Talk:Banked turn

merge
Suggest merging this article, cant (road/rail) and cant deficiency into one article "Banking / Cant (engineering)"

FengRail (talk) 21:27, 11 April 2009 (UTC)


 * There has been no support for the proposed merger between these articles, and the editor who proposed it has not been active at WP since just after the proposals were made. Hence it may be safely said that this merge proposal is now dead.
 * There is a separate proposal to merge cant deficiency into cant (road/rail). (See Talk:cant deficiency if you wish to participate.)
 * -- EdJogg (talk) 11:14, 16 October 2009 (UTC)


 * However, there currently exists four separate articles that cover this topic: Banked turn, Cant (road/rail), Cant deficiency, and Cross slope. Even Camber angle chimes in. The fact that banked turn discusses both ground-based and aerial maneuvers only suggests that that article should be split. A similar argument can be made about Cant (road/rail). The fact that the editor who proposed the merge has not been active should have nothing to do with it. I suggest we either put everything in one article with separate sections for rail, road, and aerial, or separate the topic into three articles along those same lines. -AndrewDressel (talk) 14:06, 3 April 2010 (UTC)

Banked_turn.png
The figure banked_turn.png is incorrect and should be removed or corrected. First, the figure shows both a centripetal and a centrifugal force acting on the same body, which cannot happen in any frame of reference. Second, the "vector sum of body forces" is shown as the resultant of the weight and the centrifugal force, and shown to be equal and opposite to the lift (and generally downward, which is obviously false). The only real forces acting on a plane are the weight and the lift, the latter larger by cos theta, and their resultant is the the centripetal force, directed horizontally. This figure could be corrected if the pink and red arrows were removed, and the yellow arrow shown as the sum of the green and blue.--74.104.101.121 (talk) 03:08, 23 December 2010 (UTC)


 * Why cannot the centripetal and a centrifugal force act on the same body in a frame of reference accelerating with the body? If they don't, then the body must accelerate relative to the accelerating frame, which seems pointless. Since the two body forces are weight and centrifugal force, they should equal the vector sum of body forces. Why should the sum be generally downward? Yes, your description would fit a non-accelerating reference frame, but that is not what the diagram uses. I don't necessarily like the image either, because it is not a correct free-body diagram, but then it doesn't claim to be. Perhaps you can create one. -AndrewDressel (talk) 03:31, 23 December 2010 (UTC)


 * I agree with 74.104.101.121. I don't agree with Andrew.  There are only two forces acting on the aircraft in the view in which it is shown - weight and lift.  The resultant of all forces is the centripetal force shown in yellow.  There is a centrifugal force in operation, but it is the horizontal component of the force on the atmosphere, not on the aircraft; this centrifugal force is the equal and opposite force predicted by Newton's Third Law of Motion.  A centrifugal force is shown in red, acting on the aircraft so the suffix apparent has been added.  I think showing this apparent centrifugal force serves to complicate the situation, not clarify it.  The force shown in purple and labelled vector sum of all body forces is actually the force acting on the atmosphere, caused by the aircraft generating lift, as predicted by Newton's Third Law.  It should not be shown as a force acting on the aircraft.


 * I think the diagram is trying to do too many things. It is commendably demonstrating the two forces acting on the aircraft, but in addition it is trying to show the relevance of an apparent centrifugal force and a vector sum of all body forces.  The best way ahead for Wikipedia is for the diagram to show only the two forces, and perhaps also the resultant of those two forces.  There are plenty of sources that can be referenced to allow independent verification of such a diagram.  Dolphin  ( t ) 03:48, 23 December 2010 (UTC)


 * The view is not described, so we can only infer which view it is by the forces shown. The presence of a centrifugal force implies that it is a view of the accelerating reference frame. In such a frame, aren't inertial forces acting on the body perfectly valid and even necessary for the body to be at rest with respect to the reference frame? As I state above, I don't necessary like the diagram, especailly because it includes both vectors and their sums, but I still don't see by what standards it is incorrect. Feel free to create and propose a replacement. -AndrewDressel (talk) 04:49, 23 December 2010 (UTC)


 * I agree that the reference frame is not described, so it should be an inertial reference frame. If it is not an inertial reference frame it needs to be stated.  The purpose of using a rotating reference frame, and showing an apparent centrifugal force, is to make use of D'Alembert's principle so that a problem in dynamics reduces to a problem in statics, and the vector sum of forces is zero.  However, that is not what is happening in this diagram which shows both the apparent centrifugal force and the centripetal force, and the vector sum of forces is not zero.
 * And what of the vector sum of all body forces. Should there not also be another vector called surface force?  Is there some significance to showing a body force and omitting surface forces?  I don't think there is a significance.  There is too much information and too little explanation on this diagram. I have proprosed a replacement - one which only shows the genuine forces, and perhaps also the resultant of these genuine forces.   Dolphin  ( t ) 06:23, 23 December 2010 (UTC)


 * Perfect. Sounds like concensus for replacing, not just deleting, the current diagram. Yes, the current one shows both too much and explains not enough. -AndrewDressel (talk) 14:03, 23 December 2010 (UTC)

The idea of the diagram was originally to illustrate the nomenclature of the various forces that may come to play in such scenario, but I agree that the result can be confusing to the uninitiated.

The vector sum of all body forces is a leftover of the from which the one in question has been derived (the original purpose was to show why the lift has to increase) and it is correct, as the only body forces (either real or apparent) are weight and centrifugal force, but that's irrelevant anyway.

I'm happy to edit the picture and remove centrifugal and purple resultant. I think the yellow resultant should be left though, as it is explicitly mentioned in the article (and indeed was agreed to be added to the first draft of the diagram).

--Giuliopp (talk) 17:52, 23 December 2010 (UTC)


 * Thanks Giuliopp. That is a very good proposal.  Dolphin  ( t ) 09:41, 25 December 2010 (UTC)

I see the picture has been edited. I would make one more request for editing the picture. Right now it shows three vectors, green for lift, purple for weight, and yellow for centripetal acceleration. There are a number of statements suggesting that there are only two forces, lift and weight, on the aircraft. I agree. The lift can be broken down into component vectors perpendicular and parallel to the lift vector. The centripetal acceleration is just the component of the lift vector perpendicular to the weight, meaning it is in the horizontal (constant altitude) plane. The other component is the one parallel to the weight, and it is not shown. I suggest two changes - first, show the components as dashed lines to indicate that they are just components of a force (lift), and second, indicate clearly that the component of lift opposite the weight must balance the weight out completely in order to maintain level flight. This will show the "centripetal force" as essentially an unbalanced force that is turning the aircraft in its circle. — Preceding unsigned comment added by SaintLouisEE (talk • contribs) 23:48, 10 September 2022 (UTC)

Frictionless banked turn
The first two equations in this section are wrong. Note that the equation "N cos(theta) = mg" should be "N = mg cos(theta). The posted equation implies that the normal force is always equal to or greater than the weight. If this equation holds true then as the bank angle approaches 90 degrees, the normal force approaches infinity, which is absurd.

The equation "N sin(theta) = v^2/r" is also wrong. It should be "mg sin(theta) = m v^2 cos(theta)/r".

The remaining equations are correct, however, the text needs to be changed to reflect the contents of the correct equations. Chasboson (talk) 19:08, 26 June 2014 (UTC)


 * Actually as the angle approaches 90° the force does approach infinity. It's not absurd but it has some interesting implications. At 90° the surface is vertical: a wall of death. What the equation says as the surface approaches 90° you need to go faster and faster to stay on it without friction. At 90° you can't: no speed will do it. So a wall of death cyclist needs some friction – quite a lot which is why they also need to go fast enough, as it's the speed which determines the normal force (and so the friction) by the other equation.-- JohnBlackburne wordsdeeds 19:31, 26 June 2014 (UTC)

The normal force does not increase with bank angle. It decreases. The equation for normal force has nothing to do with velocity. Neither "N = cos(theta)" nor "N cos(thata) = mg" is a function of velocity. It is the same on a vehicle resting on the road as on one moving. What does change is the centrifugal force. The task here is to balance that force with the component of the weight (not normal force) opposing it. I agree that at 90 degrees the object will fall, but that's because the normal force drops to zero and the weight prevails.

Chasboson (talk) 21:28, 26 June 2014 (UTC)
 * You are neglecting the fact that the object is not resting on the surface: it's moving along it - what you write is correct for an object resting on an inclined surface (which requires friction). The top half of the diagram shows this, while the bottom half is a cross-section view of the banked track. Because there's no friction the object (ball/car/bike) must go faster as the slope increases to not slide down it. And as it goes faster it presses down more on the surface: it's this plus the gravity which gives the normal force. E.g. you can rewrite the equations to give
 * $$N^2 = (mg)^2 + (\frac{mv^2}{r})^2$$
 * showing how they are added (as two sides of a right angled triangle, using Pythagoras's theorem) more explicitly.-- JohnBlackburne wordsdeeds 21:45, 26 June 2014 (UTC)

As I stated, the normal force has nothing to do with motion along the curve. The centrifugal force does. Evidently you either haven't read the page on "normal force" on Wikipedia, or you haven't had time to 'correct' it to match your formula. You certainly won't find the equation "N cos(theta) = mg" or the equivalent "N = mg/cos(theta)" there. I see a real mismatch here and I think it would be important to reconcile the differences. I've derived the equations for motion along a banked curve using the conventional and accepted formula for the normal force and arrived at the correct answer. The version here has two errors which cancel and still get the correct answer, but I don't think that is acceptable.

Chasboson (talk) 00:42, 27 June 2014 (UTC)


 * The equations on the Normal Force page relate to a specific example, "In a simple case such as an object resting upon a table". Those equations are different to this page because the forces are arranged in a different triangle. In the stationary example, there is a right angle between Friction and Normal Force, and Weight is the hypotenuse, so the weight is supported partly by friction and partly by normal force. In this case there is a right angle between Centripetal Force and Weight, and the Normal Force is the hypotenuse. The normal force supports the weight and provides the centripetal force as well. That's why in this case the normal force does in fact depend on the motion along the curve. Burninthruthesky (talk) 13:54, 27 June 2014 (UTC)

It's understandable how these cases can be confused as they look similar, but if you look closely the difference in both the maths and geometry is clear. To save me having to draw pictures here are two diagrams.


 * The first shows and inclined plane. The forces – the normal force, gravity and the inclined frictional force – are shown.
 * The second shows a banked turn. The forces – lift (equivalent to the normal force), gravity and the horizontal resultant centripetal force – are shown.

In both diagrams the forces form a right angled triangle. But the triangles are different.


 * In the first two of the triangle sides are at an angle, so gravity (mg) is the longest side, giving N = mg cos θ
 * In the second two of the sides are horizontal and vertical, so the normal force is the longest side, giving mg = N cos θ

Again this corresponds to the physics. On an inclined plane as the angle increases the normal force decreases as more and more of the weight is supported by friction. But on a banked curve as the slope increases the normal force goes up as the centripetal force increases. On e.g. a racetrack this corresponds to the steepest banked curves being at the sharpest/tightest corners. The sharper the corner the more centripetal force is required to make the turn, requiring more banking and more normal force.-- JohnBlackburne wordsdeeds 14:53, 27 June 2014 (UTC)

Well, it appears that our disagreement is over the definition of the normal force. I maintain that the normal force is found from "N = mg cos(theta)". You claim that the normal force is the sum of the gravitational component and the centripetal component. Where you write "N^2 = mg^2 + (mv^2/r)^2" I would write "(mg)^2 = N^2 + (mg sin(theta))^2". So if 'N' is affected by lateral forces, this may be a good place to introduce citations. In my physics texts the normal force is found by resolving the weight into two components: one along the surface and one perpendicular to it --- the perpendicular component is 'N', or rather the reactive perpendicular force is 'N'. Other forces may affect the vehicle, but they are in addition to and separate from the normal force. At least that is what I find in my texts.

Chasboson (talk) 18:28, 27 June 2014 (UTC)


 * Yes, it does seem that the definition of "normal force" is the root of the issue. The Wikipedia article you mentioned defines this as, "the component perpendicular to the surface". It goes on to give a specific example of how to calculate the normal force on an object resting on an inclined plane. That is not to say that all normal forces are calculated that way.


 * The article references Physics for Scientists and Engineers (which can be previewed on Google Books). p. 143 gives the equations for the forces on a banked roadway. As you suggest, I've added inline citations to the relevant equations in this article. Hope this clarifies. Burninthruthesky (talk) 10:55, 28 June 2014 (UTC)

There remains the difference between the two equations "N cos(theta) = mg" and "N = mg cos(theta). They can't both be right. I understand that you 'changed" the triangle, but didn't explain why. It certainly isn't necessary, because the desired result follow directly from the equation "N = mg cos(theta)". Moreover, changing the triangle causes the balancing of forces to occur along the horizontal axis rather than along the incline. Using "N = mg cos(theta)" results in the balancing to occur along the banked surface, which makes more sense. Under this approach the gravitational force pulling the vehicle downward along the surface is "mg sin(theta)" and the centripetal force pushing it up along the curve is "m v^2 cos(theta)/r". Equating these and dividing both sides by "mg cos(theta)" results in the desired relation. Simple, neat and unambiguous.

Interestingly, the normal force 'N' doesn't even appear in the result.

Chasboson (talk) 15:56, 28 June 2014 (UTC)


 * You write Using "N = mg cos(theta)" results in the balancing to occur along the banked surface, which makes more sense, but no, it doesn't. It's a frictionless banked turn. There is no friction. There is no force along the banked surface. There is only the normal force, the gravitational force, and the resultant horizontal centripetal force. Hence the different arrangement of forces around the triangle and the different formula.-- JohnBlackburne wordsdeeds 16:06, 28 June 2014 (UTC)

Hmmm. I realize the problem involves a frictionless surface. That's why the lateral forces need to cancel. I thought your first drawing above illustrated the forces nicely except that without friction the component of centrifugal force balances the component of gravitational force.

I'm willing to leave it at this point. I think there is a better way to derive the solution to the frictionless banked curve problem, but suppose getting the right answer is the main thing.

Chasboson (talk) 18:43, 28 June 2014 (UTC)


 * The equations, N cos(theta) = mg and N = mg cos(theta) are both correct in each of these two different situations.
 * For the stationary object, the normal force is simply the normal component of the weight.
 * For the turning object, the normal force is the normal component of the weight plus the normal component of centripetal force. You could write this as:


 * $$N = mg \cos \theta + {mv^2\over r} \sin \theta$$


 * This would effectively be splitting the triangle formed by N, weight, and centripetal force into two smaller triangles (one of those is identical to the triangle in the stationary example). Since the larger triangle already contains a right angle it's easier to deal with it as it is. Burninthruthesky (talk) 13:44, 29 June 2014 (UTC)


 * This gives a less direct way to get to the result. Doing it down the surface, i.e. considering the parallel components of the weight plus centripetal force, gives you


 * $$F = mg \sin \theta - {mv^2\over r} \cos \theta$$


 * But F is zero: it's frictionless. This lets you deduce the result I gave earlier. Simply square and add the two equations for for N and F . The trigonometric terms either cancel or drop out using the Pythagorean identity and you get


 * $$N^2 = (mg)^2 + (\frac{mv^2}{r})^2$$


 * Which shows the magnitude of N is greater than that of gravity, and the three forces are related as sides of a right angled triangle.-- JohnBlackburne wordsdeeds 14:25, 29 June 2014 (UTC)


 * Fascinating. Thanks for that. Burninthruthesky (talk) 17:19, 29 June 2014 (UTC)

It seems to me that taking the equation for 'F' and setting F to zero is the simplest solution to the original frictionless banked turn problem. The desired result can be found immediately with a little algebra from that. Doing it this way eliminates any need for the normal force. That makes sense because the normal force does not generate lateral components.

Chasboson (talk) 17:59, 29 June 2014 (UTC)

I decided to make one last post to recap and clarify my objections to the derivation of the 'frictionless banked turn' equation. On rereading this thread it is clear that misunderstandings, skipping around and switching topics has interfered with any cogent discussion. I want to focus on very specific points.

First, the equation:


 * $$ N \cos \theta = mg.$$

My complaint was that the normal force, as defined by this equation, will always equal or exceed the weight. As the bank angle approaches 90 degrees, N approaches infinity. I claimed that this cannot be correct. There is no mechanism for an object, solely under the influence of gravity, to exert forces of 2 time its weight or 5 times its weight etc. That's more like sorcery than physics.

When I first posted my objection, the response I got had to do with a 'wall of death' and various references to motion and velocity. However, this equation contains no term with a velocity component. It is unaffected by velocity because once the weight and bank angle are given, the normal force, N, involved is determined. It is fixed by those values and if the equation was correct, N would be the same for zero velocity or at the speed of light. I noticed that different equations for N were later invoked which included centrifugal force and which can increase N beyond the gravitational component, but no other forces are present in the equation under discussion, which merely purports to show the relation among three variables, none of which is velocity. Incidentally, for this equation it doesn't matter if the surface is frictionless or not. If sufficient friction is present, the object will exert the force and remain motionless. If the surface is frictionless, it will exert the same force and accelerate down the incline. It makes no difference if the object is in motion or not. The correct equation for the gravitational component of N is


 * $$ N = mg \cos \theta. $$

I also complained about the equation


 * $$ N \sin \theta ={m v^2 \over r} $$

This shows that the forces are being treated along the horizontal axis instead of along the surface of the banked turn. The logical place to resolve the forces is along the incline which is where the object is constrained to move. This is shown in the first figure above. The response I got to the objection was that it is a frictionless surface and "there are no forces" along the banked surface. I claim that there are two forces whose sum is to be set to zero. Note that their sum is zero, not that the forces disappear. If either of these forces disappears, the other will determine the acceleration of the vehicle up or down the incline. The gravitational force along the incline is


 * $$ mg \sin \theta. $$

The problem to solve is simply to find an equal and opposite force countering the gravitational force as is done by friction in the first figure above. Such a force, on a frictionless surface, would be available from the component of centrifugal force which lies along the incline, namely:


 * $$ {m v^2 \cos \theta \over r}. $$

This is exactly what was done in the recently posted equation for F. Equating these two forces and dividing both sides of the equation by


 * $$ mg \cos \theta $$

gives the desired result:


 * $$ \tan \theta = {v^2 \over gr}. $$

No detour through normals or switching triangles is necessary.

Other than that, the rest of the section is OK.

Chasboson (talk) 16:41, 8 July 2014 (UTC)


 * The only misunderstandings I've seen have been yours. In the discussion above, JohnBlackburne and I have repeatedly explained them to you.


 * The only forces in this situation are weight and normal force, which combine to produce centripetal force. Nobody has 'switched' triangles. The triangle of forces which exist can be used directly to confirm all the trigonometric relationships between its sides. For that reason I don't agree that the article would benefit from using the method you suggest.


 * Wikipedia policy requires a reliable reference to support any additions to articles. If you wish to improve this article I suggest you do some research rather than suggesting people who have tried to help you understand are advocating "sorcery" (Please see WP:Civil). Burninthruthesky (talk) 21:33, 8 July 2014 (UTC)

I don't doubt that there is misunderstanding on my side. But I don't think you ever explained how an object, solely under the influence of gravity, can generate forces greater than its weight. I gather you expect the centrifugal force to explain this, but it is not present in the equation I disputed. Neither is any term involving velocity. If you had been able to explain it, this thread would have stopped a long time ago. On the other hand, I understand that it is not your job to educate me, so I accept your frustration in responding. Still, I would like to see an explanation, as this would certainly improve the article.

Chasboson (talk) 15:57, 9 July 2014 (UTC)


 * Ok, I will try to answer your question. All the geometry stems from the fact that the forces must form a complete triangle in order for the forces to balance. You seem to be saying that
 * $$ N \cos \theta = mg.$$
 * does not contain the term v, so N cannot depend on v. In fact N does depend on v, because with g fixed, increasing v requires an increase of theta and an increase in N in order for the sides of the triangle to meet. This fact is expressed in the equation
 * $$ N \sin \theta ={m v^2 \over r} $$
 * You say you don't feel it's logical to resolve the components horizontally, but we can't change the fact that the centripetal force in this example is horizontal.
 * The turning object is not solely under the influence of gravity. Newton's second law dictates that an object must have centripetal force to keep it turning in a circle, which is how the equation above is derived.
 * You have referred again to the equations and the diagram from the Normal force page. These are not correct in the situation described on this page and I think you still have them confused. It is incorrect to say, "the gravitational component of N is N = mg cos (theta)", because N is defined as the total normal force, not just the gravitational component of the normal force.
 * Did you look at the reference I mentioned on | Google books? Perhaps a different explanation may help your understanding. I haven't spotted any obvious gaps in the explanation given in the article. Burninthruthesky (talk) 17:13, 9 July 2014 (UTC)

Did you accept the equation for F above as correct? It appeared to me that you did. I certainly do, and this may be the key to agreement, and the conclusion of this thread.

Chasboson (talk) 20:05, 9 July 2014 (UTC)


 * Yes, I accept the equation for F given above is correct, but I don't think it's an approppriate derivation for use in the article. As I said, the triangle of forces can be used to confirm all the relationships between its sides. That includes the tan (theta) relationship which is clearly illustrated, in fact follows directly, from the triangle of actual forces. Burninthruthesky (talk) 20:21, 9 July 2014 (UTC)

OK. I am no longer confused about your exposition. I understand why you dealt with the problem the way you did. However, I completely disagree with you. Thus, there is no longer any need to continue this session.

Thank you for maintaining civil discourse throughout. It is not only a Wikipedia policy, but brings honor to scientific debate.

Chasboson (talk) 16:40, 10 July 2014 (UTC)


 * Welcome to Wikipedia. I'm unsure from your message whether you simply disagree with my opinion on changing the article or you disagree with the way I dealt with the problem. I feel I should explain a little more about the operating principles of Wikipedia which may be unfamiliar to you.


 * Firstly, I did not write any of the text in that section of the article, nor could I claim any ownership. It has been written through the collaboration of volunteer editors like you and I. When you challenged the accuracy of the article, three of us discussed the validity of your claim. On your suggestion I added inline citations to improve the verifiability of the article.


 * During the continuing discussion it was demonstrated that the normal force is contributed to by centripetal force as well as gravity. You suggested the article be changed to use part of the calculation derived from this explanation. Original research is allowed in talk pages, but not in articles.


 * Although I agreed with JohnBlackburne that the method of resolving the forces along the slope is an indirect method of achieving the same relationships between the variables, you requested my, "agreement, and the conclusion of this thread". Editing decisions on Wikipedia are made by consensus, whereby an effort is made to satisfy all editors' legitimate concerns, but does not mean unanimity. I think we agree there is no consensus for change on this occasion.


 * Thank you for your contribution. I hope you find Wikipedia a pleasant experience. Burninthruthesky (talk) 09:01, 11 July 2014 (UTC)

Sorry I wasn't specific. I believe that the appropriate place to zero the forces is along the incline, where the vehicle is constrained to move. You said that was not appropriate and prefer to zero the forces elsewhere. That's a simple, specific disagreement which doesn't allow for compromise. Hence, my statement about terminating the session. I do agree that a three against one vote certainly trumps the one.

Chasboson (talk) 16:02, 11 July 2014 (UTC)

Normal force
In an inclined plane : the normal force N =mg cosQ where Q is angle of inclination. But as the law of banking of road says vertical component of normal force N cosQ=mg.The substituting the value of N we get mg cos^2 Q=mg! How can it be possible? Shivaram Raut (talk) 07:12, 20 February 2017 (UTC)


 * When an object is stationary on an inclined plane, the force normal to the surface of the plane is N = mg cosQ. There is also another force - parallel to the surface of the plane. It is the force of static friction and acts to prevent the object sliding down the inclined plane. This force of static friction is mg sinQ.
 * When an object is moving around an inclined, circular path (such as a cyclist riding on a banked velodrome) there is no force of static friction. In this dynamic case the normal force is mg secQ (or mg divided by cosQ). The horizontal component of the normal force is mg tanQ and is the centripetal force causing the object to follow a curved path. Dolphin  ( t ) 11:58, 20 February 2017 (UTC)

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Resolution?
I stumbled across this thread and believe the debate (argument) over the use of the normal force symbol 'N' can be resolved. In earlier physics, involving static forces, it is usual to state that N = mg cos(&theta;). This can elicit a classic difference between mathematicians and physicists because a mathematician is likely to regard this as a definition of N, rather than a current value. In such a case, it should be redefined at the beginning of this article. The confusion would not have arisen if 'N' was identified as a TOTAL force, which may be obvious to some, but which could be clarified by writing: N = Nt = Ng + Nv (vector addition). This way, Ng would be identified as the component of the normal force due to gravity and Nv would be identified as the component due to velocity.

I hesitate to suggest that anyone is 'right' or 'wrong' here, I just think some clarification would resolve the positions taken and would improve the article. — Preceding unsigned comment added by Reslox (talk • contribs) 18:15, 3 July 2019 (UTC)

Banked Turn in Aeronautics
I liked the development of the banked turn. I think the approach of starting out with a basic frictionless banked turn is a good way to teach this topic. I would like to suggest that we add a bit more on the aeronautics to describe the forces and what is necessary to achieve a banked turn. What is said in the text explaining that an aircraft needs to use its elevators or put its nose up to create more lift is correct, although we should add that any method that increases lift will do. The major piece that is missing here on the aeronautics side, although it goes a little beyond the banked turn, is that generally when lift is increased, drag is also increased. For the aircraft to keep at a constant speed an altitude, it is necessary that the aircraft apply more thrust to counteract the extra drag. The net result of this is that the engines need to increase their power to achieve the constant speed turn. I am hoping that someone who is an aerodynamicist will add in some description of this. One of the neat things about doing this is that, when in another article, someone wants to understand the flight envelope of an aircraft, they will be able to visualize the limits of maneuver capability - if you don't have enough excess thrust available in your engines to counteract the excess drag from the higher lift mechanisms needed for a banked turn, you will not be able to succeed in the constant speed maneuver. This would also be enhanced by a force diagram that would include drag as well as the lift and gravity vectors already in the figure. SaintLouisEE (talk) 23:27, 10 September 2022 (UTC)