A massive black hole of mass m and radius R i | vedclass

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Question

$(d)$ Given, $P=G c^{-5} m^{x} R^{y} \omega^{z} \quad \ldots (i)$

Here, dimensions of various physical quantities are

Angular speed, $\omega=\le

Vedclass · Accepted Answer

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

Vedclass · Answer

$(d)$ Given, $P=G c^{-5} m^{x} R^{y} \omega^{z} \quad \ldots (i)$

Here, dimensions of various physical quantities are

Angular speed, $\omega=\left[ T ^{-1}\right]$

Power, $P=\left[ ML ^{2} T ^{-3}\right]$

Mass, $m=[ M ]$

Radius, $R=[ L ]$

Speed, $c=\left[ LT ^{-1}\right]$

Universal gravitational constant,

$G=\left[ M ^{-1} L ^{3} T ^{-2}\right]$

Substituting dimensions in Eq. $(i)$, we have

${\left[ ML ^{2} T ^{-3}\right]=} {\left[ M ^{-1} L ^{3} T ^{-2}\right]\left[ L ^{-5} T ^{5}\right][ M ]^{x} }$

${\left[ L ^{y}\left[ T ^{-z}\right]\right.}$

Equating dimensions of same quantity, we get

$1=-1+x \Rightarrow x=2$

$2=3-5+y \Rightarrow y=4$

$-3=-2+5-z \Rightarrow z=6$

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