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A massive black hole of mass $m$ and radius $R$ is spinning with angular velocity $\omega .$ The power $\mathrm{P}$ radiated by it as gravitational waves is given by $\mathrm{P}=\mathrm{Gc}^{-5} \mathrm{~m}^{\mathrm{x}} \mathrm{R}^{\mathrm{y}} \omega^{\mathrm{z}}$, where $\mathrm{c}$ and $\mathrm{G}$ are speed of light in free space, and the universal gravitational constant, respectively. Then
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The correct answer is:
$x=2, y=4, z=6$
$\mathrm{P}=\mathrm{ML}^{2} \mathrm{~T}^{-3}, \mathrm{c}=\mathrm{LT}^{-1}, \omega=\mathrm{T}^{-1}, \mathrm{R}=\mathrm{L}, \mathrm{m}=\mathrm{M}$
$\mathrm{G}=\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}$
$\left[\mathrm{ML}^{2} \mathrm{~T}^{-3}\right]=\left[\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}\right]\left[\mathrm{LT}^{-1}\right]^{-5} \mathrm{M}^{\mathrm{x}} \mathrm{L}^{\mathrm{y}} \mathrm{T}^{-\mathrm{z}}$
solve we get $\mathrm{x}=2, \mathrm{y}=4, \mathrm{z}=6$
$\mathrm{G}=\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}$
$\left[\mathrm{ML}^{2} \mathrm{~T}^{-3}\right]=\left[\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}\right]\left[\mathrm{LT}^{-1}\right]^{-5} \mathrm{M}^{\mathrm{x}} \mathrm{L}^{\mathrm{y}} \mathrm{T}^{-\mathrm{z}}$
solve we get $\mathrm{x}=2, \mathrm{y}=4, \mathrm{z}=6$
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