jubateee

2021-12-30

An intergalactic spaceship arrives at a distant planet that rotates on its axis with a period of T = 34 hours. The mass of the planet is . The spaceship enters a circular orbit with an orbital period that is equal to the planets

Stella Calderon

Expert

Time period of planet in its own axis (T) = 34 hours
$=122400$ seconds
Mass of planet
let the mass of spaceship = m
Therefore,
$\text{velocity}\cdot \text{time}=\text{circumference of the orbit}$
using, orbital velocity $=\sqrt{G\frac{M}{r}}$
$\sqrt{G\frac{M}{r}}\cdot T=2\pi r$
squaring both sides
${T}^{2}\cdot G\frac{M}{r}=4{\pi }^{2}\cdot {r}^{2}$
${r}^{3}={T}^{2}\cdot G\frac{M}{4}{\pi }^{2}$
$r={\left[{T}^{2}\cdot G\frac{M}{4}{\pi }^{2}\right]}^{\frac{1}{3}}$
Substitute the given values in above expression
$r={\left[{\left(122400\right)}^{2}\cdot 6.67\cdot {10}^{-11}\cdot 2.7\cdot \frac{{10}^{25}}{4\cdot {\pi }^{2}}\right]}^{\frac{1}{3}}$

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