Question

In 2014, Damian Walters demonstrated that he could run through a vertical loop (after a lot of failed attempts). What minimum running speed did he need to do it? Assume the loop has a radius of 1.96 m, that Damian's mass is 76.5 kg, that he stays in contact with the loop at all times, and (unrealistically) that you can neglect friction and drag forces. Also, keep in mind that it's Damian's center of mass that matters, which is approximately 1.16 m above his feet.

This is a two part problem:

(1) you are dealing with energy conservation, so you want to related the speed (and hence kinetic energy) at the bottom of the loop with the kinetic & potential energies at the top

(2) since this is circular motion, there is a constraint on the speed at the top of the loop: it has to be fast enough so that Damian "stays in contact"; i.e., that there be a normal force between his feet and the loop surface at the top of the loops. Since he also has a weight force acting on him, using a force diagram you can determine how these forces provide the necessary centripetal acceleration. If the condition is minimum speed, then this is just before he loses contact; i.e,. when F_NF​N​​ = 0.

Also, be careful with computing potential energy here: what matters is how much his center of mass rises during the loop run. It may be helpful to draw a diagram here to make sure you have the right increase in height, which will be less than 2R.

Answer 1