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Wize University Physics Textbook (Master) > Circular Motion

Banked Roads

Centripetal Acceleration and Centripetal ForcePrevious Section
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Car Turning Around a Circular Section of Road



  • If the road is flat, then the centripetal force is equal to the force of friction on the car.
  • If there is no friction, there is no force that can supply the centripetal force required to make the car move in a circular path - there is no way that the car can turn!
  • If the road is banked, the horizontal component of the normal force provides the centripetal force.


Wize Concept
If a car is turning around a curve without friction, then the curve must be banked at some angle θ\theta(in radians) in order for the car to not slide out of the turn.





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Example: Car in a Flat Ramp


A car is negotiating a 160160 m diameter flat ramp and is about to slip off the road with speed of 9090 km/h. What is the coefficient of static friction between the tires and the road?

A) 0.80.8 B) 0.60.6 C) 0.40.4 D) 0.150.15



First, let's convert the speed: 90 kmhr=90⋅1000 m3600 s=25 ms90 \ \dfrac{km}{hr}=90\cdot\dfrac{1000\ m}{3600\ s}=25\ \dfrac{m}{s}

The only force toward the centre of this circular path is the static friction between tires and the road.

Since the car is about to slip off, it means that we are talking about the maximum static friction (the car is on the verge of sliding).

So, the centripetal force is equal to maximum static friction:


Horizontal direction: ∑F=mac\sum F=ma_c

Fstatic, max=mv2rF_{static, \ max}=\dfrac{mv^2}{r}
μsN=mv2r\mu_sN=\dfrac{mv^2}{r}

Vertical direction: ∑F=0\sum F=0 since there is no motion along this direction
NB−mg=0N_B-mg=0
N=mgN=mg

Combining the equations we get:

μsmg=mv2r\mu_s\cancel mg=\dfrac{\cancel mv^2}{r}

μs=v2rg\mu_s=\dfrac{v^2}{rg}

=25280×9.81=0.8=\dfrac{25^2}{80\times9.81}=0.8


A racing car is moving on a circular banked track with radius of 200200 m at a constant speed. The driver feels a normal force 3030 percent heavier than his true weight. What are the angle of the bank and the speed of the racing car?


Extra Practice
Problems on Centripetal Force
Practice: Banked Road and Centripetal Force
A curve with a 120-m radius on a level road is banked at the correct angle for a speed of 20 m/s.
(a) If an automobile rounds this curve at 30 m/s, what is the minimum coefficient of static friction needed between tires and road to prevent skidding?
A 20 m radius curve is banked in such a way that vehicles can drive on the road at speeds up to 65 km/h, and make the turn even when there is no friction. At what angle should the curve be banked? (Enter your answer in degrees, two sig figs)
Problems on Centripetal Force
(Heads up - this is a tougher problem! If you can follow it start to finish, then you are doing great :) )
A curve with a 120-m radius on a level road is banked at the correct angle for a speed of 20 m/s.
(a) If an automobile rounds this curve at 30 m/s, what is the minimum coefficient of static friction needed between tires and road to prevent skidding?
Consider a banked road where there is a coefficient of static friction of 0.30 and a coefficient of kinetic friction of 0.25 between the tires and the roadway.The radius of the curve is R =50 m. (a) If the banking angle is b=25∘b=25^{\circ}, what is the maximum speed the automobile can have before sliding up the banking ? (b) What is the minimum speed the automobile can have before sliding down the baking?
Enter the numerical answer with no units as your answer (units are m/s)
A curve with a 120-m radius on a level road is banked at the correct angle for a speed of 20 m/s. If an automobile rounds this curve at 30 m/s, what is the minimum coefficient of static friction needed between tires and road to prevent skidding?
A bus moving at 40 km/h negotiates a 130m radius banked turn designed for 60 km/h. What coefficient of friction is needed to keep the bus on the road?
If there is no friction between the tires and the road at what angle should the road bed be banked so that a car traveling at 30 m/s can safely drive without sliding if the radius of the curve is 100 m?
A turn around has a radius of 70 meters and a banking angle of 10o. What is the critical, speed (the speed for which no friction is required between the car's tires and the surface) for a car on this curve?
A 20 m radius curve is banked in such a way that vehicles can drive on the road at speeds up to 65 km/h, and make the turn even when there is no friction. At what angle should the curve be banked?
A turn around has a radius of 70 meters and a banking angle of 10o. What is the critical, speed (the speed for which no friction is required between the car's tires and the surface) for a car on this curve? (Pick the closest number)