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Physics · Units, Dimensions & Error Analysis

JEE Advanced 2023 — Paper 2 — Question 15

Young's modulus of elasticity Y is expressed in terms of three derived quantities, namely, the gravitational constant G, Planck's constant hh and the speed of light cc, as Y=cαhβGγY=c^{\alpha} h^{\beta} \mathrm{G}^{\gamma}. Which of the following is the correct option?

  1. Option A:

    α=7,β=−1,γ=−2\alpha=7, \beta=-1, \gamma=-2

    Correct
  2. Option B:

    α=−7,β=−1,γ=−2\alpha=-7, \beta=-1, \gamma=-2

  3. Option C:

    α=−7,β=−1,γ=−2\alpha=-7, \beta=-1, \gamma=-2

  4. Option D:

    α=−7,β=1,γ=−2\alpha=-7, \beta=1, \gamma=-2

Answer: A

Step-by-step solution

Y=cαhβGγ\mathrm{Y}=\mathrm{c}^{\alpha} \mathrm{h}^{\beta} \mathrm{G}^{\gamma} Y→\mathrm{Y} \rightarrow Young's modulus of elasticity c→\mathrm{c} \rightarrow speed of light h→h \rightarrow plank's constant G →\rightarrow Gravitational constant [ML−1 T−2]=[LT−1]α[ML2 T−1]β[M−1 L3 T−2]γ\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]=\left[\mathrm{LT}^{-1}\right]^{\alpha}\left[\mathrm{ML}^{2} \mathrm{~T}^{-1}\right]^{\beta}\left[\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}\right]^{\gamma}

\beta-\gamma=1 …(i) \end{gathered}$$ $$\begin{gathered} \alpha+2 \beta+3 \gamma=-1 …(ii) \end{gathered}$$ $-\alpha-\beta-2 \gamma=-2$ $$\begin{gathered} \alpha+\beta+2 \gamma=2 …(iii) \end{gathered}$$ From (i), (ii) and (iii) $\alpha=7, \beta=-1, \gamma=-2$

Answer key and solution verified before publishing.

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Exam
JEE Advanced 2023
Paper
Paper 2
Subject
Physics
Chapter
Units, Dimensions & Error Analysis
Topic
Units and Dimensions Analysis