An airplane is flying in a horizontal circle at a speed of 514.8 km/h. The plane is banked for this turn, its wings tilted at an angle of 41.6° from the horizontal. Assume that a lift force acting perpendicular to the wings acts on the aircraft as it moves through the air. What is the radius of the circle in which the plane is flying?
km
12
To find the radius of the circle in which the plane is flying, we can use the concept of centripetal force.
The centripetal force acting on an object moving in a circular path is given by the equation:
F = (m * v^2) / r
Where:
F = Centripetal force
m = Mass of the object
v = Velocity of the object
r = Radius of the circular path
In this case, the lift force acting on the airplane provides the centripetal force:
F = Lift force
To start, we need to convert the speed from km/h to m/s since the SI units are commonly used in physics calculations. We know that 1 km/h is equal to 1000/3600 m/s.
So, the speed of the airplane (v) in m/s is:
v = (514.8 km/h) * (1000/3600) m/s
v ≈ 143 m/s
Now, we can use the lift force and the speed to find the radius (r) of the circular path.
The lift force (F) acting on the airplane is equal to the gravitational force component perpendicular to the wings. It can be calculated using the equation:
F = m * g * cos(θ)
Where:
m = Mass of the airplane
g = Gravitational acceleration (approximately 9.8 m/s^2)
θ = Angle of bank (41.6°)
The gravitational force (mg) can be calculated using the mass of the airplane and the gravitational acceleration.
Now, we can rearrange the centripetal force equation to solve for the radius (r):
r = (m * v^2) / F
Substituting the values we have:
r = (m * v^2) / (m * g * cos(θ))
m cancels out in the equation, simplifying it to:
r = v^2 / (g * cos(θ))
Plugging in the values:
r = (143 m/s)^2 / (9.8 m/s^2 * cos(41.6°))
Now, we can calculate the radius:
r ≈ 1739 meters
Therefore, the radius of the circle in which the plane is flying is approximately 1739 meters.