Gravitation

Physics Part I

Physics

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A spaceship is stationed on Mars. How much energy must be expended on the spaceship to launch it out of the solar system? Mass of the space ship = 1000 kg; mass of the Sun = 2 Ã— 10^{30}Â kg; mass of mars = 6.4 Ã— 10^{23}Â kg; radius of mars = 3395 km; radius of the orbit of mars = 2.28 Ã— 10^{8}kg; G= 6.67 Ã— 10^{â€“11}Â m^{2}kg^{â€“2}.

Mass of the spaceship,Â

Mass of the Sun,Â

Mass of Mars,Â

Orbital radius of Mars,Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â =2.28 Ã— 10

Radius of Mars,Â

Â Â Â Â Â Â Â Â Â Â Â Â Â = 3.395 Ã— 10

Universal gravitational constant, G = 6.67 Ã— 10

Potential energy of the spaceship due to the gravitational attraction of the Sun =

Potential energy of the spaceship due to the gravitational attraction of Mars = -

Since the spaceship is stationed on Mars, its velocity and hence, its kinetic energy will be zero.Â

Total energy of the spaceship =Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = Â

The negative sign indicates that the system is in bound state.Â

Energy required for launching the spaceship out of the solar system,Â

= â€“ (Total energy of the spaceship)Â

= Â

= 6.67Â Ã—Â 10

=Â 596.97Â Ã—Â 10

=Â 6Â Ã—Â 10

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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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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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Is Geodesic always a straight line?

No, Geodesic is a straight line if and only if, Â the two points lie on the flat surface. If the two points lie on the curved surface then it is a curved line.

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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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