Answer:
The mass of Jupiter is [tex]1.92x10^{27}Kg[/tex].
Explanation:
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]
By replacing (2) in equation (1) it is gotten:
[tex]m\frac{v^{2}}{r} = G\frac{Mm}{r^{2}}[/tex] (3)
Therefore, the mass of Jupiter can be determined if M is isolated from equation (3):
[tex]M = \frac{rv^{2}}{G}[/tex] (4)
But r is the distance between Jupiter and Io ([tex]4.22x10^{8}m[/tex]).
However, it is necessary to know the 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] (5)
Notice that it is necessary to express T in units of seconds.
[tex]T = 1.77days . \frac{24hrs}{1day}[/tex] ⇒ [tex]42.48hours.\frac{3600s}{1hour}[/tex] ⇒ [tex]152928s[/tex]
Where T is the orbital period of Io ([tex]1.52x10^{5}s[/tex]).
[tex]v = \frac{2\pi(4.22x10^{8}m)}{1.52x10^{5}s}[/tex]
[tex]v = 17444m/s[/tex]
Finally, equation (4) can be used.
[tex]M = \frac{(4.22x10^{8}km)(17444m/s)^{2}}{(6.67x10^{-11}N.m^{2})}[/tex]
[tex]M = 1.92x10^{27}Kg[/tex]
Hence, the mass of Jupiter is [tex]1.92x10^{27}Kg[/tex].
