Mechanical Properties of Solids

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 \(\mathrm{P}\times {10}^{11}{\mathrm{Nm}}^{-2}\), where the value of P is: ( Take \(\mathrm{g}=3\pi \mathrm{m}/{\mathrm{s}}^{2}\))

[JEE Main 2025, 8 Apr (Shift 1)]

Young's modulus is determined by the equation given by \(Y =49000 \frac{ M }{ l } \frac{\text { dyne }}{ cm ^2}\) where \(M\) is the mass and \(l\) is the extension of wire used in the experiment. Now error in Young modulus \((Y)\) is estimated by taking data from \(M-l\) plot in graph paper. The smallest scale divisions are \(5 g\) and \(0.02 cm\) along load axis and extension axis respectively. If the value of \(M\) and \(l\) are \(500 g\) and \(2 cm\) respectively then percentage error of \(Y\) is :

[JEE Main 2024, 08 Apr (Shift 1)]

Young's modulus is determined by the equation given by \(Y =49000 \frac{ M }{ l } \frac{\text { dyne }}{ cm ^2}\) where \(M\) is the mass and \(l\) is the extension of wire used in the experiment. Now error in Young modulus \((Y)\) is estimated by taking data from \(M-l\) plot in graph paper. The smallest scale divisions are \(5 g\) and \(0.02 cm\) along load axis and extension axis respectively. If the value of \(M\) and \(l\) are \(500 g\) and \(2 cm\) respectively then percentage error of \(Y\) is :

[JEE Main 2024, 08 Apr (Shift 1)]

A string A of length 0.314 m and Young's modulus \(2\times {10}^{10}N/{m}^{2}\) is connected to another string B of length and Young's modulus both twice of those of A. this series combination of string is then suspended from a rigid support and its free end is fixed to a load of mass 0.8 kg . The net change in length of the combination is _____ mm (radius of both the strings is 0.2 mm and acceleration due to gravity \(=10m/{s}^{2})\)
(Mass of both strings is to be neglected as compared to the mass of load)

[04 April, 2026 (Shift-I)]

Two wires A and B are made of same material having ratio of lengths \(\frac{{\mathrm{L}}_{\mathrm{A}}}{{\mathrm{L}}_{\mathrm{B}}}=\frac{1}{3}\) and their diameters ratio \(\frac{{\mathrm{d}}_{\mathrm{A}}}{{\mathrm{d}}_{\mathrm{B}}}=2\). If both the wires are stretched using same force, what would be the ratio of their respective elongations?

[JEE Main 2025, 7 Apr (Shift 1)]

A ball descends to the bottom of a sea with a depth of 2.5 km, experiencing a decrease of x% in its volume due to the surrounding pressure. Given that the bulk modulus of the ball's material is \(2\times 1{0}^{9}\), determine the value of \(x\).

The increase in the pressure required to decrease the volume \((∆V)\) of water is \(6.3\times {10}^{7}N/{m}^{2}\). The percentage decrease in the volume is _____. (Bulk modulus of water = \(2.1\times {10}^{9}N/{m}^{2}).\)

[04 April, 2026 (Shift-I)]

A cube of side 10 cm having bulk modulus of \(1.4 \times 10^{11} \mathrm{~Pa}\) is placed in atmosphere. Now it is subjected to extra pressure of \(7 \times 10^6 \mathrm{~Pa}\) then magnitude of change in volume of cube is

(Shift - II Memory Based)

A cube of side 10 cm having bulk modulus of \(1.4 \times 10^{11} \mathrm{~Pa}\) is placed in atmosphere. Now it is subjected to extra pressure of \(7 \times 10^6 \mathrm{~Pa}\) then magnitude of change in volume of cube is

(Shift - II Memory Based)

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 \(\mathrm{P}\times {10}^{11}{\mathrm{Nm}}^{-2}\), where the value of P is:

(Take \(\mathrm{g}=3\pi \mathrm{m}/{\mathrm{s}}^{2}\))

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.

[JEE Main 2026, 6 Apr (Shift 1)]

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

[JEE Main 2024, 01 Feb (Shift 1)]

A ball descends to the bottom of a sea with a depth of 2.5 km, experiencing a decrease of x% in its volume due to the surrounding pressure. Given that the bulk modulus of the ball's material is \(2\times 1{0}^{9}\), determine the value of \(x\).

The fractional compression \(\left(\frac{\Delta V}{V}\right)\) of water at the depth of 2.5 km below the sea level is ______ %.

Given, the Bulk modulus of water \(=2\times {10}^{9}N{m}^{-2}\), density of water \(={10}^{3}\) \(kg{m}^{-3}\), acceleration due to gravity \(=g=10m{s}^{-2}\).

[JEE Main 2025, 29 Jan (Shift 1)]

Two wires A and B are made of same material having ratio of lengths \(\frac{{\mathrm{L}}_{\mathrm{A}}}{{\mathrm{L}}_{\mathrm{B}}}=\frac{1}{3}\) and their diameters ratio \(\frac{{\mathrm{d}}_{\mathrm{A}}}{{\mathrm{d}}_{\mathrm{B}}}=2\). If both the wires are stretched using same force, what would be the ratio of their respective elongations?

When a force of 5N is applied extension in the spring is found to be \({x}_{1}\), extension becomes \({x}_{2}\) when the force applied is 7N. Calculate the tension in spring when extension is \(5{x}_{1}-2{x}_{2}\)

(Shift - II Memory Based)

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:

[JEE Main 2026, 5 Apr (Shift 1)]

The fractional compression \(\left(\frac{\Delta V}{V}\right)\) of water at the depth of 2.5 km below the sea level is ______ %.

Given, the Bulk modulus of water \(=2\times {10}^{9}N{m}^{-2}\), density of water \(={10}^{3}\) \(kg{m}^{-3}\), acceleration due to gravity \(=g=10m{s}^{-2}\).

[JEE Main 2025, 29 Jan (Shift 1)]

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 :

[JEE Main 2024, 27 Jan (Shift 2)]

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 :

[JEE Main 2024, 27 Jan (Shift 2)]

A material has a bulk modulus of 25 × 10¹¹ N/m². If it undergoes a volumetric strain of 0.2%, what is the excess pressure applied ?

(Shift -II Memory based)

A material has a bulk modulus of 25 × 10¹¹ N/m². If it undergoes a volumetric strain of 0.2%, what is the excess pressure applied ?

(Shift -II Memory based)

When a force of 5N is applied extension in the spring is found to be \({x}_{1}\), extension becomes \({x}_{2}\) when the force applied is 7N. Calculate the tension in spring when extension is \(5{x}_{1}-2{x}_{2}\)

(Shift - II Memory Based)

Under isothermal condition, the pressure of a gas is given by \(P=a V^{-3}\), where \(a\) is a constant and \(V\) is the volume of the gas. The bulk modulus at constant temperature is equal to

[JEE Main 2023, 13 Apr (Shift 1)]

Choose the correct relationship between Poisson ratio \((\sigma)\). bulk modulus \((K)\) and modulus of rigidity \((\eta)\) of a given solid object:

[JEE Main 2021, 25 July (Shift 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.

[JEE Main 2023, 24 Jan (Shift 2)]

The Young's modulus of a steel wire of length 6 m and cross-sectional area 3 mm², is 2 × 11¹¹ N/m². The wire is suspended from its support on a given planet. A block of mass 4 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 m/s²)

[JEE Main 2023, 1 Feb (Shift 2)]

Choose the correct relationship between Poisson ratio \((\sigma)\). bulk modulus \((K)\) and modulus of rigidity \((\eta)\) of a given solid object:

[JEE Main 2023, 30 Jan (Shift 1)]

At \( 40^{\circ} \mathrm{C} \), a brass wire of \( 1 \mathrm{~mm} \) radius is hung from the ceiling. A small mass \( M \) is hung from the free end of the wire. When the wire is cooled down from \( 40^{\circ} \mathrm{C} \) to \( 20^{\circ} \mathrm{C} \), it regains its original length of \( 0.2 \mathrm{~m} \). The value of \( M \) is close to [Coefficient of linear expansion and Youngs modulus of brass are \( 10^{-5} /{ }^{\circ} \mathrm{C} \) and \( 10^{11} \mathrm{~N} / \mathrm{m}^{2} \) respectively, \( g=10 \mathrm{~ms}^{-2} \) ]

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

[JEE Main 2021, 20 July (Shift 2)]

Under isothermal condition, the pressure of a gas is given by \(P=a V^{-3}\), where \(a\) is a constant and \(V\) is the volume of the gas. The bulk modulus at constant temperature is equal to:

[JEE Main 2021, 31 August (Shift 2)]

The Young's modulus of a steel wire of length \(6 \mathrm{~m}\) and cross-sectional area \(3 \mathrm{~mm}^{2}\), is \(2 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}\). The wire is suspended from its support on a given planet. A block of mass \(4 \mathrm{~kg}\) 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 \(\left.=10 \mathrm{~m} / \mathrm{s}^{2}\right)\)

[JEE Main 2024, 31 Jan (Shift 2)]

The restoring force per unit area is known as:

[JEE Main 2021, 27 August (Shift 1)]

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]

[JEE Main 2021, 27 July (Shift 1)]

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

[JEE Main 2022, 26 July (Shift 2)]