Commercial aircrafts usually bank at an angle of no more than 30.0 degrees when turning. At 30.0 degrees of bank, how many g's of lift must the aircraft pull if it is to maintain uniform circular motion without losing altitude?
To determine the number of g's of lift an aircraft must pull at a 30.0-degree bank angle, we need to understand the relationship between bank angle, gravitational force (g), and lift force.
In a coordinated turn, the lift force provides both vertical lift to counteract the weight of the aircraft and horizontal lift to change its direction. The total lift force can be calculated using the formula:
Lift = Weight / cos(bank angle)
In this case, since the aircraft is maintaining uniform circular motion without losing altitude, the lift force must equal the weight of the aircraft. Therefore, we can set the formula equal to weight:
Weight = Weight / cos(bank angle)
Simplifying and rearranging the equation, we get:
1 = 1 / cos(bank angle)
To find the value of cos(bank angle), we take the inverse cosine (cos^(-1)) of both sides:
cos^(-1)(1) = cos^(-1)(1 / cos(bank angle))
Cos^(-1)(1) is 0°, so the equation becomes:
0° = bank angle
Therefore, in order to maintain a 30.0-degree bank angle without losing altitude, an aircraft must pull exactly 1g of lift.