Curved Banks of Aeroplanes: Why is Banking Necessary?

[ehabmozart]

Homework Statement

My question is. An aeroplane of mass 1.25 x 10^5 kg is turning horizontally in the air, following a circular path of radius 15km at a speed of 245 m/s . It's centripetal acceleration is 4.0 m/s^2 and the centrepital force on it is 5 x 10^5 N. In order to negotiate this path, the plane banks. Explain why this is necessary and calculate a value of the angle of banking necessary. Well, according to the rules, this is NOT a homework. It's a worked example in my book. However, the answers shows the aeroplane itself is NOT curved. I guess that's not right. The free diagram shows the plane .. Vertical line is L cos alpha.. Horizontal is L sin alpha and the diagonal is L... I need clarification. Thanks!

Homework Equations

The Attempt at a Solution

 

[thisischris]

Hi there. I've got a similar question in my textbook.
What do you need clarification of? What do you mean by the airplane isn't curved - I'm guessing you mean 'banking/tilting', in which case you get a value of theta which indicates that it's 'tilting'.

 

[NemoReally]

The dynamics of an aeroplane are a bit more complicated than many texts imply. However, for the purposes of this question it is sufficient to only consider a simple model and not to think about it too deeply.

When an aircraft wants to change direction, it rolls (banks) about its longitudinal axis so that the wing in the desired direction of turn is lower than the other one (assuming a standard monoplane). The lift force now has both a vertical and a horizontal component, the latter being in the direction of the turn. It is this component that provides the centripetal acceleration. But don't forget that the vertical component is reduced so, unless it increases to match the aircraft weight, the aircraft will drop.

 

[HallsofIvy]

What do you mean by "the aeroplane itself is NOT curved. I guess that's not right."? Of course, the airplane itself is not curved! It would be a mechanical disaster if it were! In a large picture we might think of the airplane as a point on the curve so its "curvature" doesn't come up. Closer, the airplane is, at every point, tangent to its path.

 

[azizlwl]

The aircraft is presented as a point at centre of mass with its weight.
The force is perpendicular to the wings.
So when banking the force applied must be at certain angle to horizontal.

The only curved is its path, circular motion.

 

[rcgldr]

The plane has to bank in order to peform a coordinated turn, so that the pilot only feels a force that appears to be "downwards" from the pilot's perspective. This is probably what the original question is asking for.

Some aerobatic planes can turn (or even loop) usuing just rudder inputs, using the fuselage as a wing (called knife edge flight).

 

[truesearch]

The way your question is worded and the data supplied is explained well by Nemoreally.
LCosα is the vertical component opf the lift (equals the weight of the plane) and LSinα is the horizontal component of the lift (equals the centripetal force).
From your data you should be able to find L if you combine these 2 equations.