different materials of lengths 6.0 cm and 5.4 cm, respectively and area of cross sections $3.0 \times 10^{-5}\text{ m}^2$ and $4.5 \times 10^{-5}\text{ m}^2$, respectively are stretched by the same magnitude under a given load. The ratio of the Young's modulus of $A$ to that of $B$ is $x : 3$. The value of $x$ is ______

A 3 m long wire of radius 3 mm shows an extension of 0.1 mm when loaded vertically by a mass of 50 kg in an experiment to determine Young's modulus. The value of Young's modulus of the wire as per this experiment is $P \times 10^{11}$ Nm${}^{-2}$, where the value of $P$ is: (Take $g = 3\pi$ m/s${}^{2}$)

Two wires A and B are made of same material having ratio of lengths $\frac{L_A}{L_B} = \frac{1}{3}$ and their diameters ratio $\frac{d_A}{d_B} = 2$. If both the wires are stretched using same force, what would be the ratio of their respective elongations?

Young's modulus of material of a wire of length '$L$' and cross-sectional area $A$ is $Y$. If the length of the wire is doubled and cross-sectional area is halved then Young's modulus will be:

A wire of length $L$ and radius $r$ is clamped at one end. If its other end is pulled by a force $F$, its length increases by $l$. If the radius of the wire and the applied force both are reduced to half of their original values keeping original length constant, the increase in length will become:

The maximum elongation of a steel wire of $1\text{ m}$ length if the elastic limit of steel and its Young's modulus, respectively, are $8 \times 10^{8}\text{ N m}^{-2}$ and $2 \times 10^{11}\text{ N m}^{-2}$, is:

Two metallic wires $P$ and $Q$ have same volume and are made up of same material. If their area of cross sections are in the ratio $4 : 1$ and force $F_1$ is applied to $P$, an extension of $\Delta l$ is produced. The force which is required to produce same extension in $Q$ is $F_2$.

The value of $\frac{F_1}{F_2}$ is _________.

For a solid rod, the Young's modulus of elasticity is $3.2 \times 10^{11}\text{ N m}^{-2}$ and density is $8 \times 10^{3}\text{ kg m}^{-3}$. The velocity of longitudinal wave in the rod will be.

A 100 m long wire having cross-sectional area $6.25 \times 10^{-4}\text{ m}^2$ and Young's modulus is $10^{10}\text{ N m}^{-2}$ is subjected to a load of 250 N, then the elongation in the wire will be:

A wire of length ' $L$ ' and radius ' $r$ ' is clamped rigidly at one end. When the other end of the wire is pulled by a force $f$, its length increases by ' $l$ '. Another wire of same material of length ' $2L$ ' and radius ' $2r$ ' is pulled by a force ' $2f$ '. Then the increase in its length will be:

A force is applied to a steel wire 'A', rigidly clamped at one end. As a result elongation in the wire is $0.2\text{ mm}$. If same force is applied to another steel wire ' $B$ ' of double the length and a diameter $2.4$ times that of the wire ' $A$ ', the elongation in the wire ' $B$ ' will be (wires having uniform circular cross sections)

Young's moduli of the material of wires A and B are in the ratio of $1 : 4$, while its area of cross sections are in the ratio of $1 : 3$. If the same amount of load is applied to both the wires, the amount of elongation produced in the wires $A$ and $B$ will be in the ratio of

[Assume length of wires A and B are same]

The Young's modulus of a steel wire of length $6\text{ m}$ and cross-sectional area $3\text{ mm}^2$, is $2 \times 10^{11}\text{ N/m}^2$. The wire is suspended from its support on a given planet. A block of mass $4\text{ kg}$ is attached to the free end of the wire. The acceleration due to gravity on the planet is $\frac{1}{4}$ of its value on the earth. The elongation of wire is (Take $g$ on the earth $= 10\text{ m/s}^2$):

Under the same load, wire A having length $5.0\text{ m}$ and cross section $2.5 \times 10^{-5}\text{ m}^2$ stretches uniformly by the same amount as another wire B of length $6.0\text{ m}$ and a cross section of $3.0 \times 10^{-5}\text{ m}^2$ stretches. The ratio of the Young's modulus of wire A to that of wire $B$ will be:

If the length of a wire is made double and radius is halved of its respective values. Then, the Young's modulus of the material of the wire will:

The force required to stretch a wire of cross-section $1\text{ cm}^2$ to double its length will be: (Given Young's modulus of the wire $= 2 \times 10^{11}\text{ N/m}^2$)

A wire of length $L$ and radius $r$ is clamped rigidly at one end. When the other end of the wire is pulled by a force $F$, its length increases by $5\text{ cm}$. Another wire of the same material of length $4L$ and radius $4r$ is pulled by a force $4F$ under same conditions. The increase in length of this wire is __________________ $\text{cm}$.

The elongation of a wire on the surface of the earth is 10−4 m. The same wire of same dimensions is elongated by 6 $\times$ 10−5 m on another planet. The acceleration due to gravity on the planet will be ____________ ms−2. (Take acceleration due to gravity on the surface of earth = 10 ms−2)

A uniform metallic wire is elongated by 0.04 m when subjected to a linear force F. The elongation, if its length and diameter is doubled and subjected to the same force will be ________ cm.

Two separate wires A and B are stretched by 2 mm and 4 mm respectively, when they are subjected to a force of 2 N. Assume that both the wires are made up of same material and the radius of wire B is 4 times that of the radius of wire A. The length of the wires A and B are in the ratio of a: b. Then a/b can be expressed as 1/x where x is _________.

A wire of length L, area of cross section A is hanging from a fixed support. The length of the wire changed to L1 when mass M is suspended from its free end. The expression for Young's modulus is:

Two rods of same material and volume having circular cross-section are subjected to tension $T$. Within the elastic limit, same force is applied to both the rods. Diameter of the first rod is half of the second rod, then the extensions of first rod to second rod will be in the ratio

A wire of length $L$ and radius $r$ is rigidly fixed at one end. On stretching the other end of the wire with a force $F$, the increase in length is $l$. If another wire of the same material but double the length and radius is stretched with a force $2F$, then increase in length is

Young's moduli of two wires A and B are in the ratio 7: 4. Wire A is 2 m long and has radius R. Wire B is 1.5 m long and has radius 2 mm. If the two wires stretch by the same length for a given load, then the value of R is close to :-

Two wires are made of the same material and have the same volume. The first wire has cross- sectional area A and the second wire has cross-sectional area 3A. If the length of the first wire is increased by $\Delta$l on applying a force F, how much force is needed to stretch the second wire by the same amount?

The Young's modulus of steel is twice that of brass. Two wires of same length and of same area of cross section, one of steel and another of brass are suspended from the same roof. If we want the lower ends of the wires to be at the same level, then the weights added to the steel and brass wires must be in the ratio of

Copper of fixed volume V is drawn into wire of length $l$. When this wire is subjected to a constant force F, the extension produced in the wire is $\Delta l$. Which of the following graphs is a straight line?

The following four wires are made of the same material. Which of these will have the largest extension when the same tension is applied?

Two wires are made of the same material and have the same volume. However wire $1$ has cross- sectional area $A$ and wire $2$ has cross-sectional area $3A.$ If the length of wire $1$ increases by $\Delta x$ on applying force $F,$ how much force is needed to stretch wire $2$ by the same amount?

A metal string $A$ is suspended from a rigid support and its free end is attached to a block of mass $M$. Second block having mass 2 M is suspended at the bottom of the first block using a string $B$. The area of cross sections of strings $A$ and $B$ are same. The ratio of lengths of strings of $A$ to B is 2 and the ratio of their Young's moduli $(Y_A/Y_B)$ is 0.5. The ratio of elongations in $A$ to $B$ is ______.