A system of three moons orbits a gas giant Moon A has a mass of 1.45 x 10^22kg a

Question

A system of three moons orbits a gas giant

Moon A has a mass of 1.45 x 10^22kg and a period of 30 days and a radius of orbit (to center of gas giant) of 2.3 x 10^8 m

Moon B has a mass of 8.98 x 10^22 kg and an unknown period and a radius of 4.5 x 10K^8 m

Moon C has a mass of 1.77 x 10^22 kg and a period of 120 days and an unknown orbit

Determine how many days it takes moon B to orbit the Gas giant

Determine the radius of moon C's orbit

Calculate velocity of Moon A as it orbits Gas giant

There is a 4th moon, moon D with 3x the mass of moon A that orbits the Gas Giant with a radius that is 3x the radius of moon A's orbit Determine the velocity of moon D

Explanation / Answer

let mass of the star be M kg, mass of the moon is m kg, radius of the orbit r m, and orbital velocity is v m/s

then balancing gravitational attraction by centripetal force,

G*M*m/r^2=m*v^2/r

==>v=sqrt(G*M/r)....(1)

where G=universal gravitational constant

time period of rotation=circumeference of the orbit/speed

==>T=2*pi*r/v....(2)

for moon A:

T and r is given .

then orbital speed=2*pi*r/T

=2*pi*2.3*10^8 /(30*24*3600 seconds)

=557.54 m/s

using this value of orbital speed in equation 1,

mass of the gas giant=r*v^2/G

=2.3*10^8*557.54^2/(6.674*10^(-11))

=1.0713*10^24 kg

Q1.

for moon B, mass m is given, and radius r is given

then speed =v=sqrt(G*M/r)

=sqrt(6.673*10^(-11)*1.0713*10^24/(4.5*10^8))

=398.57 m/s

then time period=2*pi*r/v

=2*pi*4.5*10^8/398.57

=7.094*10^6 seconds

=82.106 days

Q2.for moon C, mass m and period T is given

as we know, T=2*pi*r/v

=2*pi*r/sqrt(G*M/r)

==>T^2=4*pi^2*r^3/(G*M)

==>r=(T^2*G*M/(4*pi^2))^(1/3)

=((120*24*3600)^2*6.674*10^(-11)*1.0713*10^24/(4*pi^2))^(1/3)

=5.7957*10^8 m

hence radius of moon C orbit is 5.7957*10^8 m

Q3.

velocity of moon A=2*pi*r/T

=2*pi*2.3*10^8/(30*24*3600)

=557.54 m/s

Q4.moon D's radius=3*2.3*10^8=6.9*10^8 m

then its velocity=sqrt(G*M/r)

=sqrt(6.674*10^(-11)*1.0713*10^24/(6.9*10^8))

=321.9 m/s

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.