Question

An aircraft loops the loop of radius $R=500m$ with a constant velocity $v=360km/h$. Find the weight of the flyer of mass $m=70kg$ in the lower, upper and middle points of the loop.

Answer 1

While moving in a loop, normal reaction exerted by the flyer on the loop at different points and uncompensated weight if any contribute to the weight of flyer at those points.
(a) when the aircraft is at the lower most point, Newton's second law of motion in projection form $F_n=m{w}_n$ gives $N-mg=\frac{m{v}^2}{R}$
or, $N=mg+\frac{m{v}^2}{R}=2.09kN$
(b) when it is at the upper most point, again from $F_n=m{w}_n$ we get
$N^{\prime \prime }+mg=\frac{m{v}^2}{R}$
$N^{\prime \prime }=\frac{m{v}^2}{R}-mg=0.7kN$
(c) When the aircraft is at the middle point of the loop, again from $F_n=m{w}_n$
$N^{\prime }=\frac{m{v}^2}{R}=1.4kN$
The uncompensated weight is $mg$. Thus effective weight $=\sqrt{{N^{\prime }}^2+m^2{g}^2}=1.56kN$ acts obliquely.