How can I calculate the angle of an airplane's wings during a circular motion?

In summary, A pilot flying an airplane at a speed of v in a turn of radius r will experience a force of (VxV)/gr, resulting in an angle of the wings of the airplane to the horizontal of tanQ=(VxV)/gr. This can be explained by considering the balance between the weight of the plane and the altered lifting force caused by banking the wings.
  • #1
Thiendrah
13
0

Homework Statement


A pilot with mass m fles an airplane at a speed of v in a turn of radius r. Prove that angle of the wings of the airplane to the horizontal is tanQ=(VxV)/gr


Homework Equations





The Attempt at a Solution


This was asked on my first exam last week.
Since it's tangent. All we need to get is the force, result will equal to (FxcosQ)/(FxSinQ)=(VxV)/gr?

I'm not getting anywhere with this? Can someone help me?
 
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  • #2
The issue we are concerned with here is how we get the plane to move along a circle of radius r. Start with thinking about the plane flying with the wings level (in a horizontal plane). There are two forces acting on the plane in the vertical direction: its weight, Mg, and the "lift", L, supplied by the wings. With the plane in level flight, these vertical forces must balance.

Now we want the plane to make a turn, so we "bank" the wings by an angle Q. So the "lifting force" will be off-vertical by an angle Q; since it must still balance the weight of the plane vertically, its magnitude will change to L' . (How is it related to Mg?)

This altered lifting force now has a horizontal component as well. This component of L', which will equal L' sin Q , is what supplies the centripetal force to pull the plane into a circular path (for the interval of the wing-banking). So we have

L' sin Q = M·(v^2)/r .

Try things from there.

BTW, a similar argument can be used to explain why a cyclist (leg- or motor-powered) must "lean into the turn" when they want to go round a corner...
 
Last edited:
  • #3
oh, I think i got it.

So we got our force FnXsinQ=m.a=m.(vXv)/r, also the other component is "mg", so

when you divide those two, you will get mvv/gr.

thanks a lot.
 
  • #4
Thiendrah said:
So we got our force FnXsinQ=m.a=m.(vXv)/r, also the other component is "mg", so

when you divide those two, you will get mvv/gr.

I'm presuming that what you are calling 'Fn' corresponds to the lift force; there is no normal force for an aviation problem. The lift force when the plane is banked by an angle Q will be L' = mg/cos Q .

In your last sentence, shouldn't the mass m have divided out?
 

Related to How can I calculate the angle of an airplane's wings during a circular motion?

1. What is circular motion in the context of an airplane?

Circular motion in an airplane refers to the movement of an airplane along a curved path or trajectory, rather than a straight line. This type of motion is typically seen when an airplane is turning, banking, or following a circular flight pattern.

2. How does an airplane maintain circular motion?

In order to maintain circular motion, an airplane needs to have a centripetal force acting on it. This force is directed towards the center of the circular path and is provided by the wing design and the pilot's control inputs. The airplane's velocity and angle of bank also play a role in maintaining circular motion.

3. What factors affect the radius of an airplane's circular motion?

The radius of an airplane's circular motion is affected by the airplane's speed, angle of bank, and the strength of the centripetal force. A higher speed or steeper angle of bank will result in a smaller radius, while a lower speed or shallower angle of bank will result in a larger radius.

4. How does circular motion affect an airplane's lift and weight?

In circular motion, the centripetal force acts perpendicular to the velocity of the airplane, while the weight acts vertically downwards. This means that the centripetal force does not contribute to the lift of the airplane, but rather helps to balance out the weight. The lift of the airplane is still provided by the wings, which need to generate enough lift to counteract the weight and provide the necessary centripetal force for circular motion.

5. Can an airplane experience circular motion in the vertical direction?

Yes, an airplane can experience circular motion in the vertical direction, such as when performing a vertical loop. In this case, the lift force from the wings must be greater than the weight of the airplane in order to provide the necessary centripetal force for circular motion. However, this type of circular motion is typically not sustained for extended periods of time due to the high energy requirements.

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