Two particles of identical mass are moving in circular orbits under a potential given by V ( r )= Kr ^ n , where K is a constant. If the radii of their orbits are r₁, r₂ and their speeds are v₁, v₂ , respectively, then

Step-by-step solution

array l V ( r )= Kr ^ - n gravitational field = E =- d V dr =(- K ) d dr ( r ^ - n ) =(- K )(- n ) r ^ - n -1 = Kn r ^ n +1 array force on mass = E M , where M = mass of body array l M E₁= M V₁² r₁ M E₂= M V₂² r₂ V₁² V₂² = r₁ E₁ r₂ E₂ V₁² V₂² = r₁ r₂ K n r₁^ n+1 r₂^ n+1 K n V₁² V₂² = r₂^ n r₁^ n V₁² r₁^ n =V₂² r₂^ n array