An airplane turns by banking the wings such that the lift of the wings has a component in the direction of the centre of the turn. A pilot wishes to turn the aircraft in a radius of 800.0m while flying at a constant speed of 55.6 m/s. Determine the angle of bank needed.
centipetal acceleration = v^2/R in -x direction
g in -y direction
tan angle = v^2 / gR
To determine the angle of bank needed, we can use the equation for the centripetal force:
F = m * v^2 / r
Where:
F = centripetal force
m = mass of the aircraft
v = velocity of the aircraft
r = radius of the turn
Since the mass of the aircraft cancels out, we can rewrite the equation as:
F = v^2 / r
And since the lift also provides the centripetal force, we can equate the lift to the centripetal force:
L = v^2 / r
Where:
L = lift
We know that lift is given by:
L = W * cos(theta)
Where:
W = weight of the aircraft
theta = angle of bank
Since we're given the lift equation and we know that the weight is equal to mg, we can substitute into the lift equation to solve for theta:
mg * cos(theta) = v^2 / r
Simplifying the equation further, we get:
cos(theta) = v^2 / (rg)
Now, we can substitute the given values:
v = 55.6 m/s
r = 800.0 m
g = 9.8 m/s^2
cos(theta) = (55.6^2) / (800.0 * 9.8)
cos(theta) = 0.39945
To find the angle of bank, we take the inverse cosine of both sides:
theta = cos^(-1)(0.39945)
Using a calculator, we find:
theta = 66.76 degrees
Therefore, the angle of bank needed is approximately 66.76 degrees.
To determine the angle of bank needed, we can use the centripetal force formula. The centripetal force is provided by the component of the lift force acting towards the center of the turn.
The formula for the centripetal force is:
F = (m * v^2) / r
Where:
F is the centripetal force
m is the mass of the airplane
v is the speed of the airplane
r is the radius of the turn
In this case, we are given the speed of the airplane (v = 55.6 m/s) and the radius of the turn (r = 800.0 m).
To find the centripetal force, we need the mass of the airplane. Unfortunately, we don't have that information. However, we can still proceed with finding the angle of bank, assuming a typical value for the mass of a commercial airplane (~50,000 kg).
The lift force (L) can be calculated using the equation:
L = m * g
Where:
L is the lift force
m is the mass of the airplane
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Again, assuming the mass of the airplane is 50,000 kg, we can calculate the lift force:
L = (50,000 kg) * (9.8 m/s^2)
L = 490,000 N
Now, we can calculate the component of the lift force (F_c) acting towards the center of the turn:
F_c = L * sin(θ)
Where:
F_c is the component of the lift force towards the center of the turn
θ is the angle of bank
Since we need to find the angle of bank, we rearrange the equation:
θ = arcsin(F_c / L)
Let's calculate the component of the lift force:
F_c = (m * v^2) / r
Assuming the mass of the airplane is 50,000 kg:
F_c = (50,000 kg) * (55.6 m/s)^2 / 800.0 m
F_c ≈ 1,935,200 N
Now, substituting the values into the equation for the angle of bank:
θ = arcsin(1,935,200 N / 490,000 N)
Calculating the angle of bank (θ):
θ ≈ arcsin(3.95)
Using a scientific calculator or online arc sin calculator, we find:
θ ≈ 75.86 degrees
Therefore, the angle of bank needed to turn the aircraft in a radius of 800.0 m while flying at a constant speed of 55.6 m/s is approximately 75.86 degrees.