Young's modulus of elasticity Y is expressed | vedclass

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(A) The dimensional formula for Young's modulus $Y$ is $[M^1 L^{-1} T^{-2}]$.The dimensional formulas for the given constants are:Speed of light $c =

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$\alpha = 7, \beta = -1, \gamma = -2$

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(A) The dimensional formula for Young's modulus $Y$ is $[M^1 L^{-1} T^{-2}]$.The dimensional formulas for the given constants are:Speed of light $c = [L T^{-1}]$Planck's constant $h = [M L^2 T^{-1}]$Gravitational constant $G = [M^{-1} L^3 T^{-2}]$Given the relation $Y = c^\alpha h^\beta G^\gamma$,we equate the dimensions:$[M^1 L^{-1} T^{-2}] = [L T^{-1}]^\alpha [M L^2 T^{-1}]^\beta [M^{-1} L^3 T^{-2}]^\gamma$$[M^1 L^{-1} T^{-2}] = [M^{\beta - \gamma} L^{\alpha + 2\beta + 3\gamma} T^{-\alpha - \beta - 2\gamma}]$Comparing the powers of $M, L,$ and $T$:$1$) $\beta - \gamma = 1$$2$) $\alpha + 2\beta + 3\gamma = -1$$3$) $-\alpha - \beta - 2\gamma = -2$Adding equations $(2)$ and $(3)$:$(\alpha + 2\beta + 3\gamma) + (-\alpha - \beta - 2\gamma) = -1 + (-2)$$\beta + \gamma = -3$Now,solving $\beta - \gamma = 1$ and $\beta + \gamma = -3$:Adding them gives $2\beta = -2 \Rightarrow \beta = -1$.Substituting $\beta = -1$ into $\beta - \gamma = 1$ gives $-1 - \gamma = 1 \Rightarrow \gamma = -2$.Substituting $\beta = -1$ and $\gamma = -2$ into equation $(2)$:$\alpha + 2(-1) + 3(-2) = -1$$\alpha - 2 - 6 = -1$$\alpha - 8 = -1 \Rightarrow \alpha = 7$.Thus,the correct values are $\alpha = 7, \beta = -1, \gamma = -2$.

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?

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