Gravitation

Question : 4 of 25

Marks: +1, -0

One of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is

4.22 \times 10^{8}

m. Show that the mass of Jupiter is about one-thousandth that of the sun.

Solution:

For Jupiter’s satellites

Orbital period,

T_J = 1.769 \text{ days} = 1.769 \times 24 \times 60 \times 60 \text{ s}

Radius of the orbit,

R_J = 4.22 \times 10^{8} \text{ m}

Mass of jupiter,

M_{J}= \frac{4 \pi^{2} R_{I}^{3}}{G T_{I}^{2}}

....(i)

Now, orbital period of Earth around the sun

T=1

year

=365.25 \times 24 \times 60 \times 60 \text{ s}

Orbital radius,

R_{S}=1.496 \times 10^{11} \text{ m}

Mass of sun

M_{S}= \frac{4 \pi^{2} R^{3}}{G T_{E}^{2}}

....(ii)

Dividing (ii) by (i)

\frac{M_{S}}{M_{J}} = \frac{R^{3}}{R_{J}^{3}} \times \frac{T_{J}^{2}}{T_{E}^{2}}

= \frac{ (1.496 \times 10^{11})^{3} }{ (4.22 \times 10^{8})^{3} } \times \frac{ (1.77 \times 24 \times 60 \times 60)^{2} }{ (365.25 \times 24 \times 60 \times 60)^{2} }

=1046

\frac{M_{J}}{M_{S}} = \frac{1}{1046} \simeq \frac{1}{1000}

M_{J}= \frac{1}{1000} M_{S}

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