Inclination of road or surface other than flat
A
banked turn
(or
banking turn
) is a turn or change of direction in which the vehicle banks or inclines, usually towards the inside of the turn. For a road or railroad this is usually due to the roadbed having a transverse down-slope towards the inside of the curve. The bank angle is the angle at which the vehicle is
inclined
about its longitudinal axis with respect to the horizontal.
Turn on flat surfaces
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If the bank angle is zero, the surface is flat and the
normal force
is vertically upward. The only force keeping the vehicle turning on its path is
friction
, or
traction
. This must be large enough to provide the
centripetal force
, a relationship that can be expressed as an inequality, assuming the car is driving in a circle of radius
:
![{\displaystyle \mu mg>{mv^{2} \over r}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47053e0c382b5422f83326a0e065c359c43ac035)
The expression on the right hand side is the centripetal acceleration multiplied by mass, the force required to turn the vehicle. The left hand side is the maximum frictional force, which equals the
coefficient of friction
multiplied by the normal force. Rearranging the maximum cornering speed is
![{\displaystyle v<{\sqrt {r\mu g}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4bcfdf8b423d99f7f81bd1f0ee7ba874d4bcf64)
Note that
can be the coefficient for static or dynamic friction. In the latter case, where the vehicle is skidding around a bend, the friction is at its limit and the inequalities becomes equations. This also ignores effects such as
downforce
, which can increase the normal force and cornering speed.
Frictionless banked turn
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Upper panel: Ball on a banked circular track moving with constant speed
; Lower panel: Forces on the ball. The resultant or
net force
on the ball found by
vector addition
of the
normal force
exerted by the road and vertical force due to
gravity
must equal the required force for centripetal acceleration dictated by the need to travel a circular path.
As opposed to a vehicle riding along a flat circle, inclined edges add an additional force that keeps the vehicle in its path and prevents a car from being "dragged into" or "pushed out of" the circle (or a railroad wheel from moving sideways so as to nearly rub on the wheel
flange
). This force is the horizontal component of the vehicle's normal force (N). In the absence of friction, the normal force is the only one acting on the vehicle in the direction of the center of the circle. Therefore, as per Newton's second law, we can set the horizontal component of the normal force equal to mass multiplied by centripetal acceleration:
[1]
![{\displaystyle {mv^{2} \over r}=N\sin \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c90950d82bd087c2e54f9a0ac1b830d98235e4c0)
Because there is no motion in the vertical direction, the sum of all vertical forces acting on the system must be zero. Therefore, we can set the vertical component of the vehicle's normal force equal to its weight:
[1]
![{\displaystyle N\cos \theta =mg}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c58da4e77573441d1600dacb3b701f33c356e237)
Solving the above equation for the normal force and substituting this value into our previous equation, we get:
![{\displaystyle {mv^{2} \over r}={mg\tan \theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62b43abefd151095907a7bd2837ee1fc3eafb51c)
This is equivalent to:
![{\displaystyle {v^{2} \over r}={g\tan \theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7f72f2c0e75f891ec3888d612947e3de0da32e)
Solving for velocity we have:
![{\displaystyle v={\sqrt {rg\tan \theta }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f908405f2c74b4773a5975570655cad306203f62)
This provides the velocity that in the absence of friction and with a given angle of incline and
radius of curvature
, will ensure that the vehicle will remain in its designated path. The magnitude of this velocity is also known as the "rated speed" (or "balancing speed" for railroads) of a turn or curve.
[2]
Notice that the rated speed of the curve is the same for all massive objects, and a curve that is not inclined will have a rated speed of 0.
Banked turn with friction
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A steeply banked turn on a ramp connecting eastbound
California State Route 92
to northbound
Interstate 880
in
Hayward, California
.
Cyclists take a tight downhill corner in the Beanpot Criterium at Tufts University.
When considering the effects of friction on the system, once again we need to note which way the friction force is pointing. When calculating a maximum velocity for our automobile, friction will point down the incline and towards the center of the circle. Therefore, we must add the horizontal component of friction to that of the normal force. The sum of these two forces is our new net force in the direction of the center of the turn (the centripetal force):
![{\displaystyle \underbrace {{mv^{2} \over r}=N\sin \theta } _{\text{Frictionless formula}}+\underbrace {\mu _{s}N\cos \theta } _{\text{Friction term}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5079497e85f86452a1dabfe868d52a059973b23d)
Once again, there is no motion in the vertical direction, allowing us to set all opposing vertical forces equal to one another. These forces include the vertical component of the normal force pointing upwards and both the car's weight and the vertical component of friction pointing downwards:
![{\displaystyle \underbrace {N\cos \theta =mg} _{\text{Frictionless formula}}+\underbrace {\mu _{s}N\sin \theta } _{\text{Friction term}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6590cad872224a1d8250f0e78574832c157e3679)
By solving the above equation for mass and substituting this value into our previous equation we get:
![{\displaystyle {\frac {{\frac {N\cos \theta -\mu _{s}N\sin \theta }{g}}v^{2}}{r}}=N\sin \theta +\mu _{s}N\cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5993743108892071ba869cb6234a1587bcbcbaa2)
Solving for
we get:
![{\displaystyle v_{\mathrm {max} }={\sqrt {rg\left(\sin \theta +\mu _{s}\cos \theta \right) \over \cos \theta -\mu _{s}\sin \theta }}={\sqrt {rg{\frac {\tan \theta +\mu _{s}}{1-\mu _{s}\tan \theta }}}}={\sqrt {rg\tan(\theta +\theta _{\mathrm {crit} })}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40c1be94e1b99da208228cf3639f7ad760719cd5)
Where
is the critical angle, such that
. This equation provides the maximum velocity for the automobile with the given angle of incline,
coefficient of static friction
and radius of curvature. By a similar analysis of minimum velocity, the following equation is rendered:
![{\displaystyle v_{\mathrm {min} }={\sqrt {rg\left(\sin \theta -\mu _{s}\cos \theta \right) \over \cos \theta +\mu _{s}\sin \theta }}={\sqrt {rg{\frac {\tan \theta -\mu _{s}}{1+\mu _{s}\tan \theta }}}}={\sqrt {rg\tan(\theta -\theta _{\mathrm {crit} })}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61198275f27df381c2e1a2c17677dc133a5542f1)
Notice
![{\displaystyle {\frac {v_{\mathrm {min} }}{v_{\mathrm {max} }}}={\sqrt {\frac {\tan(\theta -\theta _{\mathrm {crit} })}{\tan(\theta +\theta _{\mathrm {crit} })}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a497ed75a04d875a47db7a1e00de5b55042bf06d)
The difference in the latter analysis comes when considering the direction of friction for the minimum velocity of the automobile (towards the outside of the circle). Consequently, opposite operations are performed when inserting friction into equations for forces in the centripetal and vertical directions.
Improperly banked road curves increase the risk of run-off-road and head-on crashes. A 2% deficiency in superelevation (say, 4% superelevation on a curve that should have 6%) can be expected to increase crash frequency by 6%, and a 5% deficiency will increase it by 15%.
[3]
Up until now, highway engineers have been without efficient tools to identify improperly banked curves and to design relevant mitigating road actions. A modern
profilograph
can provide data of both road
curvature
and
cross slope
(angle of incline). A practical demonstration of how to evaluate improperly banked turns was developed in the EU Roadex III project. See the linked referenced document below.
Banked turn in aeronautics
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Douglas DC-3
banking to make a left turn.
When a
fixed-wing aircraft
is making a turn (changing its direction) the aircraft must roll to a banked position so that its
wings
are angled towards the desired direction of the turn. When the turn has been completed the aircraft must roll back to the wings-level position in order to resume straight flight.
[4]
When any moving vehicle is making a turn, it is necessary for the forces acting on the vehicle to add up to a net inward force, to cause
centripetal acceleration
. In the case of an aircraft making a turn, the force causing centripetal acceleration is the horizontal component of the
lift
acting on the aircraft.
In straight, level flight, the lift acting on the aircraft acts vertically upwards to counteract the weight of the aircraft which acts downwards. If the aircraft is to continue in level flight (i.e. at constant
altitude
), the vertical component must continue to equal the weight of the aircraft and so the pilot must pull back on the stick to apply the
elevators
to
pitch
the nose up, and therefore increase the
angle of attack
, generating an increase in the lift of the wing. The total (now angled) lift is greater than the weight of the aircraft, The excess lift is the horizontal component of the total lift, which is the
net force
causing the aircraft to accelerate inward and execute the turn.
Vector diagram showing lift and weight acting on a fixed-wing aircraft during a banked turn. The yellow force depicted represents the net resultant force that causes centripetal acceleration.
Because centripetal acceleration is:
![{\displaystyle a={v^{2} \over r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa7d07a6d6f6ee15e06ff2d877b50deb9e9557c2)
During a balanced turn where the angle of bank is
the lift acts at an angle
away from the vertical. It is useful to resolve the lift into a vertical component and a horizontal component.
Newton's second law in the horizontal direction can be expressed mathematically as:
![{\displaystyle L\sin \theta ={mv^{2} \over r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/031d21680b44a050d7262cacb748848e2ca3ec79)
where:
is the lift acting on the aircraft
is the angle of bank of the aircraft
is the
mass
of the aircraft
is the
true airspeed
of the aircraft
is the radius of the turn
In straight level flight, lift is equal to the aircraft weight. In turning flight the lift exceeds the aircraft weight, and is equal to the weight of the aircraft (
) divided by the
cosine
of the angle of bank:
![{\displaystyle L={mg \over {\cos \theta }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23ce861c8a04d4abfb67f0e6fa3e16182685e033)
where
is the gravitational field strength.
The radius of the turn can now be calculated:
[5]
![{\displaystyle r={v^{2} \over {g\tan \theta }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7736a0611208c4e5267a9ec8572ecff44660247e)
This formula shows that the radius of turn is proportional to the square of the aircraft's
true airspeed
. With a higher airspeed the radius of turn is larger, and with a lower airspeed the radius is smaller.
This formula also shows that the radius of turn decreases with the angle of bank. With a higher angle of bank the radius of turn is smaller, and with a lower angle of bank the radius is greater.
In a banked turn at constant altitude, the
load factor
is equal to
. We can see that the load factor in straight and level flight is
, since
, and to generate sufficient lift to maintain constant altitude, the load factor must approach infinity as the bank angle approaches
and
approaches
. This is physically impossible, because structural limitations of the aircraft or physical endurance of the occupants will be exceeded well before then.
Banked turn in athletics
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Most
indoor track and field
venues have banked turns since the tracks are smaller than
outdoor
tracks. The tight turns on these small tracks are usually banked to allow athletes to lean inward and neutralize the
centrifugal force
as they race around the curve; the lean is especially noticeable on
sprint
events.
[6]
See also
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References
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]
- ^
a
b
Serway, p. 143
- ^
Beer, Ferdinand P.
; Johnston, E. Russell (July 11, 2003).
Vector Mechanics for Engineers: Dynamics
. Science/Engineering/Math (7 ed.). McGraw-Hill.
ISBN
978-0-07-293079-5
.
- ^
D.W. Harwood, et al.,
Prediction of the Expected Safety Performance of Rural Two-Lane Highways
, Turner-Fairbank Highway Research Center, McLean, Virginia, December 2000, page 39,
http://www.fhwa.dot.gov/publications/research/safety/99207/99207.pdf
- ^
Federal Aviation Administration (2007).
Pilot's Encyclopedia of Aeronautical Knowledge
. Oklahoma City OK: Skyhorse Publishing Inc. Figure 3?21.
ISBN
978-1-60239-034-8
.
- ^
Clancy, L.J, Equation 14.9
- ^
Greene, Peter (February 1987).
"Sprinting with banked turns"
.
Journal of Biomechanics
.
20
(7): 667?80.
doi
:
10.1016/0021-9290(87)90033-9
.
PMID
3654665
.
Further reading
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]
- Surface vehicles
- Serway, Raymond.
Physics for Scientists and Engineers.
Cengage Learning, 2010.
- Health and Safety Issues
, the EU Roadex III project on health and safety issues raised by poorly maintained road networks.
- Aeronautics
- Kermode, A.C. (1972)
Mechanics of Flight
, Chapter 8, 10th Edition, Longman Group Limited, London
ISBN
0-582-23740-8
- Clancy, L.J. (1975),
Aerodynamics
, Pitman Publishing Limited, London
ISBN
0-273-01120-0
- Hurt, H.H. Jr, (1960),
Aerodynamics for Naval Aviators
, A National Flightshop Reprint, Florida
External links
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- Surface vehicles
- Aeronautics
https://edu-physics.com/2021/05/08/how-banking-of-road-will-help-the-vehicle-to-travel-along-a-circular-path-2/