Banked Curves
A car goes round a curve on a banked road. Find the speed at which it needs no friction, set it, then start the simulation to test your answer.
Swipe the picture sideways to see all of it.
Your task: work out the speed at which the car needs no friction to stay on the bank for the angle and radius you have chosen. Type it in, then start the simulation. Use g = 9.8 m s-2. Remember that the car is not in equilibrium: its net force points to the centre of the curve.
Show the answer and the working
Without friction, only the weight and the normal force act on the car. The normal force is perpendicular to the road, so it leans towards the centre of the curve.
Vertical: N cosθ = mg
Horizontal (towards the centre): N sinθ = mv2/r
Divide: tanθ = v2/(rg), so v = √(rg tanθ)
The mass cancels, so the design speed is the same for every car. Faster than this, the car tends to slide up the bank (to the outside). Slower, it tends to slide down (to the inside).
With friction, the fastest and slowest speeds with no slipping are:
vmax = √[rg(tanθ + μ)/(1 - μ tanθ)] and vmin = √[rg(tanθ - μ)/(1 + μ tanθ)]
Learn more in the Circular motion applications lesson.