Suppose a planet is revolving around the sun in a elliptical orbit at any instant of time it is at distance r with a speed v and f is the gravitational force between the sun and the planet. Choose the incorrect relationship among the following . (1) f~1\r^2 (2) f~v^2\r (3) v^2~1\r. (4) v^2~1\r^2

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Solution

Consider a planet of mass m rotating around the sun (mass M >> m) in a circular orbit of radius r with velocity v. Then, by applying Newton's law of gravitation and the second law of motion, we can write

Gravitational force = mass × centripetal acceleration

F=GMm/r^2

f= m(v^2/r) —- (1) Simplifying we get

v^2 = GMm/r ——(2)

So f~1/r^2 is wrong by (1)

And v^2 ~1/r^2 is also wrong by (2(

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