Answer :
To solve this problem related to the satellite orbiting the planet Cruton, we can break down the problem into three parts: finding the radius of the orbit, the kinetic energy of the satellite, and the mass of planet Cruton.
(a) What is the radius of the orbit?
We will use the formula for gravitational force to find the radius. The gravitational force [tex]F[/tex] can be expressed as:
[tex]F = \frac{G M m}{r^2}[/tex]
Where:
- [tex]F = 85.9 \, \text{N}[/tex] (the gravitational force exerted on the satellite),
- [tex]m = 38.1 \, \text{kg}[/tex] (mass of the satellite),
- [tex]M[/tex] is the mass of planet Cruton,
- [tex]r[/tex] is the radius of the orbit,
- [tex]G[/tex] is the gravitational constant [tex]6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2[/tex].
The centripetal force required to keep the satellite in its circular orbit is provided by the gravitational force:
[tex]F = \frac{m v^2}{r}[/tex]
Since the satellite completes one orbit in 7.25 hours, we convert this period (T) to seconds: [tex]T = 7.25 \times 3600 \, \text{s}[/tex].
The velocity [tex]v[/tex] can be expressed in terms of the period and radius as:
[tex]v = \frac{2 \pi r}{T}[/tex]
Setting the two expressions for the force equal gives:
[tex]\frac{G M m}{r^2} = \frac{m (\frac{2 \pi r}{T})^2}{r}[/tex]
After canceling [tex]m[/tex] and simplifying, we solve for [tex]r[/tex]:
[tex]r^3 = \frac{G M T^2}{4 \pi^2}[/tex]
However, without the mass [tex]M[/tex] of the planet, we must first solve part (c) to find [tex]r[/tex].
(b) What is the kinetic energy of the satellite?
The kinetic energy ([tex]KE[/tex]) of the satellite is given by:
[tex]KE = \frac{1}{2} m v^2[/tex]
Substituting [tex]v = \frac{2 \pi r}{T}[/tex], [tex]KE[/tex] becomes:
[tex]KE = \frac{1}{2} m \left(\frac{2 \pi r}{T}\right)^2[/tex]
This equation will be computed after solving for [tex]r[/tex].
(c) What is the mass of planet Cruton?
We can estimate the mass using Kepler’s third law in terms of the gravitational parameter. From our earlier formula substitution, equate Kepler's third law:
[tex]T^2 = \frac{4 \pi^2 r^3}{G M}[/tex]
Reordering, solve for [tex]M[/tex]:
[tex]M = \frac{4 \pi^2 r^3}{G T^2}[/tex]
To find [tex]r[/tex] and [tex]M[/tex] together, use combined equations and iterative methods or numerical techniques with the provided force [tex]F[/tex]. Here use assumed values or measured parameters obtained in computational physics for precision, merging results sequentially for actual problem resolution.
This problem combines Newton's laws of motion and gravitation with orbital mechanics, requiring careful calculations often done with computational assistance or simplifying assumptions for student-friendly manual solving.
Without constructing individual numeric conclusions without computational trial, understanding variables’ structure and representation provides groundwork for composing actual solution process iteratively.
