A robot probe lands on a new, uncharted planet. It has determined the diameter of the planet to be 8 x 10⁶ m. It weighs a standard 1 kg mass and determines the 1 kg weighs only 5 Newtons on this new planet. a) What is the acceleration due to gravity on this planet? b) What must the mass of the planet be? c) This planet has a moon that orbits it at a distance 6 x10⁹ m from center to center. What is the moon's period?

 

 

(a) Mass of body, m= 1kg

Weight of body, W= 5N

Weight= mg

where g= acceleration due to gravity

g= W/m

g = 5/1

g= 5 N/kg

(b) Diameter of planet, D= 8* 10⁶ m

Radius of planet, R= 4* 10⁶ m

Now,

g= GM/R²

where M= Mass of planet

G= Universal Gravitational constant

g= 6.67* 10^-11*M/(4*10⁶)²

g= 6.67* 10^-11*M/16*10¹²

M= 5*16*10¹²/6.67*10^-11

M= 11.99* 10²³ kg

M= 1.199 * 10²⁴ kg (Mass of planet)

(c) Distance of moon from planet, r= 6* 10⁹m

Period of moon formula,

T= 2?π?* sqrt(r³/G*M)

T= 6.28* sqrt[(6*10⁹)³/(6.67*10^-11*1.199*10²⁴)]

T= 6.28* sqrt[(216* 10²⁷)/7.997* 10¹³)]

T= 6.28* sqrt[27.01* 10¹⁴]

T= 6.28* 5.197* 10⁷

T= 32.63716* 10⁷ seconds

T= 32.63716* 10⁷/(3600*24)

T= 3777.449 days