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← JEE Advanced 2023 · Paper 2
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JEE Advanced 2023 · Paper 2, Physics question 2

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

Choose one option
Answer and worked solution

Official answer: A

Dimensions.

  • [Y]=ML−1T−2[Y]=ML^{-1}T^{-2};
  • [c]=LT−1[c]=LT^{-1}, [h]=ML2T−1[h]=ML^2T^{-1}, [G]=M−1L3T−2[G]=M^{-1}L^3T^{-2}.

Matching powers.

  • M: β−γ=1\beta-\gamma=1;
  • L: α+2β+3γ=−1\alpha+2\beta+3\gamma=-1;
  • T: −α−β−2γ=−2-\alpha-\beta-2\gamma=-2.

Solve. Adding the L and T equations gives β+γ=−3\beta+\gamma=-3. With β−γ=1\beta-\gamma=1: β=−1\beta=-1, γ=−2\gamma=-2, and then α=7\alpha=7.

Y=c7h−1G−2Y=c^7h^{-1}G^{-2}.

Topics:Dimensional analysis
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