An airplane is circling an airport by traveling a horizontal circular path at a speed of 400 km/h. The bank angle of the wings is 20 degrees. What is the radius of the circular path the plane is flying?
I honestly do not know where to start this. And it is an algebra-based Physics 1 AP class I'm taking.
centripetal acceleration = v^2/R
Lift of wings = F
weight = m g = F cos 20
so F = m g/cos 20
centripetal force = F sin 20
= m g sin 20 / cos 20 = m g tan 20
centripetal force = m Ac = m v^2/R
so
m v^2/R = m g tan 20
so
R = v^2/(g tan 20)
To solve this problem, we can use the concept of centripetal force and the force of gravity acting on the airplane.
First, let's identify the forces involved:
1. Centripetal force (Fc): This is the force that keeps an object moving in a circular path. It is directed towards the center of the circular path and is equal to the mass of the object multiplied by its acceleration toward the center of the circle.
2. Force of gravity (Fg): This is the force pulling the airplane downwards due to gravity. It acts vertically downwards.
Now, we need to find the relationship between these forces. The centripetal force is provided by the component of the lift force (generated by the wings' bank angle) perpendicular to the horizontal plane.
Let's break it down into steps:
Step 1: Find the lift force (Fl) exerted by the wings of the airplane.
- The lift force is given by the equation: Fl = m * g, where m is the mass of the airplane and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Step 2: Find the horizontal component of the lift force (Fh).
- The lift force can be resolved into horizontal and vertical components. The horizontal component is given by Fh = Fl * sin(bank angle), where the bank angle is given as 20 degrees.
Step 3: Equate the horizontal component of the lift force (Fh) to the centripetal force (Fc).
- The centripetal force is given by the equation: Fc = m * v^2 / r, where v is the speed of the airplane and r is the radius of the circular path.
Step 4: Solve for the radius (r).
- By equating Fh to Fc, we can get the equation: Fl * sin(bank angle) = m * v^2 / r.
- Rearrange the equation to solve for r: r = m * v^2 / (Fl * sin(bank angle)).
Step 5: Convert the speed from km/h to m/s.
- Divide the given speed of 400 km/h by 3.6 to convert it to m/s.
Step 6: Plug in the values and solve for the radius (r).
I hope this step-by-step explanation helps you solve the problem in your AP Physics 1 class!