Question

In 1610, Galileo used his telescope to discover four prominent moons around Jupiter. Their mean orbital radii a and periods T are as follows: (a) Io has a mean orbital radius of 4.22 x 108 m and a period of 1.77 days. Find the mass of Jupiter from this information. (b) Europa has a mean orbital radius of 6.71 x 108 m and a period of 3.55 days. Find the mass of Jupiter from this information. (c) Ganymede has a mean orbital radius of 10.7 x 108 m and a period of 7.16 days. Find the mass of Jupiter from this information. (d) Callisto has a mean orbital radius of 18.8 x 108 m and a period of 16.7 days. Find the mass of Jupiter from this information.

Answer 1

Lets work from point of view of energies

Acceleration along a circular path is equal to the square of the velocity divided by the radius.
A=(V^2)/R
The acceleration is equal to the force that a one Kilogram object would require to keep it on a circular path of radius R
F=M*A
The Newtonian description of the force of gravity is F=G*M1*M2/(R^2)
G=6.673E-11 m^3/kg/s^2
If M1 is the Mass of Jupiter
M1=A/G*(R^2)

a)
Radius
4.22E+08
Period in Days
1.77E+00
Period in seconds
1.53E+05
circumference
2.65E+09
velocity = Circumference/Period in s
1.73E+04
acceleration = V^2/R
7.12E-01
Gravitational constant
6.67E-11
Mass = A/G*(R^2)
Mass of jupiter = 1.90E+27
b)
Radius
6.71E+08
Period in Days
3.55E+00
Period in seconds
3.07E+05
circumference
4.22E+09
velocity = Circumference/Period in s
1.37E+04
acceleration = V^2/R
2.82E-01
Gravitational constant
6.67E-11
Mass = A/G*(R^2)
1.90E+27 Kg
c)
Radius
1.07E+09
Period in Days
7.16E+00
Period in seconds
6.19E+05
circumference
6.72E+09
velocity = Circumference/Period in s
1.09E+04
acceleration = V^2/R
1.10E-01
Gravitational constant
6.67E-11
Mass = A/G*(R^2)
1.89E+27 kg
d)
Radius
1.88E+09
Period in Days
1.67E+01
Period in seconds
1.44E+06
circumference
1.18E+10
velocity = Circumference/Period in s
8.19E+03
acceleration = V^2/R
3.56E-02
Gravitational constant
6.67E-11
Mass = A/G*(R^2)
1.89E+27 Kg