Refer to figure.
In level flight, any turn will be coordinated or uncoordinated. A coordinated turn is one in which the only acceleration is along the normal axis of the airplane, towards the center of the turn, and the angle of bank is precisely correct for the aircraft speed and the radius of turn. If not all these factors are present, the turn will be uncoordinated and there will be a lateral acceleration either towards or away from the center of the turn. The total lift of a balanced turn provides the centripetal force and the force that opposes the mass of the airplane.
If there is acceleration towards the center of the turn the airplane is slipping and if it is away from the center of the turn the airplane is skidding. Either condition will be indicated on the turn and slip indicator. An overbanked airplane will slip into the turn and requires an increase of speed or a reduction in bank angle to correct this condition. An underbanked airplane will skid out of the turn, which requires a decreased speed or an increased bank angle to correct this condition. These are indicated on the turn and slip indicator by the needle one-way and the ball the other.
To maintain altitude and airspeed when executing a turn both the angle of attack and the thrust have to be increased above their level flight values.
In a turn, turn radius and rate of turn are defined as:
Turn radius = TAS 2/g tan Φ
Rate of Turn = g tan Φ/TAS
Therefore, turn radius is directly proportional to TAS squared and inversely proportional to bank angle and Rate of Turn is directly proportional to bank angle and inversely proportional to TAS.
In this exercise, TAS is equal to 200 knots and the bank angle is 45°:
First, convert 200kt to m/s:
200kt x 0.5144 = 102.88m/s
Then, calculate the turn radius
Turn radius = 102.88 2/10 x tan 45
Turn radius ≈ 1058.43m
The circumference is given by:
c = 2 * π * r
c = 2 * π * 1058.43
c = 6650.31 m
At 102.88 m/s, it would take approximately 65 seconds to complete the circumference.
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France