# How can an aircraft turn if the horizontal force component is zero?

Here is the simplest coordinated turn diagram

Being:

$L$ = aerodynamic lift

$\tau$ = angle between vertical component and lift

$Fc$ = centrifugal "force"

In many books you can read that:

$L\cdot sin(\tau) = Fc = m \cdot (V^2)/R$ where $R$ is the curvature radius.

Then, how can an aircraft turn if there is no net force? I mean, from Newton's second law we have that $F = m \cdot a$ and in this case centrifugal force compensates for lift component in the horizontal plane.

• You skipped a step. The image depicts forces acting on a pendulum, not just the plane. You can't just take random images from the internet and expect them to make sense in a different context :) The two forces are weight and lift, and they combine to make an apparent force on the pendulum. The horizontal component of the lift produces an apparent force on the contents of the airplane, which changes the deviation of a suspended object from the weight vector (in this case, by exactly the same amount as the "floor" of the aircraft, which is the point of the original question). – Luaan May 16 '17 at 14:09
• That diagram doesn't what is applying which forces to what, – ratchet freak May 17 '17 at 8:19

The problem of the picture you are looking at is that both the actual and the apparent forces are shown.

The real force is the centripetal one (that in turn is only the horizontal component of the lift force), the centrifugal one does not exists, is only an "impression" of the centripetal one as seen by a person standing in the aircraft (a non inertial reference frame).

The aircraft flies in a circle because the centripetal component pulls the aircraft towards the center of the turn, the centrifugal force is only what a person in the aircraft feels, but is not an actual force acting on the aircraft, it is a representation of the inertia of the aircraft (and person inside).

• @XF-91 yes, any person inside the aircraft – Federico May 16 '17 at 11:48
• @XF-91 the only forces are aerodynamic forces (usually decomposed as lift and drag) and weight. Nothing else act on the airplane, and thus no other forces exist. – Manu H May 16 '17 at 12:16
• Mandatory XKCD: xkcd.com/123 – Drunix May 16 '17 at 12:53
• @Jenc In any case you cannot write the the centrifugal force and the centripetal forces as if acting on the same object because they are not. The centrifugal force acts on the plane and the centripetal force on the air. If both acted on the plane, the plane would indeed be in balance and not turn. – MichaelK May 16 '17 at 13:39
• Alternatively, when you do the math in the reference frame of the plane, the centrifugal force is exactly the one that prevents the plane from turning (since the plane can't turn in its own frame). To answer the question: the plane indeed does not turn if you consider the centrifugal force, because it means you have chosen to use its own reference frame. – TonioElGringo May 16 '17 at 14:10

### It is easier if we look only at the forces experienced by the aircraft, and in an inertial frame of reference

In this revised diagram, the vertical component of the lift balances the weight, which is vertical. There is a remaining horizontal component of the lift, and this causes the turning.

### "Centrifugal force" does not exist (and is not needed) in an inertial frame of reference

The problem with the original diagram in the question is it superimposes an imaginary force, the centrifugal force, upon the real list of forces on the aircraft.

It is hard for the general public to understand Newton's first law of motion, that any object tends to travel in a straight line when no force is acting it. It is hard for them to understand that motion in a circle is dramatically different from straight-line, constant-speed motion, since both seem (in a sense) steady or continuous.

### "Centrifugal force" is a term produced by humans to describe what they think must be happening

In the case of a passenger going round a corner in the car, this inward force is applied by the outer door of the car, against which the passenger ends up leaning during the turn. It is an inward (centripetal) force, but for the general public who is not aware of the special status of inertial frames of reference will likely use the car (which seems large and solid and therefore plausible) as a frame of reference and say "something must be pushing me against the door" rather than "the door is pushing me towards the centre of the circle".

Because of this use of a non-inertial frame of reference, the general public explanation needs to an extra force to explain why objects seem to accelerate towards the outer door of the car and the passenger feels pressed against it. In an inertial frame of reference it is the outer door of the car pressing in on the passenger, and accelerating towards any objects moving freely inside the car.

### In everyday life we sometimes use special frames of reference for convenience

We say the sun "comes up" in the morning, rather than the earth "rotates bringing the sun into view". This is because it is convenient for humans to use a non-inertial, but readily identifiable, frame of reference. It does not mean that people really think that the sun orbits around a stationary earth.

It is perfectly reasonable to imagine the sun travelling around the earth, for convenience, but if we want to make calculations about space travel with multiple objects (stars, planets etc) affecting a body's motion, it will get very complicated indeed to add in all the numerous centrifugal forces etc we would have to add in. It would be simpler to use an inertial frame of reference, and all objects moving relatively simply in relation to that.

Likewise we can learn to be sophisticated about circular motion. Let the general public speak of "centrifugal force" as this simplifies conversation, but when trying to calculate things, use an inertial frame of reference to avoid agony.

### Even apparently inertial frames of reference may have unconsidered rotation in relation to a larger-scale environment

Jan Hudec in the comments below brilliantly deflates the above criticism of rotating frames of reference by pointing out that even my notionally inertial frame of reference (the earth) is in fact rotating, so that the gravitational force measured is lower than it would be for a plane moving similarly above a non-rotating earth, i.e. is attenuated by a centrifugal component. Ouch!

And of course the earth is rotating around the sun, and the sun around the galaxy, etc:

Big fleas have little fleas,
Upon their backs to bite 'em,
And little fleas have lesser fleas,


So it is just a question of how much rotation one wants to accept as part of one's frame of reference. My suggestion is that when a questioner is feeling puzzled as the original poster was, a good way to break the problem down is to keep the frame of reference outside the rotation being studied, so that the studied rotation appears as a rotation: in this case the only acceleration (and therefore force on the orbiting object) will be centripetal.

• This has already been mentioned in a comment on another answer, but here it is again: xkcd.com/123 Randall Munroe appears to have studied physics beyond the level of a first-year college course. – David K May 16 '17 at 13:51
• @Federico The physics concept is to construct Newton's laws in a rotating frame of reference, as described in the second and third frames of the comic. That's not merely subjective feeling, but the mathematics is beyond what you will get in the usual popular science treatment. – David K May 16 '17 at 14:17
• I do think this is the best answer, because of the diagram, but I'm reluctant to upvote something that says "centrifugal force is a mistaken [...] concept" unconditionally (especially in large bold text). The xkcd comic people have been linking has it right: it's a perfectly valid concept, just one that only exists in noninertial reference frames. – David Z May 16 '17 at 15:25
• " Technically invalid reference frames" is equally misleading. Any reference frame is equally valid, technically and mathematically. Some are a bit more convenient, but that is engineering and not science. – MSalters May 16 '17 at 16:03
• I suppose that's good enough for an upvote, then. I do still think this answer is a little more dismissive of noninertial reference frames than it should be, but it's not inaccurate anymore in the sense that I thought it was before. – David Z May 18 '17 at 15:09

The diagram is not a correct force diagram in the inertial frame of reference.

In the inertial frame of reference, the horizontal force on the aircraft is unbalanced. This causes the aircraft to accelerate in the direction of the net horizontal force, perpendicular to the direction of flight, and this curves the path of the aircraft.

The law of "equal and opposite reaction" does not mean that every force on every object is balanced at all times. If the forces were always balanced, nothing would ever move. The aircraft could not leave its parking space, let alone get airborne and turn.

Physicists sometimes consider non-inertial frames in which there actually is a centrifugal force on the aircraft as shown in the figure. In order to construct such a frame of reference, however, the physicist assumes that the frame of reference is rotating relative to an inertial frame of reference. During a turn, the frame of reference attached to the aircraft is such a frame of reference, but only because the aircraft is turning.

In other words, the forces shown in the figure cannot all exist in an inertial frame of reference, and cannot exist in the frame of reference of an aircraft that is flying straight (without turning), but they do all exist in the non-inertial frame of reference of an aircraft that is turning. No matter how you look at it, a horizontal component of lift by banking the aircraft must be associated with a turn.

Lift and weight are acting on the plane. If the vector sum of lift and plane is not zero, the plane will accelerate.

When the plane is banked:

• the lift is not vertical
• the lift is not parallel to the weight
• the sum of lift and weight is not zero
• the sum of lift and weight has a horizontal component (centripetal force)
• the centripetal force causes the plane to turn

The concept of centrifugal force exists only in a rotating reference frame. Therefore the drawing with centripetal and centrifugal forces acting in the same reference frame is wrong.

• the aircraft is not a closed system. – Federico May 16 '17 at 13:29
• No, but air space is (as far as the relevant forces are concerned). – Jenc May 16 '17 at 13:34
• I have issues with your definition of "closed system" then – Federico May 16 '17 at 13:35
• You may name them. – Jenc May 16 '17 at 13:37
• a closed system is a system that cannot interact with anything outside of it. air interacts with everything around it. – Federico May 16 '17 at 13:44

The simplest and best answer to the question: An aircraft can't!

Well......with a few exceptions:

Blimps and airships turn using a rudder input just like their nautical cousins do.

Airplanes can be turned using only rudder inputs as well, thereby creating a centripetal component of thrust. This type of turning is far from efficient though and not practical for conventional maneuvering.

Also remember that centrifugal force is a reaction experienced by the aircraft and its occupants to the centripetal loading from the horizontal component of lift. There is no external centrifugal force being applied to the airplane pulling it against the turn as in the diagram.

• I think what you're saying is that an airplane cannot practically turn without banking. But I think OP is asking about a banked turn, claiming that the force diagram indicates that that won't work either. – David K May 16 '17 at 13:25