Determine the radius of the circular orbit for a satellite travelling with an angular speed equal to the angular speed of the earth's rotation. The mass of the earth is 6 10²⁴ kg .

Step-by-step solution

The angular speed of the earth's rotation is = 2 T , where T = 24 hours = 86400 s . For a satellite in a circular orbit, the gravitational force provides the necessary centripetal force: GMm r^2 = m ^2 r r^3 = GM ^2 = GMT^2 4 ^2 Substituting the given values M = 6 10²⁴ kg , G = 6.67 10⁻¹¹ N m ^2 kg ⁻² , and T = 86400 s : r^3 = 6.67 10⁻¹¹ 6 10²⁴ (86400)^2 4 ^2 r^3 40.02 10¹³ 7.465 10^9 39.48 r^3 75.67 10²¹ m ^3 Taking the cube root: r 4.23 10^7 m = 42300 km