How to use the centripetal force and banked curve calculator
- Choose Force and acceleration, Ideal banking angle or Safe speed range.
- Enter the speed and the turn radius in the units you prefer. For forces, enter the mass too.
- For the speed range, enter the banking angle and a tire friction coefficient.
- Read the answer and the steps that show the formula with your numbers.
Formula
Centripetal acceleration a = v^2 / r; force F = m v^2 / r
Ideal banking angle: tan(theta) = v^2 / (g r)
Highest speed: v^2 = g r (sin t + mu cos t) / (cos t - mu sin t)
Lowest speed: v^2 = g r (sin t - mu cos t) / (cos t + mu sin t), or 0 if negative
g is 9.80665 m/s^2, mu is the friction coefficient between tires and road and t is the banking angle. If cos t - mu sin t is zero or negative, friction alone can hold the car at any speed.
Force, angle and speed range
A 1200 kg car going 15 m/s around a 50 m radius needs a centripetal acceleration of 15^2 / 50 = 4.5 m/s^2, which is 0.459 g, and a force of 5400 N. A lap of that circle takes about 20.94 seconds. On a flat road the tires need a friction coefficient of at least 0.459.
To go 20 m/s around a 100 m curve with no help from friction, the bank must satisfy tan(theta) = 400 / 980.665 = 0.4079, so theta = 22.19 degrees. On a 15 degree bank with a friction coefficient of 0.5 and a 100 m radius, the car can travel from rest up to 106.16 km/h before sliding outward.
Why a bank helps
A car in a turn needs a force toward the center of the circle. On a flat road only tire friction provides it. Tilting the road toward the inside lets part of the normal force from the road point inward, so less friction is needed. At exactly the ideal angle the car needs no friction at all, and a slower or faster car must rely on friction to avoid sliding down or up the bank.
Compare with straight-line motion in the kinematics calculator, collisions in the collision calculator, and boat speed limits in the hull speed calculator. Average speeds are in the average speed calculator.
Using the safe speed range
The highest speed is the point where friction is at its limit pushing the car down the bank; the lowest is where friction holds the car up it. Between them the car stays on the road. The numbers depend a lot on the friction coefficient, which is much lower on a wet or icy surface, so use a value for the real conditions.
Assumptions and limits
- The vehicle is treated as a point mass moving at a constant speed on a circle of constant radius.
- Air drag, lift, suspension, tire slip angles and load transfer are ignored.
- g = 9.80665 m/s^2. Speeds, radii and masses can be entered in common units and are converted for you.
- This is a physics exercise, not a safety rating for any road, and it is not a recommended speed.
Frequently asked questions
How do I calculate centripetal force?
Multiply the mass by the square of the speed and divide by the radius. A 1200 kg car at 15 m/s on a 50 m radius needs 1200 x 225 / 50 = 5400 newtons.
What is the ideal banking angle?
It is the angle where the car needs no friction: tan(theta) = v^2 / (g r). It does not depend on the mass of the car, only on the speed and the radius.
What happens above the highest speed?
Friction can no longer hold the car on the road and it slides outward, up the bank. The highest speed rises with the friction coefficient and with the bank angle.
Why is there sometimes no highest speed?
If the friction coefficient times the tangent of the bank is at least 1, friction can hold the car against the bank at any speed in this model. The page then says there is no limit.