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(D) Given the formula for power: $P = G c^{-5} m^{x} R^{y} \omega^{z} \quad \dots (i)$The dimensions of the physical quantities are:Power $P = [M L^{2

Vedclass · Accepted Answer

$x=2, y=4, z=6$

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(D) Given the formula for power: $P = G c^{-5} m^{x} R^{y} \omega^{z} \quad \dots (i)$The dimensions of the physical quantities are:Power $P = [M L^{2} T^{-3}]$Gravitational constant $G = [M^{-1} L^{3} T^{-2}]$Speed of light $c = [L T^{-1}]$Mass $m = [M]$Radius $R = [L]$Angular velocity $\omega = [T^{-1}]$Substituting these dimensions into equation $(i)$:$[M L^{2} T^{-3}] = [M^{-1} L^{3} T^{-2}] [L T^{-1}]^{-5} [M]^{x} [L]^{y} [T^{-1}]^{z}$$[M L^{2} T^{-3}] = [M^{-1} L^{3} T^{-2}] [L^{-5} T^{5}] [M^{x}] [L^{y}] [T^{-z}]$$[M L^{2} T^{-3}] = [M^{x-1}] [L^{3-5+y}] [T^{-2+5-z}]$$[M L^{2} T^{-3}] = [M^{x-1}] [L^{y-2}] [T^{3-z}]$Comparing the powers of $M, L,$ and $T$ on both sides:For $M$: $x - 1 = 1 \Rightarrow x = 2$For $L$: $y - 2 = 2 \Rightarrow y = 4$For $T$: $3 - z = -3 \Rightarrow z = 6$Thus,the values are $x=2, y=4, z=6$.

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