Question

The weight of spaceman Speff at the surface of planet X, solely due to its gravitational pull, is 389 N. If he moves to a distance of 1.86 × 104 km above the planet's surface, his weight changes to 24.31 N. What is the mass of planet X, if Speff's mass is 75.0 kg? (G = 6.67 × 10-11 N · m2/kg2)

Answer 1

given,

W = weight of speff at surface of planet = 384 N

h= height = 1.86 x 10⁴ Km

W' = weight of speff at height h from the planet = 24.31 N

Ms = speff mass = 75 kg

M= mass of planet = ?

R = radius of planet

G = 6.67 x 10⁻¹¹ m³/kg.s²

`g = (W)/(M)`

`g = (389)/(75)`

g = 5.187 m/s²

we know,

`g = (GM)/(r^2)`

`5.19 = (GM)/(r^2)`.......(1)

now,

`g' = (W)/(M)`

`g' = (24.31)/(75)`

g' = 0.324 m/s²

`g = (GM)/((r+h)^2)`

`0.324= (GM)/((r+h)^2)`.........(2)

dividing equation (1)/(2)

`(5.19)/(0.324)=(((r+h))/(r))^2`

`4=1+(h)/(r)`

`r = (1.86* 10^4)/(3)`

r = 6.2 x 10⁶ m

mass of planet X

from equation 1

`M = (g.r^2)/(G)`

`M = (5.19 * (6.2* 10^6)^2)/(6.2 * 10^(-11))`

M = 3.21 x 10²⁴ kg