Uniform circular motion - airplane

In summary, the question is asking for the radius of a horizontal circle that a plane is flying in at a speed of 480 km/h with its wings tilted 40 degrees to the horizontal. The required force is provided by an "aerodynamic lift" that is perpendicular to the wing surface. To find the radius, the formula R= (v^2/g)cot(θ) can be used, where v is the speed, θ is the angle of the wings, and g is the acceleration due to gravity. After converting the speed to m/s and plugging in the values for v, θ, and g, the radius can be calculated.
  • #1
missrikku
I have been having trouble getting key components of this problem:

A plan is flying in a horizontal circle at a speed of 480 km/h. The wings are tilted 40 degrees to the horizontal. What is the radius of the circle in which the plane is flying? Assume that the required force is provided entirely by an "aerodynamic lift" that is perpendicular to the wing surface.

okay, so I have:

V = 480 km/h
@ = 40 deg
Find: R

That required force would look much like the Normal force of a box on an incline plane, right?

So, could i use:

Fy = N - mgcos@ = ma ?

with a = V^2/R

but, I don't have a mass to use. So I'm unsure as how to approach this. Would I also need to find Fx?

Fx = mgcos@ = ma, right?
 
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  • #2
You don't need to know the mass because you don't really have to find the force. You only need to know the acceleration toward the center of the circle. That together with the speed around the circle will give you the radius. Either calculate N/m and then cancel m later or just assume m=1.

Fy = N - mgcos@ = ma ?

with a = V^2/R

but, I don't have a mass to use. So I'm unsure as how to approach this. Would I also need to find Fx?

Fx = mgcos@ = ma, right?

Well, I don't know. What are Fx and Fy? How do you have your coordinate system set up?
 
  • #3
+y is in the direction of N
+x is along the wing of the plane that is up 40degrees from the horizontal

Since the plane is flying in a horizontal direction, I was thinking I just had to use:

Fx = mgsin@ = ma

Then, I can see how the mass doesn't matter:

gsin@ = a

and since a = v^2/R:

gsin@ = V^2/R

Then:

R = V^2/gsin@

Since:

V = 480 km/h = 133.3333 m/s
@ = 40

I got R = 2822.18 m = 2.8 km

But, I checked my book and the answer is supposed to be 2.2 km.

I was having trouble at first because I forgot to change the unit of measurement, but now .. I still don't get the right answer
 
  • #4
+y is in the direction of N
+x is along the wing of the plane that is up 40degrees from the horizontal
Then there is no net force in either the x or y directions- the plane isn't moving thos directions!
Since the plane is flying in a horizontal direction, I was thinking I just had to use:

Fx = mgsin@ = ma
But you just said that Fx is NOT horizontal! There is a "lift" in the "y" direction (normal to the airplane): you need to break that into vertical and horizontal components. Calling the normal force N, I get that the vertical component is N cos(θ) and the horizontal component is N sin(θ). Since the airplane is going neither up nor down, the vertical component must of set gravity: N cos(&theta)= mg which tells us that N= mg/cos(θ). (I think that's the part you missed.)
It is the horizontal component that causes the airplane to turn.
From "f= ma", since the horizontal force, f, is N sin(θ), we must have N sin(&theta)= ma. (this is the part you have!)

Since N= mg/cos(θ), ma= (mg/cos(θ))sin(θ) so
a= g tan(θ)= v2/R. Solving for R,
R= (v2/g)cot(θ).

I see that you did correctly convert the speed to m/s (I almost missed that myself!). If you put v= 133.3 m/s, θ= 40 degrees and g= 9.8 m/s2, you should get the right answer.
 

1. What is uniform circular motion?

Uniform circular motion is a type of motion in which an object moves in a circular path with a constant speed. This means that the object covers the same distance in the same amount of time at every point in the circular path.

2. How does an airplane experience uniform circular motion?

An airplane experiences uniform circular motion when it flies in a circular path at a constant speed. This could occur during a turn or when flying in a circular holding pattern.

3. What factors affect the uniform circular motion of an airplane?

The uniform circular motion of an airplane can be affected by factors such as the speed of the airplane, the radius of the circular path, and the mass of the airplane. These factors can impact the centripetal force acting on the airplane, which is necessary to maintain the circular motion.

4. How is the centripetal force calculated in uniform circular motion?

The centripetal force in uniform circular motion is calculated using the formula Fc = mv^2/r, where Fc is the centripetal force, m is the mass of the object, v is the speed of the object, and r is the radius of the circular path.

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

Yes, an airplane can experience uniform circular motion in a vertical direction if it is flying in a vertical circular path, such as during a loop maneuver. In this case, the centripetal force is provided by the lift force of the airplane's wings, which is perpendicular to the circular path.

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