Weight and Mass (Astronomy)

Suppose your mass on Earth is 70 kg. Now consider a planet with a radius 2times that of Earth and 3 times the mass.

(a) What is the average density of the planet?

(b) What is your weight on Earth?

(c) What is your mass on the planet?

(d) What is your weight on the planet?

(e) What is the orbital speed at the surface of the planet?

(f) What is the orbital period at the surface of the planet?

(g) A rocket orbits the planet at a distance of 3R(Earth) from the SURFACE of the planet [i.e., its altitude is 3R(Earth)]. What is its orbital velocity and period?

Answer

Mass of earth $M_{earth} =5.9736*10^{24}$ Kg

Earth Radius $R_{earth} = 6371$ Km

$M_{planet} =1.791*10^{25}$ Kg

$R_planet = 12742$ Km

a)

$Density of planet = M_{planet}/V_{planet} =M_{planet}*3/4*\pi*R_{planet}^3 =$ $=3*1.79*10^{25}/(4\pi*12742^3) =2.07*10^{12} (Kg/Km^3) = 2.07*10^3 (Kg/m^3)$

b)

$G_{earth} = m*M_{earth}*k/R_{earth}^2 =m*g =70*9.81 =686.7$ N

c) The mass on the any place (on any planet) does not depend on the sizes of the planet and remains the same $m= 70$ Kg

d)

$G_{planet} =m*M_{planet}*k/R_{planet}^2 =m*3/4 *g =3/4*G_{earth} =515$ N

e)

$v^2/R_{planet} =M_{planet}*k/R_{planet}^2$

$v^2 =M_{planet}*k/R_{planet} = 1.79*10^{25}*6.673*10^{-11}/(12742*10^3) =9,374*10^7 m^2/s^2$

$v =\sqrt{v^2} =9.68$ Km/s

f)

Orbital period

$T =2*\pi*R_{planet}/v =2*\pi*12742/9.68 =8270.7$ sec

g)

$v_{Rocket}^2/(5R_{earth}) =M_{planet}*k/(5R_{earth})^2$

$v_{Rocket}^2 = M_{planet}*k/(5R_{earth}) =5/2 v^2 =5/2*9.374*10^7 =23.435*10^7 m^2/s^2$

$v_{Rocket} = \sqrt{v_{Rocket^2}} =15.3 km/s$

$T_{rocket} = 2*pi*5R_{earth}/v_{Rocket} = 2*\pi*5*6371/15.3 =1.31*10^4 sec$