Cars Traveling Around a Banked Curve (no friction)
NOTE: In these types of problems, always choose a coordinate system such that x is parallel or antiparallel to ar.
Ex.
Find the exact speed a car of mass m traveling along a banked curve (whose path is the shape of a
circle of radius r) must have in order to make the curve without sliding up or down the incline.
N
Nx
Ny
θ
ar
v2
ar =
r
N x = N sin θ
N y = N cos θ
or
θ
W
Determine the motion in each direction using Newton’s 2nd law and the force diagram.
∑ Fx = −mar
∑ Fy = 0
− N x = − mar
N y −W = 0
Substituting & solving for the Normal Force:
 v2 
− N sin θ = − m  
 r 
2
mv
N=
r sin θ
N cos θ − mg = 0
N=
mg
cos θ
Equating the two expressions for the Normal force:
mv 2
mg
=
r sin θ cos θ
v2 =
rg sin θ
cos θ
v = rg tan θ
In terms of the angle:
v2
tan θ =
rg
W/o friction, this is the restriction on the speed of the car to go
around a banked curve without sliding up or down the incline.
If vcar > v, the car will slide up the incline
If vcar < v, the car will slide down the incline