(a)When an object is in a circular orbit of radius r

Solution 86CP Step 1 a) The time period of the object can be given as, 3 T = 2r2 Gm E Gm E The orbital speed of the obv =t, r 2(r+r) 2 The new time periodT + T = Gm E 3 Then, we can rewrite the t(r + r)2as, 3 3 3 (r + r) = r 2(1 + r) 2 r 3 By binomial expansion, we can wrrt2(1 + r) 2 = r2(1 + 3r) r 2 r Or, r2 1 + 3r = r + 3 r r ( 2 r) 2 Therefore, the equation will change as, 3 3 r r 2(r + 2 ) T + T = Gm E 3 (2r 2) 2(3 r r) That isT + T = + 2 Gm E Gm E 3 (2r 2) We can write, = T Gm E 3 r r Then, T + T = T + Gm E Rearranging, T + T = T + 3 r Gm E r Gm E Cancel out the terms T on both sides, and we can write,r = v Therefore, the change in time period of the object if we raise the orbit by a distance of r, 3 r T = v