Circular Motion Problem

Question

At a radius of 161.178 m, and a centripetal acceleration of 6.58 times free-fall acceleration, what is the magnitude of the net force that maintains circular motion exerted on the 111 kg pilot by the seat belts, the friction against the seat, and so forth? Answer in units of N.

Answer 1

Let's try to understand the basic concepts required to understand circular motion.

In order to move a body of mass m moving at constant speed v in a circular path, there needs to be a force of constant magnitude that acts on the body perpendicular to its instantaneous direction of motion (which, by virtue of its going around in a circle, is constantly changing). This force is referred to as the Centripetal force. You may take its definition as any force that is always acting ⊥ to the instantaneous direction of motion. Such a force will always curve the motion of the body into a circular path. Moreover, the direction of the centripetal force will always be inward (pointing radially towards the center of the circle)

For instance, when you tie a stone at the end of a string and swing this apparatus around, the (always taut) string applies a radially inward centripetal force on the stone, which results in its circular motion. In this problem, the necessary force is provided by all the forces that hold the pilot in place as the spaceship moves in a circle (Newton's first law : without forces, everything wants to move in straight lines at constant speeds)

Next, we are interested in the radius of this circular path. If the magnitude of the centripetal force is F_cp , then, the resultant circular path is of radius r satisfying F_cp = m v² / r , or r = m v² / F_cp.

In your question, what is given is the centripetal acceleration (as some multiple of acceleration due to gravity g) i.e. given : v² / r = α g for some α , so r = v² / α g.

If you have followed this far, you'll be able to finish the exercise yourself.

Hope this helps.

PS: Little note about frames of reference. This analysis of circular motion is made w.r.t the fixed ('lab') frame with respect to which the circular motion is observed. We could also analyse this motion in the rotating frame of the plane as well. In this frame, the pilot is just sitting still. Accordingly, there is no circular motion of the pilot that is observed (the world around will be observed to be rotating though). So, there is no inward centripetal force F_cp acting on the pilot in this frame. However, now, the pilot feels a fictitious force that tends to push them outward. This is called centrifugal force F_cg, which has the same magnitude as F_cp, but feels as if it is pushing outward instead of pulling in. It is a fictitious force, in the sense that the pilot feels it, but its not actually there (it causes no motion). It's presence is due to the fact that the frame of the pilot is an accelerating frame (non-inertial), and hence, Newton's laws do not hold in them.