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Assume that the force of gravitation F...

Assume that the force of gravitation `F prop 1/(r^(n))` . Then show that the orbital speed in a circular orbit of radius r is proportional to ` 1/(r^((n-1)//2))` , while its period T is proportional to ` r^((n+1)//2)`

Text Solution

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` F = K/(r^(n)) = (mv_(0)^(2))/r " " or , v_(0) = sqrt(K/m).1/(r^((n-1)2)), " " :. v_(0) prop 1/(r^((n-1)//2))`
` T = (2pir)/0 = (2pir)/(sqrt(K/m).1/(r^((n-1)//2))) " or", T prop r^((n+1)//2 )`
Note : If n = 1 , i.e , `F prop 1/r , v_(0)` becomes independent of the orbital radius and `T prop r ` .
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