JEE Advanced Hooke’s Law & Young’s Modulus PYQ with Solutions

Q 1 :

One end of a horizontal thick copper wire of length 2L and radius 2R is welded to an end of another horizontal thin copper wire of length L and radius R. When the arrangement is stretched by applying forces at two ends, the ratio of the elongation in the thin wire to that in the thick wire is [2013]

0.25
0.50
2.00
4.00

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(3)

$Using, Y = \frac{F / A}{Δ ℓ / ℓ_{0}}$

$Y = \frac{F / π {(2 R)}^{2}}{Δ ℓ_{1} / 2 L} = \frac{F / π R^{2}}{Δ ℓ_{2} / L}$

$∴$ $\frac{Δ ℓ_{2}}{Δ ℓ_{1}} = 2$

Q 2 :

The adjacent graph shows the extension $(Δ ℓ)$ of a wire of length 1 m suspended from the top of a roof at one end and with a load W connected to the other end. If the cross-sectional area of the wire is $10^{- 6} m^{2}$ , calculate the Young's modulus of the material of the wire. [2003]

[IMAGE 373]

$2 \times 10^{11} N/m$
$2 \times 10^{- 11} N/m$
$3 \times 10^{- 12} N/m$
$2 \times 10^{- 13} N/m$

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(1)

$Using, Y = \frac{F / A}{Δ ℓ / ℓ} = \frac{F}{A} \cdot \frac{ℓ}{Δ ℓ} = \frac{20 \times 1}{10^{- 6} \times 10^{- 4}}$

$= 2 \times 10^{11} {N/m}^{2}$

Q 3 :

A block of weight 100 N is suspended by copper and steel wires of same cross sectional area $0.5 {cm}^{2}$ and, length m and 1 m, respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are $30 °$ and $60 °$ , respectively. If elongation in copper wire is $(Δ l_{c})$ and elongation in steel wire is $(Δ l_{s})$ , then the ratio $\frac{Δ l_{c}}{Δ l_{s}}$ is _______.

[Young's modulus for copper and steel are $1 \times 10^{11} {N/m}^{2}$ and $2 \times 10^{11}$ ${N/m}^{2}$ , respectively.] [2019]

[IMAGE 374]

(2)

[IMAGE 375]

$Given : l_{c} = \sqrt{3} m, l_{s} = 1 m, Y_{c} = 1 \times 10^{11} {N/m}^{2}$ $and Y_{s} = 2 \times 10^{11} {N/m}^{2}$

$At equilibrium, T_{s} \cos 60 ° = T_{c} \cos 30 °$

$\Rightarrow \frac{T_{s}}{2} = \frac{T_{c} \sqrt{3}}{2}$

$\Rightarrow T_{s} = \sqrt{3} T_{c} \Rightarrow \frac{T_{c}}{T_{s}} = \frac{1}{\sqrt{3}}$

$∴$ $\frac{l_{c}}{l_{s}} = \frac{\sqrt{3}}{1}$

$and \frac{Y_{c}}{Y_{s}} = \frac{1 \times 10^{11}}{2 \times 10^{11}} = \frac{1}{2}$

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

$Here, A_{s} = A_{c}$

$∴$ $\frac{Δ l_{c}}{Δ l_{s}} = (\frac{T_{c}}{T_{s}}) \times (\frac{l_{c}}{l_{s}}) \times (\frac{Y_{s}}{Y_{c}}) = (\frac{1}{\sqrt{3}}) \times (\frac{\sqrt{3}}{1}) \times (\frac{2}{1}) = 2$

Q 4 :

In plotting stress versus strain curves for two materials P and Q, a student by mistake puts strain on the y-axis and stress on the x-axis as shown in the figure. Then the correct statement(s) is (are) [2015]

[IMAGE 376]

P has more tensile strength than Q
P is more ductile than Q
P is more brittle than Q
The Young's modulus of P is more than that of Q

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Select one or more options

(1, 2)

$From graph, the maximum stress that P can withstand$ $before breaking is greater than Q .$

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$The strain of P is more than Q therefore P is more ductile.$

$∵$ $Y = \frac{stress}{strain}$ $So for a given strain, stress is more for Q .$

$∴$ $Y_{Q} > Y_{P}$

Topic Question Set

Q 4 :

In plotting stress versus strain curves for two materials P and Q, a student by mistake puts strain on the y-axis and stress on the x-axis as shown in the figure. Then the correct statement(s) is (are) [2015]

[IMAGE 376]

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Topic Question Set

Q 2 : The adjacent graph shows the extension (Δℓ) of a wire of length 1 m suspended from the top of a roof at one end and with a load W connected to the other end. If the cross-sectional area of the wire is 10-6 m2, calculate the Young's modulus of the material of the wire. [2003] [IMAGE 373]

Q 4 : In plotting stress versus strain curves for two materials P and Q, a student by mistake puts strain on the y-axis and stress on the x-axis as shown in the figure. Then the correct statement(s) is (are) [2015] [IMAGE 376]

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Q 4 :

In plotting stress versus strain curves for two materials P and Q, a student by mistake puts strain on the y-axis and stress on the x-axis as shown in the figure. Then the correct statement(s) is (are) [2015]

[IMAGE 376]