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 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 4,800 km/h,has a turning radius of 290. Km. Find its banking angle.

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 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 4,800 km/h,has a turning radius of 290. Km. Find its banking angle.

Answer 1

I think I got it... 32.0 degrees?

Answer 2

V = 4800 km/h = 1333 m/s

Lift*cos A = M*g
Lift*sin A = M V^2/R

Tan A = V^2/(gR) = 0.6255
A = 32.0 degrees

You are correct. Congratulations!

Answer 3

To find the banking angle of the SR-71 Blackbird given its speed and turning radius, we can use the centripetal force equation.

The centripetal force equation is given by:

F = m * (v^2 / r)

Where:
F is the centripetal force,
m is the mass of the object,
v is the velocity of the object, and
r is the radius of the turn.

In this case, we need to solve for the banking angle, which is represented by the Greek letter φ (phi).

The banking angle is related to the lift force and the weight of the aircraft. It can be calculated using the equation:

tan(φ) = (v^2 / (g * r))

Where:
tan(φ) is the tangent of the banking angle,
v is the velocity of the aircraft,
g is the acceleration due to gravity (approximately 9.8 m/s^2), and
r is the radius of the turn.

Let's start by converting the given values into SI units:

Speed: 4800 km/h (kilometers per hour)
Converting it to meters per second:
4800 km/h = (4800 * 1000) / (60 * 60) = 1333.33 m/s (approximately)

Turning radius: 290 km (kilometers)
Converting it to meters:
290 km = 290,000 meters

Now, we can substitute the values into the equation to find the banking angle:

tan(φ) = (1333.33^2) / (9.8 * 290,000)

Calculating the values:

tan(φ) = 1777777.72 / 28420000
tan(φ) ≈ 0.0626

To solve for the banking angle (φ), we can take the inverse tangent (arctan) of both sides:

φ = arctan(0.0626)

Using a calculator or software that supports trigonometric functions:

φ ≈ 3.58 degrees

Therefore, the banking angle of the SR-71 Blackbird while turning at 4800 km/h with a radius of 290 km is approximately 3.58 degrees.