# Car on banked turn - net force

beginner16
hello

Suppose there is no friction between the car and the road -> if car is on a banked turn , the normal force ( which is always perpendicular to the road's surface ) is no longer vertical . The normal force now has a horizontal component , and this component can act as the centripetal force on the car

But doesn't gravity force have a component that is of equal size , but in opposite direction than normal force ? In which case net force pulling on car would be zero and thus couldn't act as centripetal force ?
So how can net force equal horizontal component of normal force ?

Also , what must be angles between the components of normal force and how are they related to the banking angle between the road and the horizontal ?

thank you

Homework Helper
beginner16 said:
hello

Suppose there is no friction between the car and the road -> if car is on a banked turn , the normal force ( which is always perpendicular to the road's surface ) is no longer vertical . The normal force now has a horizontal component , and this component can act as the centripetal force on the car

But doesn't gravity force have a component that is of equal size , but in opposite direction than normal force ? In which case net force pulling on car would be zero and thus couldn't act as centripetal force ?
So how can net force equal horizontal component of normal force ?
Gravity is equal to the normal force if the road is horizontal. If you bank the road, the component of gravity that is perpendicular to the road surface is reduced but gravity now has a component that is parallel to the surface (ie pulling it sideways). The horizontal centripetal force has a component that is perpendicular to the road and a component that is parallel to the road. If there is no parallel or perpendicular motion, the parallel forces must sum to 0 and the perpendicular forces must sum to 0.

Resolve all forces into components parallel to and perpendicular to the surface of the road. The perpendicular components are: $\vec{N}$ being the Normal force (pointing upward), $mgsin\theta$ and $cos\theta mv^2/R$ where $\theta$ is the angle of the road surface to the vertical. The parallel components are $mgcos\theta$ and $sin\theta mv^2/R$

Since there is no motion in either the parallel or perpendicular direction. So:

(1)$N - mgsin\theta - cos\theta mv^2/R = 0$ and

(2)$mgcos\theta - sin\theta mv^2/R = 0$

AM

Homework Helper
There is a "Normal Force" by a surface because
(after compressing a bit, if it doesn't break,)
it reacts to keep the object from sinking into it.

in a jump, while you straighten your legs,
you think the "Normal Force" is equal mg ?
If so, how would anything get vertical speed?
Standing up needs N>mg ; tossing a pencil ...

In the case of a banked road, the Surface Force
is "convinced" to become stronger by curving the road
as well as banking it. The road then must push inward
so the car doesn't go in a straight line thru the road.

AndrewM presumed yours is level automatically.
Here it is clearest to use axes horizontal & vertical.

The vertical Forces are gravity (mg) and enough
surface Force to so the car doesn't sink in (Ncos theta).
The horizontal Force is only Nsin(theta).

The driver wants Nsin(theta) to provide mv^2/R .
But with no friction, the road must push perp to its Area.

(A.M. has an extra "N")

beginner16
Uh , could you pretend I'm mildly retarded and please break it into little steps ?! I'm sooo confused I don't even know exactly what to ask ... but I will try

In the case of a banked road, the Surface Force
is "convinced" to become stronger by curving the road
as well as banking it. The road then must push inward
so the car doesn't go in a straight line thru the road.<
So somehow forces are alligned in such way that net force becomes centripetal force ?!

The vertical Forces are gravity (mg) and enough
surface Force to so the car doesn't sink in (Ncos theta).
The horizontal Force is only Nsin(theta).

You mean forces perpendicular to road - wouldn't that be a normal force and a component of gravity force ?

But if normal force is perpendicular to surface area and at the same time it is greater than component of the opposite gravity force also perpendicular to surface area , then resulting vector should point upwards and perpendicular to surface ?! In that case car would start moving to the side and also horizontally

The driver wants Nsin(theta) to provide mv^2/R .

Theta being angle of the road surface to the vertical ?

Resolve all forces into components parallel to and perpendicular to the surface of the road. The perpendicular components are: $mgsin\theta$ and $cos\theta mv^2/R$ ... The parallel components are $mgcos\theta$

I got it just the other way around . For component of gravity force parallel to the road I got sin(angle)*m*g and component perpendicular to road I got cos(angle)*m*g .

Since there is no motion in either the parallel or perpendicular direction. So:

(2)$mgcos\theta - sin\theta mv^2/R = 0$

AM
In any case , notice that components of both centripetal force and gravity force parallel to the road have the same direction so I don't see how they could cancel each other out

(1)$N - mgsin\theta - cos\theta mv^2/R = 0$

Dont's the parallel components of gravity force and centripetal force have opposite directions ?

beginner16
nevermind my initial questions . I forgot that since N is greater in magnitude its vertical component also has component perpendicular to the roaqd , but it's not of the same size as normal force

My question now would instead be what must an angle between banking road and horizontal be in order for all this to work ( assuming there is no friction on the road ) ?

Mentor
There are only two forces acting on the car: the normal force and the weight. Set up the equations of motion, noting that the car accelerates centripetally:
(1) Vertical forces must add to zero
(2) Horizontal forces must provide the centripetal force

Combine these equations to solve for the angle of banking for a given speed and curve radius.

Homework Helper
Gold Member
Dearly Missed
Since you call yourself "beginner", it will be a quite educative experience to try the following approach after you've followed Doc Al's approach and solved the problem in that manner:
$\vec{F}=m\vec{a}$ is valid however you choose to decompose your vectors.
Thus, instead of decomposing forces and accelerations in the vertical and horizontal directions, try to decompose them in the directions normal to and tangential to the banked turn.
Your answers will, of course, agree (at least if you do it correctly).

beginner16
got it

I hope you can help me with just one more question

It seems that on banked turn (no friction) the resulting force doesn't depend on the speed of the car

F=m*g*tan (angle)

that would suggest if car goes into banked turn with speed less than necessary for centripetal force at that angle, car will be thrown in a direction of force (meaning it won't succed in making a turn) ?

thank you very much for all your help

Homework Helper
if a car enters a steeply-banked road way too slow,
then it can slide down the bank ... I've seen cars
on icy roads do this if drivers are "too cautious".

If the driver is too aggressive, the car slips the bank
and they end up on the outside of the curve.
did you ever play with "slot-car" racers?

It is certainly BEST in statics and dynamics questions
to begin with a Free-Body-Diagram, then
add the Force vectors tail-to-tip so that the
resultant points in the direction of the acceleration.
|- - - ->/
| . . . /
| . . /
|. . /
|. /
|/
That way, you can more wisely choose axis directions
(trying to do as few vector components as possible)

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beginner16
Though I'm a bit confused why on banked turn (no friction) the resulting force doesn't depend on the speed of the car . I mean , common sense would suggest otherwise , wouldn't it ?

Homework Helper
beginner16 said:
Though I'm a bit confused why on banked turn (no friction) the resulting force doesn't depend on the speed of the car . I mean , common sense would suggest otherwise , wouldn't it ?
I am not sure why you think that it is independent of speed. The centripetal force that is required to make the turn has to be supplied by the road and gravity:

(1) $$Nsin\theta = mv^2/r$$

The vertical component of the normal force is supplied by the road to counter gravity:

(2)$$Ncos\theta = mg$$

Substituting 2 into 1:

$$mg\frac{sin\theta}{cos\theta} = mv^2/r$$

$$v = \sqrt{rgtan\theta}$$

Any speed higher than this, the car slips up the bank. Any speed lower than this, the car slides down the bank.

AM

Mentor
beginner16 said:
Though I'm a bit confused why on banked turn (no friction) the resulting force doesn't depend on the speed of the car .
Since the net force is a centripetal force, it certainly depends on the speed. ($F_{net} = m v^2/r$) The greater the speed, the greater the required centripetal force. (And, as lightgrav and Andrew Mason explain, the banking angle is tuned to a particular speed. Change the speed and you'll need a new angle.)

beginner16
What I was trying to say is that if at particular angle of a banked curve car has certain mass m , and radius is R then net force (centripetal force ) will have the same magnitude (say 10 N) , no matter how fast car is going .
So whether car has speed of 100 km/hour or 200 km/hour , centripetal force will be the same in both cases ( 10 N ) . Only difference is that at say 100 km/hour centripetal force will have just the right magnitude to keep the car on the road , while at speed 200 km/hour car will go off

So in short , I was implying that the magnitude of centripetal force doesn't depend on car's velocity , but on its mass , radius and angle of a banked curve . But in order for car to be able to drive trought that particular curve car has to have just the right speed

or am I wrong ?

beginner16
Not to be pushy , but could you at least tell me if , again , I'm totally off with my theory made in my las post ?

Mentor
I would say that you your statements were not totally correct. The bank angle is set so that a car can traverse the curve with zero friction and without sliding at a particular speed. If you go faster, the net force will change: the centripetal component will increase and, in addition, there will be a component of force accelerating you up the incline. (But I'll have to give this some more thought.)

Homework Helper
beginner16 said:
What I was trying to say is that if at particular angle of a banked curve car has certain mass m , and radius is R then net force (centripetal force ) will have the same magnitude (say 10 N) , no matter how fast car is going .
So whether car has speed of 100 km/hour or 200 km/hour , centripetal force will be the same in both cases ( 10 N ) . Only difference is that at say 100 km/hour centripetal force will have just the right magnitude to keep the car on the road , while at speed 200 km/hour car will go off

So in short , I was implying that the magnitude of centripetal force doesn't depend on car's velocity , but on its mass , radius and angle of a banked curve . But in order for car to be able to drive trought that particular curve car has to have just the right speed

or am I wrong ?
The centripetal force is supplied by the horizontal component of the normal force. Centripetal force is always proportional to speed. I think you are overlooking the fact that the normal force supplied by the road IS speed dependent. This can be best seen if the bank angle is 90 degrees (vertical road surface). In such case, the normal force is equal to the centripetal force as there is no vertical component. As speed increases, the normal force / centripetal force increases (as the square of the speed).

With an angle less than 90 degrees, the normal force has an upward vertical component as well so there is only one speed at which that upward component equals the force of gravity.

AM

beginner16
Andrew Mason said:
The centripetal force is supplied by the horizontal component of the normal force. Centripetal force is always proportional to speed. I think you are overlooking the fact that the normal force supplied by the road IS speed dependent.

AM

But if you look at the formula for centripetal force at particular angle of banked road

F = m*g*tan(angle)

you will noticed magnitude of force depends only on size of an angle and mass of an object

If force was dependant on speed at certain angles then I assume it would be possible to drive trough the banked road with any speed since centripetal force would increase accordingly ?

Mentor
beginner16 said:
But if you look at the formula for centripetal force at particular angle of banked road

F = m*g*tan(angle)

you will noticed magnitude of force depends only on size of an angle and mass of an object
That's only true for the speed at which the banking was designed for. (The speed that satisifies the relationship $\tan \theta = v^2/(rg)$.)

If force was dependant on speed at certain angles then I assume it would be possible to drive trough the banked road with any speed since centripetal force would increase accordingly ?
But if you go too fast you would not be able to negotiate the turn due to sliding off the road.

beginner16
I understand this from the formula's point of view , but somehow I can't get it into my head rationally . I mean

m*g*tan(angle) = m*v^2/r

Left sides equal right side exactly when ...

I don't see anything on the right side of equation that would predict the magnitude of horizontal component of normal force when car has certain speed

prob am not making much sense

Doc Al said:
But if you go too fast you would not be able to negotiate the turn due to sliding off the road.

exactly what kind of force does slide the car off the road if speed is too great
?

Homework Helper
beginner16 said:
But if you look at the formula for centripetal force at particular angle of banked road

F = m*g*tan(angle)
But this is not the formula for centripetal force. Where did you get this?

The centripetal force is given by $f = mv^2/r$. It is supplied by the horizontal component of the normal force. The normal force is not supplied by gravity alone. The car pushes on the road (trying to go in a straight line) and the road pushes back. If the car is making the turn and not going off the road, the normal force will 'self adjust' so that the horizontal component is equal to the centripetal force. If there is no friction, this occurs only at one speed for a given angle.

AM

beginner16
Andrew Mason said:
But this is not the formula for centripetal force. Where did you get this?

AM

F - centripetal force on banked road with no friction

F = m*v^2/r and F = m*g*tan(angle)

m*v^2/r = m * g * tan(angle)
tan(angle) = v^2 / r*g etc

Mentor
beginner16 said:
But if you look at the formula for centripetal force at particular angle of banked road

F = m*g*tan(angle)
That is only equal to the centripetal force if the car is going at the designed speed.

beginner16
Like I've said I do understand this from the formula's point of view .

But in my opinion data on this is few and far between

-Neither F = m*v^2/r or F = m*g*tan(angle) doesn't tell you why a banked curve gives you the exact right centripetal force only if you go with speed v . Why doesn't it give you that much greater centripetal force needed for the car to drive trough if you go with speed twice as v .

-And even more importantly , why doesn't m*g*tan(angle) equal m*(v + 1)^2/r instead of m*v^2/r ?

Homework Helper
beginner16 said:
Like I've said I do understand this from the formula's point of view .
I am not sure you do.

$mv^2/r = mg tan\theta$ is not the 'formula' that enables you to find the centripetal force for any speed. It is a condition that must apply if the car is to make the turn: "If v is such that $mv^2/r = mg tan\theta$, then the car will make the turn and stay on the road." and "If $mv^2/r > mg tan\theta$ then the car will not make the turn because the road cannot supply sufficient centripetal force".

But in my opinion data on this is few and far between
It has nothing to do with 'data'. It is an application of Newton's laws of motion which are well established by experiment.

-Neither F = m*v^2/r or F = m*g*tan(angle) doesn't tell you why a banked curve gives you the exact right centripetal force only if you go with speed v . Why doesn't it give you that much greater centripetal force needed for the car to drive trough if you go with speed twice as v .
Because F = m*g*tan(angle) is not the formula for centripetal force. It is a particular relationship that applies only where $v = \sqrt{rg tan\theta}$

You really should stop and try to understand what we are saying here because your questions indicate that you have missed something rather basic here.

AM

beginner16
just one more time -

If banked curve (again no friction )has an angle theta , then only possible force pointing in direction to the center of circle must have a magnitude of m*g*tan(theta) . Right?

And in order for net force to have that direction and magnitude net force also has to be equal to m*v^2/r

The part that confused me was that I didn't consider that if car was able to drive trough the curve with certain speed ,then that speed had to be v since net force ( m*g*tan(theta) ) F will only move an object in circular motion if F also equals m*v^2/r .

I also thought that if car goes with speed v+x then centripetal force will be just that much greater , but since only possible force at that angle having the direction towards the middle of a circle (ie centripetal force) can only have a magnitude of m*g*tan(theta)

Is my reasoning correct ? Are there any other logical clues that would help me reach the same conclusion?

Homework Helper
beginner16 said:
just one more time -

If banked curve (again no friction )has an angle theta , then only possible force pointing in direction to the center of circle must have a magnitude of m*g*tan(theta) . Right?
Wrong. The force pointing in the direction of the centre of the circle is $Nsin\theta$. N depends on the speed as well as the angle.

AM

beginner16
Andrew Mason said:
Wrong. The force pointing in the direction of the centre of the circle is $Nsin\theta$. N depends on the speed as well as the angle.

AM

I don't understand your reply . N*sin(theta) has same in magnitude as m*g*tan(theta) . If horizontal component of N doesn't equal m*g*tan(theta) then net force won't be directed towards the center of the circle

Only in the case that $mv^2/r = mg tan\theta$. The point here is that, as a general rule, $Nsin\theta \ne mv^2/r$. They are equal only where $v = \sqrt{rg tan\theta}$
That is right. It points to the centre of the circle only where the car actually moves in the circular path defined by the curvature of the road. That only occurs for one speed: $v = \sqrt{rg tan\theta}$.