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A space station consists of two living modules attached to a central hub on opposite sides of the hub by long corridors of equal length. Each living module contains $N$ astronauts of equal mass. The mass of the space station is negligible compared to the mass of the astronauts, and the size of the central hub and living modules is negligible compared to the length of the corridors. At the beginning of the day, the space station is rotating so that the astronauts feel as if they are in gravitational field of strength $g.$ Two astronauts, one from each module, climb into the central hub, and the remaining astronauts now feel a gravitational of strength $g'.$ What is the ratio $g'/g$ in terms of $N\,?$
$\sqrt{\frac{N}{N-1}}$
$\frac{N}{N-1}$
$\sqrt{\frac{N-1}{N}}$
$(\frac{N}{N-1})^2$
Solution
$\mathrm{Nm} \omega \mathrm{L}^{2}=(\mathrm{N}-1) \mathrm{m \omega}^{\prime} \mathrm{L}^{2}$
$\frac{\omega^{\prime}}{\omega}=\frac{N}{N-1} \frac{g^{\prime}}{g}=\frac{\omega^{\prime 2} L}{\omega^{2} L}$
$\Rightarrow \frac{g^{\prime}}{g}=\left(\frac{N}{N-1}\right)^{2}$
