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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?
$\alpha=7, \beta=-1, \gamma=-2$
$\alpha=-7, \beta=-1, \gamma=-2$
$\alpha=7, \beta=-1, \gamma=2$
$\alpha=-7, \beta=1, \gamma=-2$
Solution
$Y=c^\alpha h^\beta G^\gamma$
$\mathrm{ML}^{-1} \mathrm{~T}^{-2}=\left(\mathrm{LT}^{-1}\right)^\alpha\left(\mathrm{ML}^2 \mathrm{~T}^{-1}\right)^\beta\left(\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right)^\gamma$
$1=\beta-\gamma$ $. . . . . (1)$
$-1=\alpha+2 \beta+3 \gamma$ $. . . . .(2)$
$\frac{-2=-\alpha-\beta-2 \gamma}{-3=\beta+\gamma}$ $. . . .(3)$
$\frac{1=\beta-\gamma}{-2=2 \beta} \Rightarrow \beta=-1, \gamma=-2$
$-1=\alpha-2-6 \quad \therefore \alpha=7 $
