Question

Unlike a ship, an airplane does not use its rudder to turn. It turns by banking its wings: The lift force, perpendicular to the wings, has a horizontal component, which provides the centripetal acceleration for the turn, and a vertical component, which supports the plane's weight. (The rudder counteracts yaw and thus it keeps the plane pointed in the direction it is moving.) The famous spy plane, the SR-71 Blackbird, flying at \(4800 \mathrm{~km} / \mathrm{h}\), has a turning radius of \(290 . \mathrm{km} .\) Find its banking angle.

Short answer

Answer: The banking angle of the SR-71 Blackbird is approximately 32.11°.

Step-by-step solution

Write the known values and required variable

We know the speed (v) of the airplane as \(4800 \mathrm{~km}/\mathrm{h}\), which we can convert to meters per second for SI units: \(4800 \cdot 1000 / 3600 = 1333.33 \mathrm{~m/s}\). We also know the turning radius (r) of the airplane as \(290 \mathrm{~km}\), which can be converted to meters (290000 m). The banking angle (θ) is what we're looking for.

Write the vertical and horizontal force equations

The horizontal component of the lift force can be found using the centripetal force equation: \(F_h = \frac{mv^2}{r}\), where m is the mass of the airplane. The vertical component of the lift force can be found using the weight of the airplane: \(F_v = mg\), where g is the acceleration due to gravity.

Write equations for Lift force components

Let's denote the total Lift force as L. We know that the horizontal component of the lift force is \(L\sin{\theta}\) and the vertical component is \(L\cos{\theta}\). So, we have the following equations: \(F_h = L\sin{\theta}\) \(F_v = L\cos{\theta}\)

Find the ratio of the vertical and horizontal components

Now, let's find the ratio \(\frac{F_v}{F_h}\). From the equations in Step 3, we can write the ratio as \(\frac{L\cos{\theta}}{L\sin{\theta}}\). Simplifying, we get \(\frac{\cos{\theta}}{\sin{\theta}} = \cot{\theta}\).

Find the value of the cotangent

Since we have both centripetal force and gravitational force equations from Step 2 (\(F_h = \frac{mv^2}{r}\) and \(F_v = mg\)), we can find the cotangent of the angle: \(\cot{\theta} = \frac{F_v}{F_h} = \frac{mg}{\frac{mv^2}{r}}\). Simplifying, we get \(\cot{\theta} = \frac{gr}{v^2}\).

Solve for the banking angle (θ)

Now, we can substitute the values and calculate the cotangent of the angle: \(\cot{\theta} = \frac{(9.81 \mathrm{~m/s^2})(290000 \mathrm{~m})}{(1333.33 \mathrm{~m/s})^2}\). Solving for the cotangent, we get \(\cot{\theta} \approx 0.6374\). We can now find the angle by taking the inverse cotangent (arccotangent): \(\theta = \cot^{-1}{(0.6374)}\). Solving for the angle, we get \(\theta \approx 32.11^\circ\). The banking angle of the SR-71 Blackbird is approximately \(32.11^\circ\).

Key concepts

Understanding Centripetal Force

Centripetal force is crucial when an object moves in a curved path or circle; it's directed towards the center of the curvature and is responsible for keeping the object in its curved trajectory. When an airplane turns, it relies on this force to maintain its circular motion.

Think of it as the unseen 'hand' that keeps the airplane from flying off in a straight line, adhering it to its turning path. This force can come from many physical circumstances. In the case of an airplane, the horizontal component of the lift force serves as the centripetal force during a turn.

This horizontal component is one-half of a vital duo, with the vertical component balancing the plane's weight, making sure it stays aloft while banking—a sort of aerodynamic ballet between gravity and the forces generated by the aircraft's wings.

The Role of Lift Force

Lift force is what gets an airplane off the ground and keeps it in the air. Generated by the aircraft's wings, this force acts perpendicular to the direction of the oncoming airflow and opposing gravity's pull. For a plane to ascend, the lift must exceed the weight of the aircraft, and to maintain altitude, these forces must balance.

In a turn, the lift force doesn't only act upwards; it splits into two components: horizontal and vertical. The horizontal part guides the plane into the turn, while the vertical continues to counterbalance the plane’s weight. Understanding how lift operates and varies during flight maneuvers is key to mastering airplane physics.

Calculating the Turning Radius

The turning radius of an airplane signifies the scale of its turn. It's the radius of the circular path that the plane follows while it executes a turn. The smaller the turning radius, the tighter the turn. In the SR-71 Blackbird’s case, a turning radius of 290 kilometers tells us the circle's size if you traced the airplane's path during a full 360-degree turn.

A tighter turn would require a higher centripetal force, meaning either a faster speed or a sharper banking angle. Pilots need to be keenly aware of these dynamics, as they ensure a turn is executed safely, without overstressing the airplane's structural limits or causing discomfort—or worse—to the passengers.

Exploring Airplane Physics

Airplane physics encompasses aerodynamics, mechanics, and several other fields, intertwining to enable these remarkable machines to fly. It includes how air pressure differences create lift, how engines provide thrust, and how pilots use control surfaces like ailerons and the rudder to steer and stabilize the aircraft.

A crucial aspect of airplane physics is understanding how forces act on the aircraft during various maneuvers. For example, when an airplane turns, we analyze the banking angle to understand how lift is directed not only upwards but sideways to create that centripetal acceleration necessary for the turn, perfectly balancing the outward pull experienced by the plane and passengers.