An airplane is flying in a horizontal circle at a speed of 490 km/h. If its wings are tilted at 44° to the horizontal, what is the radius of the circle in which the plane is flying? Assume that the required force is procided entirely by an "aerodynamic lift" that is perpendiculat to the wing surface.
To find the radius of the circle in which the airplane is flying, we can use the concept of centripetal force.
Centripetal force is the force that keeps an object moving in a circular path. For an airplane flying in a circular path, the centripetal force is provided entirely by the aerodynamic lift, which is perpendicular to the wing surface.
Given:
Speed of the airplane (v) = 490 km/h
Tilt angle of wings (θ) = 44°
First, we need to convert the speed of the airplane from km/h to m/s. Since 1 km/h is equal to 0.2778 m/s, the speed of the airplane can be calculated as follows:
Speed of the airplane (v) = 490 km/h × 0.2778 m/s/km/h = 490 × 0.2778 m/s
Now we can calculate the centripetal force (F) using the equation:
F = m × a
where m is the mass of the airplane and a is the acceleration towards the center of the circular path.
Since we are assuming that the aerodynamic lift is providing the centripetal force, we can express the lift force (F) as:
F = L = (m × v^2) / r
where L is the lift force, m is the mass of the airplane, v is the speed of the airplane, and r is the radius of the circular path.
Now, we can solve for the radius (r) using the given values:
L = (m × v^2) / r
Since the lift force is perpendicular to the wing surface and the tilt angle of the wings is given (θ = 44°), we can determine the vertical component of the lift force (L_vertical) using trigonometry:
L_vertical = L × cos(θ)
The vertical component of the lift force is equal to the weight of the airplane, which can be expressed as:
L_vertical = m × g
where g is the acceleration due to gravity.
Now we can replace L_vertical in the equation to solve for the mass (m):
m × g = L × cos(θ)
Since L = (m × v^2) / r, we can rewrite the equation as:
m × g = ((m × v^2) / r) × cos(θ)
Canceling out the mass (m) on both sides, we get:
g = (v^2 / r) × cos(θ)
Now solve for the radius (r):
r = v^2 / (g × cos(θ))
Substitute the given values into the equation:
r = (490 × 0.2778 m/s)^2 / (9.8 m/s^2 × cos(44°))
Calculate the value of cos(44°):
cos(44°) ≈ 0.7193
Now substitute the values into the equation to find the radius:
r = (490 × 0.2778 m/s)^2 / (9.8 m/s^2 × 0.7193)
Calculate the right-hand side of the equation to find the value of the radius. The final answer will be in meters.