Answer:
a). The mass of Jupiter is [tex]1.873x10^{27}Kg[/tex].
b). Jupiter has 313.6 Earth masses.
Explanation:
a). Estimate the mass of Jupiter from these observations.
The Universal law of gravitation shows the interaction of gravity between two bodies:
[tex]F = G\frac{Mm}{r^{2}}[/tex] (1)
Where G is the gravitational constant, M and m are the masses of the two objects and r is the distance between them.
For this particular case M is the mass of Jupiter and m is the mass of Io. Since it is a circular motion the centripetal acceleration will be:
[tex]a = \frac{v^{2}}{r}[/tex] (2)
Then Newton's second law ([tex]F = ma[/tex]) will be replaced in equation (1):
[tex]ma = G\frac{Mm}{r^{2}}[/tex] (3)
By replacing (2) in equation (3) it is gotten:
[tex]m\frac{v^{2}}{r} = G\frac{Mm}{r^{2}}[/tex] (4)
Therefore, the mass of the Jupiter can be determined if M is isolated from equation (4):
[tex]M = \frac{rv^{2}}{G}[/tex] (5)
But r is the distance between Jupiter and Io ([tex]4.2x10^{8}m[/tex]).
However, it is necessary to know the orbital velocity of Io in order to determine the mass of Jupiter.
The orbital velocity is defined as:
[tex]v = \frac{2\pi r}{T}[/tex] (6)
Where T is orbital period of Io ([tex]1.53x10^{5}s[/tex]).
[tex]v = \frac{2\pi(4.2x10^{8}m)}{1.53x10^{5}s}[/tex]
[tex]v = 17247m/s[/tex]
Finally, equation (5) can be used.
[tex]M = \frac{(4.2x10^{8}m)(17247m/s)^{2}}{(6.67x10^{-11}N.m^{2})}[/tex]
[tex]M = 1.873x10^{27}Kg[/tex]
Hence, the mass of Jupiter is [tex]1.873x10^{27}Kg[/tex].
b). How many Earth masses does Jupiter have.
[tex]\frac{1.873x10^{27}Kg}{5.972 x10^{24}kg} = 313.6[/tex]
Therefore, Jupiter has 313.6 Earth masses.
