Solution

Statement

$2.1.62^\ast$. An airplane makes a turn, moving in a horizontal circle of radius $R$ with a constant speed $v$. What angle does the plane of the wings make with the horizontal?

Solution

When an airplane executes a coordinated turn in a horizontal circle, the aerodynamic lift force $F_{\text{lift}}$, which acts perpendicular to the plane of the wings, must provide both the vertical force to balance gravity and the horizontal centripetal force to maintain circular motion.

Let $\alpha$ be the angle the wings make with the horizontal. This means the lift force is tilted by an angle $\alpha$ from the vertical.

We can write Newton's second law for the vertical and horizontal (radial) axes:
Vertical axis (equilibrium):
$$F_{\text{lift}} \cos\alpha = mg \quad (1)$$
Horizontal axis (centripetal acceleration):
$$F_{\text{lift}} \sin\alpha = m a_c = m \frac{v^2}{R} \quad (2)$$

Divide equation (2) by equation (1):
$$\frac{F_{\text{lift}} \sin\alpha}{F_{\text{lift}} \cos\alpha} = \frac{m v^2 / R}{mg}$$
$$\tan\alpha = \frac{v^2}{gR}$$

From this, we find the required angle $\alpha$:
$$\alpha = \arctan\left( \frac{v^2}{gR} \right)$$