Determine the mass of Jupiter using the data for Io, moon of Jupiter. Express your answer using three significant figures and include the appropriate units. Mass of Io =8.9×1022 kgPeriod of revolution around Jupiter = 1.77 Earth days Mean distance from Jupiter =422×103km

The mass of Jupiter is .

Explanation:

The Universal law of gravitation shows the interaction of gravity between two bodies:

(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:

 (2)

Then, Newton's second law () will be replaced in equation (1):

By replacing (2) in equation (1) it is gotten:

 (3)

Therefore, the mass of Jupiter can be determined if M is isolated from equation (3):

 (4)

But r is the distance between Jupiter and Io ().

However, it is necessary to know the velocity of Io in order to determine the mass of Jupiter.

The orbital velocity is defined as:

  (5)

Notice that it is necessary to express T in units of seconds.

⇒ ⇒

Where T is the orbital period of Io ().

Finally, equation (4) can be used.

Hence, the mass of Jupiter is .

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