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

Answer

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

$g = \dfrac{W}{M}$

$g = \dfrac{389}{75}$

g = 5.187 m/s²

we know,

$g = \dfrac{GM}{r^2}$

$5.19 = \dfrac{GM}{r^2}$.......(1)

now,

$g' = \dfrac{W}{M}$

$g' = \dfrac{24.31}{75}$

g' = 0.324 m/s²

$g = \dfrac{GM}{(r+h)^2}$

$0.324= \dfrac{GM}{(r+h)^2}$.........(2)

dividing equation (1)/(2)

$\dfrac{5.19}{0.324}=(\dfrac{(r+h)}{r})^2$

$4=1+\dfrac{h}{r}$

$r = \dfrac{1.86\times 10^4}{3}$

r = 6.2 x 10⁶ m

mass of planet X

from equation 1

$M = \dfrac{g.r^2}{G}$

$M = \dfrac{5.19 \times (6.2\times 10^6)^2}{6.2 \times 10^{-11}}$

M = 3.21 x 10²⁴ kg