Question

After the invention of the telescope, astronomers were able to determine that Io is much smaller than Jupiter, is roughly 4.2x108 m from Jupiter, and orbits Jupiter in 42.5 hours (1.53x105 s). (a) Estimate the mass of Jupiter from these observations. (b) Look up the mass of Earth (Google it!). How many Earth masses does Jupiter have

Answer 1

Answer:

a). The mass of Jupiter is $1.873x10^{27}Kg$.

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:

$F = G\frac{Mm}{r^{2}}$  (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:

$a = \frac{v^{2}}{r}$  (2)

     

Then Newton's second law ($F = ma$) will be replaced in equation (1):

$ma = G\frac{Mm}{r^{2}}$  (3)

By replacing (2) in equation (3) it is gotten:

$m\frac{v^{2}}{r} = G\frac{Mm}{r^{2}}$ (4)

Therefore, the mass of the Jupiter can be determined if M is isolated from equation (4):

$M = \frac{rv^{2}}{G}$ (5)

But r is the distance between Jupiter and Io ($4.2x10^{8}m$).

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:

$v = \frac{2\pi r}{T}$   (6)

Where T is orbital period of Io ($1.53x10^{5}s$).

$v = \frac{2\pi(4.2x10^{8}m)}{1.53x10^{5}s}$

$v = 17247m/s$

Finally, equation (5) can be used.

$M = \frac{(4.2x10^{8}m)(17247m/s)^{2}}{(6.67x10^{-11}N.m^{2})}$

$M = 1.873x10^{27}Kg$

Hence, the mass of Jupiter is $1.873x10^{27}Kg$.

b). How many Earth masses does Jupiter have.

$\frac{1.873x10^{27}Kg}{5.972 x10^{24}kg} = 313.6$

Therefore, Jupiter has 313.6 Earth masses.