a small aircraft experiences s lift force equal to 3.8 times of its weight.
a)draw a diagram to show the forces acting on the aircraft when it is flying in a horizontal circle with its wing banked at an angle to the horizontal
b)determine the maximum bank angle that the aircraft can safely maintain without threatening the safety of the aircraft
To answer this question, we need to understand the forces acting on the aircraft when it is flying in a horizontal circle with its wing banked. Here's how you can find the solution:
a) Diagram of forces:
When an aircraft is flying in a horizontal circle with its wing banked, there are four main forces acting on it: Lift (L), Weight (W), Thrust (T), and Drag (D).
1. Lift (L): Lift is the force that opposes gravity and keeps the aircraft airborne. In this case, the lift force is 3.8 times the weight of the aircraft (L = 3.8W). Lift acts vertically upward.
2. Weight (W): Weight is the force exerted by gravity on the aircraft. It acts vertically downward.
3. Thrust (T): Thrust is the force that propels the aircraft forward. Since the aircraft is flying in a circle, thrust acts tangentially to the circle inward.
4. Drag (D): Drag is the resistance force experienced by the aircraft due to air friction. It acts tangentially to the circle outward, opposing the motion of the aircraft.
To draw a diagram, you can start with a horizontal line to represent the ground. Draw an arrow pointing upward to represent lift (L), and another arrow pointing downward to represent weight (W). Then draw two arrows tangentially inward and outward to represent thrust (T) and drag (D), respectively. Make sure to label each force.
b) Maximum bank angle calculation:
To determine the maximum bank angle that the aircraft can safely maintain without threatening its safety, we need to consider the forces acting on the aircraft and the relationship between them.
The lift force (L) provides the centripetal force required for the aircraft to fly in a horizontal circle. The centripetal force equals the product of mass (m) and centripetal acceleration (ac), which is given by the formula:
centripetal force = m * ac
The centripetal force is provided by both the weight (W) and the lift force (L). When the aircraft is banked, a component of the weight force (W) acts upward and contributes to the centripetal force along with the lift force (L). The equation for centripetal force becomes:
centripetal force = W * sin(θ) + L * cos(θ)
Where:
θ = bank angle of the aircraft
Since the lift force (L) is given to be 3.8 times the weight (W), we can substitute L = 3.8W into the equation:
centripetal force = W * sin(θ) + 3.8W * cos(θ)
As the aircraft cannot safely maintain a bank angle if the centripetal force exceeds the maximum limit, we need to find the maximum value of the centripetal force.
Given that the lift force is 3.8 times the weight, we can set the centripetal force equal to the weight:
W = W * sin(θ) + 3.8W * cos(θ)
Simplifying the equation gives:
1 = sin(θ) + 3.8 * cos(θ)
To find the maximum bank angle (θ), you can solve this equation numerically using methods like iterative approximation or a graphing calculator.
By analyzing the graph or using numerical approximation methods, you can find the maximum bank angle (θ) that satisfies the equation without threatening the safety of the aircraft.