To calculate the force required to maintain an object in orbit, the centripetal force formula F = mω²r is used, where ω represents angular velocity in radians per second. Angular velocity can be derived from the period of rotation using the equation ω = 2π/T. The discussion also touches on the dynamics of a cowboy holding onto a bull, exploring how the force exerted on the cowboy affects his speed as the bull attempts to move away. Understanding these forces is crucial for analyzing motion in circular paths. Overall, the principles of centripetal force and angular velocity are essential for calculating orbital mechanics.