# An aircraft executes a horizontal loop at a speed of 720 km/h with its wings banked at 15°. What is the radius of the loop?

Though I solved the question as in the image:

Solution:

Taking Normal force on aircraft as N.

$$\begin{array} {l}{N \cos \theta=m g, \quad N \sin \theta=\frac{m v^2}{R}} \end{array}$$

which gives

$$R=\frac{v^{2}}{g \tan \theta}=\frac{200 \times 200}{10 \times \tan 15^{\circ}}=15\, \mathrm{km}.$$

But I didn't understand that why is the radius known? Can't a plane execute a loop at any radius with the same speed and angle?

If I imagine a banked road with the slope known, then two cars can move in concentric circle with same speed (therefore having different radius). Then why can't this happen in case of a plane? Are the two situations different?

• Re, "two cars can move parallel..." You are exactly right. The problem is under-specified. In order for the problem to make sense, you must assume that the "ball is centered" in the pilot's turn coordinator. The "ball" basically is a spirit level. It will be centered when the external force acting on the plane makes a 90° angle to the wings. (i.e., when the pilot and passengers would feel as if the plane was level if they were to close their eyes.) – Solomon Slow Jun 14 '19 at 16:59
• Re, "two cars moving on concentric circles on a banked turn..." If cars were equipped with turn coordinators, and if the cars were keeping abreast of each other, then it would not be possible for the ball to be centered in both cars at the same time. Only one radius allows the ball to be centered for a given bank angle and a given speed. – Solomon Slow Jun 14 '19 at 17:00
• Better phrasing would be "why is the radius known" rather than "why is the radius constant". The latter implies that you are asking why the radius isn't changing in time. – Acccumulation Jun 14 '19 at 17:08
• @Acccumulation Thanks for that advice – asks281 Jun 14 '19 at 17:09
• The “ball centered” condition is also known as a “coordinated turn”. – dmckee --- ex-moderator kitten Jun 14 '19 at 21:55

## 1 Answer

In the case of cars driving around a banked curve there are two forces that can act towards the centre - the horizontal components of the normal force and the force force of static friction. This gives the cars a wide range of possible speeds for driving around the curve.

In the case of the airplane static friction is not acting so the only option for centripetal force is the horizontal component of the lift.

Also note that if two cars are travelling in parallel around a curve their speeds are not equal.

• Thanks a lot. And I what I meant by parallel was that the cars travel in a concentric circle. I apologise if that term was misleading. – asks281 Jun 14 '19 at 16:59
• In a non-coordinated situation the craft can have non-trivial yaw and thrust can be partially centripetal. – dmckee --- ex-moderator kitten Jun 14 '19 at 21:57