If the velocity of light c,universal gravitat | vedclass

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(A) Let the dimensional formula of mass be $M = h^x c^y G^z$.The dimensional formulas are:$h = [ML^2T^{-1}]$$c = [LT^{-1}]$$G = [M^{-1}L^3T^{-2}]$Subs

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$[h^{1/2} c^{1/2} G^{-1/2}]$

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(A) Let the dimensional formula of mass be $M = h^x c^y G^z$.The dimensional formulas are:$h = [ML^2T^{-1}]$$c = [LT^{-1}]$$G = [M^{-1}L^3T^{-2}]$Substituting these into the equation:$[M^1 L^0 T^0] = [ML^2T^{-1}]^x [LT^{-1}]^y [M^{-1}L^3T^{-2}]^z$$[M^1 L^0 T^0] = M^{x-z} L^{2x+y+3z} T^{-x-y-2z}$Comparing the powers on both sides:$1$) $x - z = 1$$2$) $2x + y + 3z = 0$$3$) $-x - y - 2z = 0$Adding equations $(2)$ and $(3)$:$(2x + y + 3z) + (-x - y - 2z) = 0 + 0$$x + z = 0 \implies x = -z$Substitute $x = -z$ into equation $(1)$:$-z - z = 1 \implies -2z = 1 \implies z = -1/2$Since $x = -z$,then $x = 1/2$.Substitute $x = 1/2$ and $z = -1/2$ into equation $(3)$:$-1/2 - y - 2(-1/2) = 0$$-1/2 - y + 1 = 0 \implies y = 1/2$Thus,the dimensions of mass are $[h^{1/2} c^{1/2} G^{-1/2}]$.

If the velocity of light $c$,universal gravitational constant $G$,and Planck's constant $h$ are chosen as fundamental quantities,the dimensions of mass in the new system are:

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