If the gravitational constant (G),Planck's co | vedclass

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Question

(A) Let the radius of gyration $[k] = [L]$ be expressed as $[k] = {h^x}{c^y}{G^z}$.The dimensions of the constants are:$[h] = [M{L^2}{T^{ - 1}}]$$[c]

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${h^{1/2}}{c^{ - 3/2}}{G^{1/2}}$

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(A) Let the radius of gyration $[k] = [L]$ be expressed as $[k] = {h^x}{c^y}{G^z}$.The dimensions of the constants are:$[h] = [M{L^2}{T^{ - 1}}]$$[c] = [L{T^{ - 1}}]$$[G] = [{M^{ - 1}}{L^3}{T^{ - 2}}]$Substituting these into the equation:$[L] = {[M{L^2}{T^{ - 1}}]^x} {[L{T^{ - 1}}]^y} [{M^{ - 1}}{L^3}{T^{ - 2}}]^z$$[L] = {M^{x - z}} {L^{2x + y + 3z}} {T^{ - x - y - 2z}}$Comparing the powers on both sides:For $M$: $x - z = 0 \implies x = z$For $T$: $-x - y - 2z = 0 \implies -x - y - 2x = 0 \implies y = -3x$For $L$: $2x + y + 3z = 1$Substituting $y = -3x$ and $z = x$ into the $L$ equation:$2x - 3x + 3x = 1$$2x = 1 \implies x = 1/2$Thus,$x = 1/2$,$z = 1/2$,and $y = -3(1/2) = -3/2$.Therefore,the dimension of the radius of gyration is ${h^{1/2}}{c^{ - 3/2}}{G^{1/2}}$.

If the gravitational constant $(G)$,Planck's constant $(h)$,and the velocity of light $(c)$ are chosen as fundamental units,the dimension of the radius of gyration is:

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