eloquent logic - circular motion

Legend

a_c	=	centripetal acceleration
v	=	velocity
r	=	radius
F_c	=	centripetal force
m	=	mass
F_N	=	normal force
F_f	=	force of friction
F_f	=	force of gravity

Creating a banked road makes it easier for the cars to turn by providing more traction. The centripetal force is now provided by both friction and normal force.

Banked Curves

It is possible to bank a roadway so that the normal force provides the centripetal force rather than relying on friction to provide it. Below diagram shows that the normal force (F_N) acting on a runner has components both vertical and horizontal.

Image of a car on a banked curved surface, like a racing track often is

With no vertical acceleration:

F_y = F_Ncos - F_g = 0

There is a horizontal acceleration, so:

F_x = ma_x

With a_H = a_c:

F_Nsin = F_c = mv²/r

To solve for is to combine the two into a single expression in terms of tan. Accordingly, write the first equation as:

F_Ncos = F_g = mg

Divide this into the previous formula:

(F_Nsin) / (F_Ncos) = (mv/r) / mg

Results:

tan = v²/gr

this expression give the proper banking angle for anyone speed, despite the mass of the object. Any car can make the posted speed safely, which is the reason racing ramps are banked. It is shown by the equation that greater the banking angle, the larger tan is and greater the speed may be.

Level Curves

Interactive Lab