Worked example 7.4: Aerobatic maneuver

Question: A stunt pilot experiences weightlessness momentarily at the top of a ``loop the loop'' maneuver. Given that the speed of the stunt plane is $v=500 {\rm km/h}$, what is the radius $r$ of the loop?

Answer: Let $m$ be the mass of the pilot. Consider the radial acceleration of the pilot at the top of the loop. The pilot is subject to two radial forces: the gravitational force $m g$, which acts towards the centre of the loop, and the reaction force $R$, due to the plane, which acts away from the centre of the loop. Since the pilot experiences an acceleration $v^2/r$ towards the centre of the loop, Newton's second law of motion yields

$$m \frac{v^2}{r} = m g- R.$$

Now, the reaction $R$ is equivalent to the apparent weight of the pilot. In particular, if the pilot is ``weightless'' then he/she exerts no force on the plane, and, therefore, the plane exerts no reaction force on the pilot. Hence, if the pilot is weightless at the top of the loop then $R=0$, giving

$$r = \frac{v^2}{g} = \frac{(500\times 1000/3600)^2}{9.81} = 1.97 {\rm km}.$$