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. 

FREE Expert Solution

Centripetal forces are equal to gravitational forces. 

$\boxed{\mathbf{\Sigma }{\mathbf{F}}_{\mathbf{C}}\mathbf{=}\mathbf{m}{\mathbf{a}}_{\mathbf{C}}}$

(a)

Centripetal force:

$\boxed{{\mathbf{F}}_{\mathbf{C}}\mathbf{=}\frac{\mathbf{G}\mathbf{M}\mathbf{m}}{{\mathbf{R}}^{\mathbf{2}}}}$

Centripetal acceleration:

$\boxed{{\mathbf{a}}_{\mathbf{C}}\mathbf{=}\frac{{\mathbf{v}}^{\mathbf{2}}}{\mathbf{R}}}$

Substituting:

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

The velocity, v is:

$\boxed{\mathbf{v}\mathbf{=}\frac{\mathbf{2}\mathbf{\pi }\mathbf{R}}{\mathbf{T}}}$