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If $E, M, J$ and $G$ respectively denote energy, mass, angular momentum and universal gravitational constant, the quantity, which has the same dimensions as the dimensions of $\frac{E J^2}{M^5 G^2}$
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The correct answer is:
angle
Given quantity is $\frac{E J^2}{M^5 G^2}$
where dimensions of the various given quantities are
Dimensions of $E=\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]$
Dimensions of $J=\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]$
Dimension of $M=[\mathrm{M}]$
Dimension of $G=\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]$
Now, on putting these dimensions in Eq. (i), we have
$\begin{aligned}
& =\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]^2}{\left[\mathrm{M}^5\right]\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]^2} \\
& =\frac{\left[\mathrm{M}^3 \mathrm{~L}^6 \mathrm{~T}^{-2}\right]}{\left[\mathrm{M}^3 \mathrm{~L}^6 \mathrm{~T}^{-2}\right]}=\text { dimensionless }
\end{aligned}$
Since, angle is a dimensionless quantity
where dimensions of the various given quantities are
Dimensions of $E=\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]$
Dimensions of $J=\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]$
Dimension of $M=[\mathrm{M}]$
Dimension of $G=\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]$
Now, on putting these dimensions in Eq. (i), we have
$\begin{aligned}
& =\frac{\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]^2}{\left[\mathrm{M}^5\right]\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]^2} \\
& =\frac{\left[\mathrm{M}^3 \mathrm{~L}^6 \mathrm{~T}^{-2}\right]}{\left[\mathrm{M}^3 \mathrm{~L}^6 \mathrm{~T}^{-2}\right]}=\text { dimensionless }
\end{aligned}$
Since, angle is a dimensionless quantity
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