PLEASE SHOW ALL WORK AND COMPLETE FULL QUESTION FOR THUMBS U

PLEASE SHOW ALL WORK AND COMPLETE FULL QUESTION FOR THUMBS UP

The International Space Station (ISS) stays in a circular orbit with radius 6.7×10³ km from the center of the earth. (a) What is the period of ISS orbital motion? (b) What is the orbital speed of ISS? (c) If we can ignore the ground speed due to earth rotation, how much energy is minimally required to send 1 kg of cargo from the earth surface onto ISS?  (G = 6.67 × 10^-11 N·m²/kg² ; earth mass M_E 6.0 × 10²⁴ kg; earth radius R_E 6.4 × 10³ km).

Solution

a) for circular motion,

gravitational force on ISS = mass of ISS x radial acc.

G Me m / (Re + h)^2 = m w^2 (Re + h)

Me - mass of earth

Re - radius of earth

(6.67 x 10^-11 x 5.972 x 10^24 ) / (6371000 + 6700000)^3 = w^2

w = 4.223 x 10^-4 rad/s

Time period. T = 2pi / w = 2pi / (4.223 x 10^-4)

T = 14877.17 sec

b) v= w . (Re + h)

v = (4.223 x 10^-4) (6371000 + 6700000)

v = 5520 m/s

c) Potential energy at the surface of earth = - G Me m / Re

            = - (6.67 x 10^-11 x 5.972 x 10^24 x 1 ) / (6371000)

      = - 62522743.78 J

Potential when it reaches at the surface of iSS
= - (6.67 x 10^-11 x 5.972 x 10^24 x 1 ) / (6371000 + 6700000)

      = - 30474516.1 J

Using enery conservation,

PE+ KE at surafce of earth = at ISS

- 62522743.78 + KE = - 30474516.1

KE =3.205 x 10^7 J ...Ans