JEE Advanced 2023 Paper 2 Q02 Physics Errors & Measurements Significant Figures Hard

JEE Advanced 2023 Paper 2 · Q02 · Significant Figures

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

  1. A. $\alpha = 7, \beta = -1, \gamma = -2$
  2. B. $\alpha = -7, \beta = -1, \gamma = -2$
  3. C. $\alpha = 7, \beta = -1, \gamma = 2$
  4. D. $\alpha = -7, \beta = 1, \gamma = -2$
Reveal answer + step-by-step solution

Correct answer:A

Solution

Dimensions: $[Y]=ML^{-1}T^{-2}$, $[c]=LT^{-1}$, $[h]=ML^{2}T^{-1}$, $[G]=M^{-1}L^{3}T^{-2}$. Equating powers in $Y=c^{\alpha}h^{\beta}G^{\gamma}$: $$M:\ \beta-\gamma=1,\qquad L:\ \alpha+2\beta+3\gamma=-1,\qquad T:\ -\alpha-\beta-2\gamma=-2.$$ From $M$: $\beta = 1+\gamma$. Substitute into $L$: $\alpha + 2(1+\gamma) + 3\gamma = -1 \Rightarrow \alpha = -3-5\gamma$. Substitute both into $T$: $-(-3-5\gamma) - (1+\gamma) - 2\gamma = -2 \Rightarrow 2+2\gamma = -2 \Rightarrow \gamma=-2$.

Hence $\beta = -1$ and $\alpha = -3 - 5(-2) = 7$. So $\alpha=7,\ \beta=-1,\ \gamma=-2$.

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