Problem: An intergalactic spaceship arrives at a distant planet that rotates on its axis with a period of T = 45 hours. The spaceship enters a geosynchronous orbit at a distance of R = 8.7 × 108 m Part (a) From the given information, write an equation for the mass of the planet. Part (b) Calculate the mass of the planet in kilograms.

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Centripetal forces are equal to gravitational forces.

$Σ F_{C} = m a_{C}$

(a)

Centripetal force:

$F_{C} = \frac{G M m}{R^{2}}$

Centripetal acceleration:

$a_{C} = \frac{v^{2}}{R}$

Substituting:

$\begin{array}{rcl} \frac{GM m}{R^{2}} & = & \frac{m v^{2}}{R} \\ v & = & \sqrt{\frac{G M}{R}} \end{array}$

The velocity, v is:

$v = \frac{2 π R}{T}$

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Problem Details

An intergalactic spaceship arrives at a distant planet that rotates on its axis with a period of T = 45 hours. The spaceship enters a geosynchronous orbit at a distance of R = 8.7 × 10⁸ m

Part (a) From the given information, write an equation for the mass of the planet.

Part (b) Calculate the mass of the planet in kilograms.

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