Elasticity | JEE Main previous year questions with Solutions

Q 1 :
If average depth of an ocean is 4000m and the bulk modulus of water is $2 \times 10^{9}$ $N m^{- 2}$ , then fractional compression $\frac{∆ V}{V}$ of water at the bottom of ocean is $α \times 10^{- 2}$ . The value of $α$ is ______ . $(G i v e n, g = 10 m s^{- 2}, ρ = 1000 k g$ $m^{- 3})$ [2024]

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(2) $B = \frac{Δ P}{(\frac{Δ V}{V})}$

$- (\frac{Δ V}{V}) = \frac{ρ g h}{B} = \frac{1000 \times 10 \times 4000}{2 \times 10^{9}}$

$= 2 \times 10^{- 2} [- ve sign represents compression]$

Q 2 :
The depth below the surface of sea to which a rubber ball be taken so as to decrease its volume by 0.02% is ______ m.

$(T a k e d e n s i t y o f s e a w a t e r = 10^{3} k g m^{- 3}, B u l k m o d u l u s o f r u b b e r = 9 \times 10^{8} N m^{- 2}, a n d g = 10 m s^{- 2})$   [2024]

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(18) $β = \frac{- Δ P}{\frac{Δ V}{V}} \Rightarrow Δ P = - β \frac{Δ V}{V}$

$ρ g h = - β \frac{Δ V}{V}$

$10^{3} \times 10 \times h = - 9 \times 10^{8} \times (- \frac{0.02}{100}) \Rightarrow h = 18 m$

Q 3 :
Match List-I with List-II

List - I List - II

A A force that restores an elastic body of unit area to its original state (I) Bulk modulus

B Two equal and opposite forces parallel to opposite faces (II) Young’s modulus

C Forces perpendicular everywhere to the surface per unit area same everywhere (III) Stress

D Two equal and opposite forces perpendicular to opposite faces (IV) Shear modulus

Choose the correct answer from the options given below: [2024]

(A)-(III), (B)-(IV), (C)-(I), (D)-(II)
(A)-(II), (B)-(IV), (C)-(I), (D)-(III)
(A)-(IV), (B)-(II), (C)-(III), (D)-(I)
(A)-(III), (B)-(I), (C)-(II), (D)-(IV)

1

2

3

4

(1)

Q 4 :
The fractional compression $(\frac{△ V}{V})$ of water at the depth of 2.5 km below the sea level is _____%. Given, the Bulk modulus of water = $2 \times 10^{9} {Nm}^{- 2}$ , density of water $= 10^{3} kg m^{- 3}$ , acceleration due to gravity g = 10 ${ms}^{- 2}$ . [2025]

1.75
1.0
1.5
1.25

1

2

3

4

(4)

$B = \frac{△ P}{- (\frac{△ V}{V})}$

$\Rightarrow - (\frac{△ V}{V}) = \frac{△ P}{B} = \frac{ρ g h}{B}$

$= \frac{10^{3} \times 10 \times 2.5 \times 10^{3}}{2 \times 10^{9}} = 1.25 %$

Q 5 :
A cylindrical rod of length 1 m and radius 4 cm is mounted vertically. It is subjected to a shear force of $10^{5} N$ at the top. Considering infinitesimally small displacement in the upper edge, the angular displacement $θ$ of the rod axis from its original position would be: (shear moduli, $G = 10^{10} N / m^{2}$ ) [2025]

1/160 $π$
1/4 $π$
1/40 $π$
1/2 $π$

1

2

3

4

(1)

$σ_{shear} = \frac{F}{A} = η θ$

$Shear moduli = \frac{σ_{shear}}{θ}$

$10^{10} = \frac{10^{5}}{π \times 16 \times 10^{- 4}} \times \frac{1}{θ}$

$θ = \frac{1}{160 π} Radian$

Q 6 :
The increase in pressure required to decrease the volume of a water sample by 0.2% is $P \times 10^{5} {Nm}^{- 2}$ . Bulk modulus of water is $2.15 \times 10^{9} {Nm}^{- 2}$ . The value of P is __________. [2025]

0%

(43)

Since bulk modulus is given as

$B = \frac{- △ P}{(\frac{△ V}{V})}$

$2.15 \times 10^{9} = \frac{- △ P}{- (\frac{0.2}{100})}$

$△ P = 2.15 \times 10^{9} \times 2 \times 10^{- 3} = 4.3 \times 10^{6} = 43 \times 10^{5} N / m^{2}$

Q 7 :
The volume contraction of a solid copper cube of edge length 10 cm, when subjected to a hydraulic pressure of $7 \times 10^{6} Pa$ , would be _______ ${mm}^{3}$ .

(Given bulk modulus of copper = $1.4 \times 10^{11}$ ${Nm}^{- 2}$ ) [2025]

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(50)

$B = - \frac{△ P}{\frac{△ V}{V}} = - \frac{V △ P}{△ V}$

$\Rightarrow △ V = - V \frac{△ P}{B} = {- (100 mm)}^{3} \times \frac{7 \times 10^{6} Pa}{1.4 \times 10^{11} N / m^{2}} = - 50 {mm}^{3}$

$\Rightarrow | △ V | = 50 {mm}^{3}$

Q 8 :
Two slabs with square cross section of different materials (1, 2) with equal sides ( $l$ ) and thickness $d_{1}$ and $d_{2}$ such that $d_{2} = 2 d_{1}$ and $l > d_{2}$ . Considering lower edges of these slabs are fixed to the floor, we apply equal shearing force on the narrow faces. The angle of deformation is $θ_{2} = 2 θ_{1}$ . If the shear moduli of material 1 is $4 \times 10^{9} N / m^{2}$ , then shear moduli of material 2 is $x \times 10^{9} N / m^{2}$ , where value of x is __________. [2025]

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(1)

Deformation angle

$2 θ_{1} = θ_{2} \Rightarrow 2 \frac{σ_{1}}{η_{1}} = \frac{σ_{2}}{η_{2}}$

$\Rightarrow 2 (\frac{F}{A_{1} η_{1}}) = \frac{F}{A_{2} η_{2}}$

$\Rightarrow 2 (\frac{F}{l d_{1} η_{1}}) = \frac{F}{l d_{2} η_{2}}$

$\Rightarrow η_{2} = \frac{η_{1}}{4} = 1 \times 10^{9} \Rightarrow x = 1$

Q 9 :
A sample of a liquid is kept at 1 atm. It is compressed to 5 atm which leads to change of volume of 0.8 ${cm}^{3}$ . If the bulk modulus of the liquid is 2 GPa, the initial volume of the liquid was ______ litre. (Take 1 atm = $10^{5}$ Pa) [2025]

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(4)

Given, Initial pressure of liquid ( $P_{i}$ ) = 1 atm Final pressure of liquid ( $P_{f}$ ) = 5 atm.

Change in pressure (dP) = $P_{f} - P_{i} =$ 4 atm = $4 \times 10^{5} Pa$

Change in volume (dV) = – 0.8 ${cm}^{3}$

Bulk modulus (B) = $2 \times 10^{9} Pa$

Now, $B = \frac{- d P}{(d V / V)} \Rightarrow V = - B (\frac{d V}{d P})$

$\Rightarrow V = - 2 \times 10^{9} \times \frac{(- 0.8 \times 10^{- 6})}{4 \times 10^{5}}$

$= 4 \times 10^{- 3} m^{3} = 4 litre$

Topic Question Set

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Q 2 :
The depth below the surface of sea to which a rubber ball be taken so as to decrease its volume by 0.02% is ______ m.

$(T a k e d e n s i t y o f s e a w a t e r = 10^{3} k g m^{- 3}, B u l k m o d u l u s o f r u b b e r = 9 \times 10^{8} N m^{- 2}, a n d g = 10 m s^{- 2})$   [2024]

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Q 4 :
The fractional compression $(\frac{△ V}{V})$ of water at the depth of 2.5 km below the sea level is _____%. Given, the Bulk modulus of water = $2 \times 10^{9} {Nm}^{- 2}$ , density of water $= 10^{3} kg m^{- 3}$ , acceleration due to gravity g = 10 ${ms}^{- 2}$ . [2025]

Q 6 :
The increase in pressure required to decrease the volume of a water sample by 0.2% is $P \times 10^{5} {Nm}^{- 2}$ . Bulk modulus of water is $2.15 \times 10^{9} {Nm}^{- 2}$ . The value of P is __________. [2025]

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Q 7 :
The volume contraction of a solid copper cube of edge length 10 cm, when subjected to a hydraulic pressure of $7 \times 10^{6} Pa$ , would be _______ ${mm}^{3}$ .

(Given bulk modulus of copper = $1.4 \times 10^{11}$ ${Nm}^{- 2}$ ) [2025]

0%

0%

Q 9 :
A sample of a liquid is kept at 1 atm. It is compressed to 5 atm which leads to change of volume of 0.8 ${cm}^{3}$ . If the bulk modulus of the liquid is 2 GPa, the initial volume of the liquid was ______ litre. (Take 1 atm = $10^{5}$ Pa) [2025]

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	List - I		List - II
A	A force that restores an elastic body of unit area to its original state	(I)	Bulk modulus
B	Two equal and opposite forces parallel to opposite faces	(II)	Young’s modulus
C	Forces perpendicular everywhere to the surface per unit area same everywhere	(III)	Stress
D	Two equal and opposite forces perpendicular to opposite faces	(IV)	Shear modulus

Topic Question Set

Q 1 : If average depth of an ocean is 4000m and the bulk modulus of water is 2×109 Nm-2, then fractional compression ∆VV of water at the bottom of ocean is α×10-2. The value of α is ______ .(Given, g=10ms-2,ρ=1000kg m-3) [2024]

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Q 2 : The depth below the surface of sea to which a rubber ball be taken so as to decrease its volume by 0.02% is ______ m. (Take density of sea water=103kgm-3,Bulk modulus of rubber=9×108Nm-2,and g=10ms-2) [2024]

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Q 4 : The fractional compression (△VV) of water at the depth of 2.5 km below the sea level is _____%. Given, the Bulk modulus of water = 2×109 Nm–2, density of water =103 kg m–3, acceleration due to gravity g = 10 ms–2. [2025]

Q 6 : The increase in pressure required to decrease the volume of a water sample by 0.2% is P×105 Nm–2. Bulk modulus of water is 2.15×109 Nm–2. The value of P is __________. [2025]

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Q 7 : The volume contraction of a solid copper cube of edge length 10 cm, when subjected to a hydraulic pressure of 7×106Pa, would be _______ mm3. (Given bulk modulus of copper = 1.4×1011 Nm-2) [2025]

0%

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Q 9 : A sample of a liquid is kept at 1 atm. It is compressed to 5 atm which leads to change of volume of 0.8 cm3. If the bulk modulus of the liquid is 2 GPa, the initial volume of the liquid was ______ litre. (Take 1 atm = 105Pa) [2025]

0%

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Q 2 :
The depth below the surface of sea to which a rubber ball be taken so as to decrease its volume by 0.02% is ______ m.

$(T a k e d e n s i t y o f s e a w a t e r = 10^{3} k g m^{- 3}, B u l k m o d u l u s o f r u b b e r = 9 \times 10^{8} N m^{- 2}, a n d g = 10 m s^{- 2})$ [2024]

Q 4 :
The fractional compression $(\frac{△ V}{V})$ of water at the depth of 2.5 km below the sea level is _____%. Given, the Bulk modulus of water = $2 \times 10^{9} {Nm}^{- 2}$ , density of water $= 10^{3} kg m^{- 3}$ , acceleration due to gravity g = 10 ${ms}^{- 2}$ . [2025]

Q 6 :
The increase in pressure required to decrease the volume of a water sample by 0.2% is $P \times 10^{5} {Nm}^{- 2}$ . Bulk modulus of water is $2.15 \times 10^{9} {Nm}^{- 2}$ . The value of P is __________. [2025]

Q 7 :
The volume contraction of a solid copper cube of edge length 10 cm, when subjected to a hydraulic pressure of $7 \times 10^{6} Pa$ , would be _______ ${mm}^{3}$ .

(Given bulk modulus of copper = $1.4 \times 10^{11}$ ${Nm}^{- 2}$ ) [2025]

Q 9 :
A sample of a liquid is kept at 1 atm. It is compressed to 5 atm which leads to change of volume of 0.8 ${cm}^{3}$ . If the bulk modulus of the liquid is 2 GPa, the initial volume of the liquid was ______ litre. (Take 1 atm = $10^{5}$ Pa) [2025]