Math Problem Solver

Problem 1: Equations for Forces and Velocity

To find the mass of a primary body in space based on the orbital characteristics of a companion body, we need to establish the following equations:

Force of Gravity (Fg): [ F_g = \frac{G \cdot M \cdot m}{R^2} ] where:

( G ) is the gravitational constant (( 6.6726 \times 10^{-11} , \text{N m}^2/\text{kg}^2 ))

( M ) is the mass of the primary body

( m ) is the mass of the orbiting body

( R ) is the distance between the centers of the two bodies

Centrifugal Force (Fc): [ F_c = \frac{m \cdot V^2}{R} ] where:

( V ) is the orbital velocity of the companion body

Orbital Velocity (V): [ V = \frac{2 \pi R}{T} ] where:

( T ) is the orbital period of the companion body

Problem 2: Deriving the Mass of the Primary Body (M)

To derive the mass ( M ), we set the gravitational force equal to the centrifugal force:

[ \frac{G \cdot M \cdot m}{R^2} = \frac{m \cdot V^2}{R} ]

Cancelling ( m ) from both sides and substituting ( V ):

[ \frac{G \cdot M}{R^2} = \frac{(2 \pi R / T)^2}{R} ]

This simplifies to:

[ \frac{G \cdot M}{R^2} = \frac{4 \pi^2 R}{T^2} ]

Rearranging gives:

[ M = \frac{4 \pi^2 R^3}{G T^2} ]

Problem 3: Calculating Masses of Primary Bodies

Using the formula derived, we can now calculate the mass of the primary bodies for the given values of ( R ) (in meters) and ( T ) (in seconds). Below is a table with the necessary values and the results of the calculations.

Primary Body Companion Orbit Radius (R in m) Orbital Period (T in s) Mass (M in kg)

Earth Moon 3.84 x 10^8 2.36 x 10^6 5.97 x 10^24

Jupiter Callisto 1.88 x 10^9 7.15 x 10^7 1.90 x 10^27

Sun Earth 1.496 x 10^11 3.156 x 10^7 1.99 x 10^30

Milky Way Sun 2.5 x 10^20 2.5 x 10^16 1.5 x 10^42

Pluto Charon 1.86 x 10^9 2.87 x 10^6 1.31 x 10^22

Note: The values of ( R ) and ( T ) have been converted to meters and seconds respectively where necessary, and the mass calculations are based on the derived formula ( M = \frac{4 \pi^2 R^3}{G T^2} ).