Elasticity | JEE Main previous year questions with Solutions

Q 1 :

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 [2024]

3 times
4 times
$\frac{3}{2}$ times
2 times

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(4)

Young's modulus, $Y = \frac{s t r e s s}{s t r a i n} = \frac{F / A}{∆ l / L} = \frac{F L}{A ∆ l}$

In case I : $Y = \frac{F L}{{πr}^{2} l}$ ...(i)

In case II : $Y = \frac{(F / 2) L}{π {(\frac{r}{2})}^{2} ∆ l} = \frac{2 F L}{π r^{2} ∆ l}$ ...(ii)

From (i) and (ii), $\frac{F L}{π r^{2} l} = \frac{2 F L}{π r^{2} ∆ l} \Rightarrow ∆ l = 2 l$

Q 2 :

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 [2024]

$Y / 4$
$4 Y$
$Y$
$2 Y$

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(3)

Young's modulus depends on the material not on length and cross-sectional area. So young's modulus remains same.

Q 3 :

With rise in temperature, the Young's modulus of elasticity [2024]

changes erratically
decreases
increases
remains unchanged

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3

4

(2)

$Y = \frac{F / A}{∆ l / l} = \frac{F / A}{α ∆ T}$ as $∆ T$ increases, $Y$ decreases

Q 4 :

A wire of cross-sectional area A, modulus of elasticity $2 \times 10^{11}$ ${Nm}^{- 2}$ and length 2 m is stretched between two vertical rigid supports. When a mass of 2 kg is suspended at the middle, it sags lower from its original position, making an angle $θ = \frac{1}{100}$ radian on the points of support. The value of A is _____ $\times 10^{- 4}$ $m^{2}$ (consider $x < < L$ ). [2024]

(1)

$\tan θ = \frac{x}{L} \approx θ, As θ is small$

$2 T \sin θ = 20 \Rightarrow T θ = 10$

$T \cdot \frac{1}{100} = 10 \Rightarrow T = 1000 N$

Change in length, $Δ L = 2 \sqrt{x^{2} + L^{2}} - 2 L$

$\Rightarrow Δ L = 2 L [1 + \frac{x^{2}}{2 L^{2}} - 1] = \frac{x^{2}}{L}$

$Stress = Y strain \Rightarrow \frac{T}{A} = Y \frac{Δ L}{(2 L)}$

$\frac{T}{A} = Y \cdot \frac{x^{2}}{L (2 L)} = \frac{Y}{2} \cdot (\frac{x^{2}}{L^{2}}) = \frac{Y}{2} \cdot θ^{2}$

$\frac{1000}{A} = \frac{Y θ^{2}}{2} = \frac{2 \times 10^{11}}{2} \times \frac{1}{100 \times 100}$

$\Rightarrow A = 1 \times 10^{- 4} m^{2}$

Q 5 :

Two persons pull a wire towards themselves. Each person exerts a force of 200 N on the wire.

Young’s modulus of the material of the wire is $1 \times 10^{11}$ ${Nm}^{- 2}$ . Original length of the wire is 2 m and the area of cross-section is 2 ${cm}^{2}$ . The wire will extend in length by _____ μm. [2024]

(20)

$\frac{F}{A} = Y \frac{Δ ℓ}{ℓ} \Rightarrow Δ ℓ = \frac{F ℓ}{A Y}$

$Δ ℓ = \frac{200 \times 2}{2 \times 10^{- 4} \times 10^{11}} = 2 \times 10^{- 5} = 20 μ m$

Q 6 :

Two metallic wires P and Q have the same volume and are made up of the 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 $∆ λ$ is produced. The force required to produce the same extension in Q is $F_{2}$ . The value of $\frac{F_{1}}{F_{2}}$ is _____. [2024]

(16)

$Y = \frac{Stress}{Strain} = \frac{F / A}{Δ ℓ / ℓ} = \frac{F ℓ}{A Δ ℓ}$

$Δ ℓ = \frac{F ℓ}{A Y}$

$V = A ℓ \Rightarrow ℓ = \frac{V}{A}$

$Δ ℓ = \frac{F V}{A^{2} Y}$

Y and V is same for both the wires

$Δ ℓ \propto \frac{F}{A^{2}}$

$\frac{Δ ℓ_{1}}{Δ ℓ_{2}} = \frac{F_{1}}{A_{1}^{2}} \times \frac{A_{2}^{2}}{F_{2}}$

$Δ ℓ_{1} = Δ ℓ_{2}$

$\frac{F_{1}}{F_{2}} = \frac{A_{1}^{2}}{A_{2}^{2}} = {(\frac{4}{1})}^{2} = 16$

Q 7 :

Each of three blocks P, Q, and R shown in the figure has a mass of 3 kg. Each of the wires A and B has a cross-sectional area of 0.005 ${cm}^{2}$ and Young’s modulus $2 \times 10^{11}$ $N m^{2}$ . Neglecting friction, the longitudinal strain on wire B is _____ $\times 10^{- 4}$ . (Take g = 10 $m / s^{2}$ ). [2024]

(2)

$3 g = (3 + 3 + 3) a \Rightarrow a = \frac{3}{9} g = \frac{1}{3} g$

$Now, 3 g - T = 3 a$

$\Rightarrow T = 3 (g - a) = 3 (g - \frac{g}{3}) = 2 g = 20 N$

$Y = \frac{longitudinal stress}{longitudinal strain}$

$longitudinal strain = \frac{longitudinal stress}{Y}$

$= \frac{T}{A Y} = \frac{20}{5 \times 10^{- 7} \times 2 \times 10^{11}} = 2 \times 10^{- 4}$

Q 8 :

Two blocks of mass 2 kg and 4 kg are connected by a metal wire going over a smooth pulley as shown in the figure. The radius of the wire is $4.0 \times 10^{- 5}$ m and Young’s modulus of the metal is $2.0 \times 10^{11}$ $N / m^{2}$ . The longitudinal strain developed in the wire is $\frac{1}{α π}$ . The value of $α$ is _______ .

[Use g = 10 $m / s^{2}$ ] [2024]

(12)

$T = (\frac{2 m_{1} m_{2}}{m_{1} + m_{2}}) g = \frac{80}{3} N$

$A = π r^{2} = 16 π \times 10^{- 10} m^{2}$

$Strain = \frac{Δ ℓ}{ℓ} = \frac{F}{A Y} = \frac{T}{A Y}$

$= \frac{\frac{80}{3}}{16 π \times 10^{- 10} \times 2 \times 10^{11}} = \frac{1}{12 π}$

$α = 12$

Q 9 :

One end of a metal wire is fixed to a ceiling, and a load of 2 kg hangs from the other end. A similar wire is attached to the bottom of the load, and another load of 1 kg hangs from this lower wire. Then the ratio of longitudinal strain of the upper wire to that of the lower wire will be _____.

[Area of cross-section of wire = 0.005 ${cm}^{2}$ , Y = $2 \times 10^{11}$ ${Nm}^{- 2}$ and g = 10 ${ms}^{- 2}$ ] [2024]

(3)

$\sum f_{net} = 0$

$T_{2} = T_{1} + 2 g = g + 2 g = 3 g$

$T_{1} = g$ ...(i)

${(stress)}_{L} = Y \times {(strain)}_{L}$

${(strain)}_{L} = \frac{stress}{Y} = \frac{F}{Y A}$

${(strain)}_{upper} = \frac{T_{2}}{Y A}$

${(strain)}_{lower} = \frac{T_{1}}{Y A}$

$\frac{{(strain)}_{upper}}{{(strain)}_{lower}} = \frac{T_{2}}{T_{1}} = \frac{3 g}{g} = 3$

Q 10 :

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? [2025]

1 : 6
1 : 12
3 : 4
1 : 3

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(2)

$\frac{L_{A}}{L_{B}} = \frac{1}{3}$ and $\frac{d_{A}}{d_{B}} = 2$

$Strain = \frac{△ L}{L} = \frac{Stress}{Y} = \frac{F}{A Y} \Rightarrow △ L = \frac{F L}{A Y}$

$△ L_{A} = \frac{F_{A} L_{A}}{A_{A} Y_{A}} and △ L_{B} = \frac{F_{B} L_{B}}{A_{B} Y_{B}}$

Given $F_{A} = F_{B}$ and $Y_{A} = Y_{B}$

$\frac{△ L_{A}}{△ L_{B}} = \frac{\frac{F_{A} L_{A}}{A_{A} Y_{A}}}{\frac{F_{B} L_{B}}{A_{B} Y_{B}}} = (\frac{L_{A}}{L_{B}}) (\frac{A_{B}}{A_{A}})$

$\frac{△ L_{A}}{△ L_{B}} = (\frac{L_{A}}{L_{B}}) (\frac{\frac{π}{4} d_{B}^{2}}{\frac{π}{4} d_{A}^{2}}) = (\frac{L_{A}}{L_{B}}) {(\frac{d_{B}}{d_{A}})}^{2}$

$\frac{△ L_{A}}{△ L_{B}} = (\frac{1}{3}) {(\frac{1}{2})}^{2} = \frac{1}{12}$

Q 11 :

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 [2023]

Both A and R are correct but R is NOT the correct explanation of A
Both A and R are correct and R is the correct explanation of A
A is not correct but R is correct
A is correct but R is not correct

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(2)

Steel is more elastic and has high elastic limit.

Q 12 :

For a solid rod, the Young’s modulus of elasticity is $3.2 \times 10^{11} {Nm}^{- 2}$ and density is $8 \times 10^{3} {kg m}^{- 3}$ . The velocity of longitudinal wave in the rod will be [2023]

$145.75 \times 10^{3} {ms}^{- 1}$
$3.65 \times 10^{3} {ms}^{- 1}$
$18.96 \times 10^{3} {ms}^{- 1}$
$6.32 \times 10^{3} {ms}^{- 1}$

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(4)

$v = \sqrt{\frac{Y}{ρ}} = \sqrt{\frac{3.2 \times 10^{11}}{8 \times 10^{3}}} = 2 \times 10^{3} \sqrt{10} = 6.32 \times 10^{3} m/s$

Q 13 :

Under the same load, wire A having length 5.0 m and cross section $2.5 \times 10^{- 5} m^{2}$ stretches uniformly by the same amount as another wire B of length 6.0 m and a cross section of $3.0 \times 10^{- 5} m^{2}$ stretches. The ratio of the Young’s modulus of wire A to that of wire B will be [2023]

1 : 4
1 : 1
1 : 10
1 : 2

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2

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4

(2)

$Δ l = \frac{F l}{S Y}$

$Since F is same for both wires and Δ l is also same$

$\frac{Δ l}{F} = \frac{l}{S Y} \Rightarrow \frac{l_{A}}{S_{A} Y_{A}} = \frac{l_{B}}{S_{B} Y_{B}}$

$\Rightarrow \frac{5}{2.5 \times Y_{A}} = \frac{6}{3 \times Y_{B}}$

$\Rightarrow \frac{Y_{A}}{Y_{B}} = 1$

Q 14 :

The Young’s modulus of a steel wire of length 6 m and cross-sectional area $3 {mm}^{2}$ is $2 \times 10^{11} {N/m}^{2}$ . 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 the wire is (Take g on the Earth $= 10 {m/s}^{2}$ ) [2023]

1 cm
1 mm
0.1 mm
0.1 cm

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4

(3)

$Tension (F) = m g = 4 \times \frac{10}{4} = 10 N$

$Δ Y = \frac{F L}{A Y} = \frac{10 \times 6}{3 \times 10^{- 6} \times 2 \times 10^{11}} = 10^{- 4} m = 0.1 mm$

Q 15 :

Young’s moduli of the material of wires A and B are in the ratio of 1 : 4, while their areas of cross-section 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 wires A and B will be in the ratio of [Assume length of wires A and B are same] [2023]

36 : 1
12 : 1
1 : 36
1 : 12

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2

3

4

(2)

$Δ L = \frac{F L}{A Y}$

$\frac{Δ L_{A}}{Δ L_{B}} = \frac{A_{B}}{A_{A}} \frac{Y_{B}}{Y_{A}} = 12$

Q 16 :

Two wires each of radius 0.2 cm and negligible mass, one made of steel and the other made of brass, are loaded as shown in the figure. The elongation of the steel wire is _______ $\times 10^{- 6} m$ . [Young’s modulus for steel $= 2 \times 10^{11} {Nm}^{- 2}$ and $g = 10 {ms}^{- 2}$ ] [2023]

(20)

$Tension in steel wire T_{2} = 2 g + T_{1}$

$T_{2} = 20 + 11.4 = 31.4 N$

$Elongation in steel wire Δ L = \frac{T_{2} L}{A Y}$

$Δ L = \frac{31.4 \times 1.6}{π {(0.2 \times 10^{- 2})}^{2} \times 2 \times 10^{11}}$

$Δ L = \frac{16}{2 \times 4 \times 10^{- 6} \times 10^{11}} = 20 \times 10^{- 6} m$

Q 17 :

The strain–stress plot for materials A, B, C, and D is shown in the figure. Which material has the largest Young’s modulus? [2026]

B
A
D
C

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(4)

$\frac{Strain}{Stress} = \frac{1}{Y} = Slope$

Q 18 :

Two wires A and B made of different materials of lengths 6.0 cm and 5.4 cm, respectively and area of cross sections $3.0 \times 10^{- 5} m^{2}$ and $4.5 \times 10^{- 5} m^{2}$ 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 ______. [2026]

4
5
1
2

1

2

3

4

(2)

$T = \frac{F / A}{Δ ℓ / ℓ} \Rightarrow Y = \frac{F ℓ}{A Δ ℓ}$

$\frac{Y_{A}}{Y_{B}} = \frac{ℓ_{A}}{ℓ_{B}} (\frac{A_{B}}{A_{A}})$

$= \frac{6}{5.4} (\frac{4.5 \times 10^{- 5}}{3 \times 10^{- 5}}) = \frac{9}{5.4} = \frac{5}{3} \Rightarrow \frac{x}{3} = \frac{5}{3}$

$x = 5$

Topic Question Set

Q 2 :

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 [2024]

Q 3 :

With rise in temperature, the Young's modulus of elasticity [2024]

Q 10 :

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? [2025]

Q 12 :

For a solid rod, the Young’s modulus of elasticity is $3.2 \times 10^{11} {Nm}^{- 2}$ and density is $8 \times 10^{3} {kg m}^{- 3}$ . The velocity of longitudinal wave in the rod will be [2023]

Q 17 :

The strain–stress plot for materials A, B, C, and D is shown in the figure. Which material has the largest Young’s modulus? [2026]

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Topic Question Set

Q 2 : 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 [2024]

Q 3 : With rise in temperature, the Young's modulus of elasticity [2024]

Q 5 : Two persons pull a wire towards themselves. Each person exerts a force of 200 N on the wire. Young’s modulus of the material of the wire is 1×1011 Nm-2. Original length of the wire is 2 m and the area of cross-section is 2 cm2. The wire will extend in length by _____ μm. [2024]

Q 7 : Each of three blocks P, Q, and R shown in the figure has a mass of 3 kg. Each of the wires A and B has a cross-sectional area of 0.005 cm2 and Young’s modulus 2×1011 Nm2. Neglecting friction, the longitudinal strain on wire B is _____ ×10-4. (Take g = 10 m/s2). [2024]

Q 10 : Two wires A and B are made of same material having ratio of lengths LALB=13 and their diameters ratio dAdB=2. If both the wires are stretched using same force, what would be the ratio of their respective elongations? [2025]

Q 12 : For a solid rod, the Young’s modulus of elasticity is 3.2×1011 Nm-2 and density is 8×103 kg m-3. The velocity of longitudinal wave in the rod will be [2023]

Q 13 : Under the same load, wire A having length 5.0 m and cross section 2.5×10-5 m2 stretches uniformly by the same amount as another wire B of length 6.0 m and a cross section of 3.0×10-5 m2 stretches. The ratio of the Young’s modulus of wire A to that of wire B will be [2023]

Q 16 : Two wires each of radius 0.2 cm and negligible mass, one made of steel and the other made of brass, are loaded as shown in the figure. The elongation of the steel wire is _______ ×10-6 m. [Young’s modulus for steel =2×1011 Nm-2 and g=10 ms-2] [2023]

Q 17 : The strain–stress plot for materials A, B, C, and D is shown in the figure. Which material has the largest Young’s modulus? [2026]

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Q 2 :

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 [2024]

Q 3 :

With rise in temperature, the Young's modulus of elasticity [2024]

Q 10 :

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? [2025]

Q 12 :

For a solid rod, the Young’s modulus of elasticity is $3.2 \times 10^{11} {Nm}^{- 2}$ and density is $8 \times 10^{3} {kg m}^{- 3}$ . The velocity of longitudinal wave in the rod will be [2023]

Q 17 :

The strain–stress plot for materials A, B, C, and D is shown in the figure. Which material has the largest Young’s modulus? [2026]