Gravitation

Physics Part I

Physics

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Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 Ã— 10^{8}Â m. Show that the mass of Jupiter is about one-thousandth that of the sun.

Orbital period ofÂ I_{0}Â ,Â *T*_{I0}Â = 1.769 daysÂ

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â =Â 1.769Â Ã—Â 24Â Ã—Â 60Â Ã—Â 60 sÂ

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â =Â 1.769Â Ã—Â 24Â Ã—Â 60Â Ã—Â 60 sÂ

Orbital radius ofÂ I_{0}Â ,Â *R*_{I}_{0}Â = 4.22Â Ã—Â 10^{8}Â mÂ

Satellite I_{0}Â is revolving around the JupiterÂ

Mass of the Jupiter is given by,Â

where,Â

G = Universal gravitational constant,

Orbital period of the earth,

T_{eÂ }= 365.25 days

Â Â = 365.25Â Ã—Â 24Â Ã—Â 60Â Ã—Â 60 sÂ

Â Â = 365.25Â Ã—Â 24Â Ã—Â 60Â Ã—Â 60 sÂ

Orbital radius of the Earth,

R

Mass of sun is given as,Â

Therefore,

*M*_{s}Â /Â *M*_{J}Â =Â

Â Â Â Â Â Â Â =Â Â

Â Â Â Â Â Â Â =Â Â

Substituting the values,Â

= ^{}

Â = 1045.04

âˆ´Â *M*_{s}Â /Â *M*_{J}Â Â ~ 1000Â

Hence, it can be inferred that the mass of Jupiter is about one-thousandth that of the Sun.

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What is Heliocentric theory?

According to the Heliocentric theory, the sun is at the centre and various planets revolve around the sun at their axis.Â

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What is Geocentric theory?

According to the geocentric theory, all the astronomical bodies like the moon, the sun and stars revolve around the earth, and the earth is at the centre of the universe.Â

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What is the celestial sphere?

At night, if we see the planets and the stars in the sky, all appear to lie in the hemisphere (rest of the hemisphere we are unable to see because of being on the other side of the earth). This sphere is called theÂ celestial sphere.

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The position co-ordinates of two particles of masses m_{1}Â and m_{2}are (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) respectively. Find the coordinates of the centre of mass.

The position vectors of masses m

Let the position coordinates of the centre of mass be (X, Y, Z).

Therefore the position vector of centre of mass is,

Since, Â Â

Â

Comparing the coefficients of Â , we get

Â

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