Cars Traveling Around a Banked Curve (w/ friction)
Ex.
Find the maximum speed a car of mass m traveling along a banked curve (whose path is the
shape of a circle of radius r) can have in order to make the curve without sliding up the incline.
Nx
Potential Sliding Motion
N
Ny

N y  N cos 
fx
f x  f s cos    s N cos 
fy
fs
v2
r
N x  N sin 
ar
or
ar 

W
f y  f s sin    s N sin 
Determine the motion in each direction using Newton’s 2nd law and the force diagram.
 Fx  mar
 Fy  0
 N x  f x  mar
N y W  f y  0
Substitute & solve for the Normal Force from the y component:
 v2 
N cos   mg   s N sin   0
 N sin    s N cos    m  
 r 
 v2 
N  sin    s cos    m  
 r 
N  cos    s sin    mg
N
mg
cos    s sin 
Substitute the expression for the Normal force into the x component equation and solve for v:


 v2 
mg
m


sin


cos

  


s
 r 
 cos    s sin  
v 2 g  sin    s cos  

cos    s sin 
r
vmax 
gr  sin    s cos  
cos    s sin 
Including friction, this is the restriction on the speed of the car
to go around a banked curve without sliding up the incline.
If vcar > vmax, the car will slide up the incline
In terms of the angle:
 v2 
   s
gr
tan     2
 v 
1  s 
 gr 
* There is also a minimum speed the car must have when you include friction or
it will start to slide down the incline.
For this scenario, the potential motion is down the incline. This means the static friction force
points up the incline.
Ex.
Find the minimum speed a car traveling along a banked curve (whose path is the shape of a
circle) can have in order to make the curve without sliding down the incline.
Nx
N
fx
Ny
fs
fy

ar 
v2
r
N x  N sin 
ar
N y  N cos 
or
f x  f s cos    s N cos 
Potential Sliding Motion

W
f y  f s sin    s N sin 
Determine the motion in each direction using Newton’s 2nd law and the force diagram.
 Fx  mar
 Fy  0
 N x  f x   mar
N y W  f y  0
Substitute & solve for the Normal Force from the y component:
 v2 
N cos   mg   s N sin   0
 N sin    s N cos   m  
 r 
 v2 
N  sin    s cos    m  
 r 
N  cos    s sin    mg
N
mg
cos    s sin 
Substitute the expression for the Normal force into the x component equation and solve for v:


 v2 
mg

  sin    s cos    m  
 r 
 cos    s sin  
v 2 g  sin    s cos  

r
cos    s sin 
vmin 
gr  sin    s cos  
cos    s sin 
Including friction, this is the restriction on the speed of the car
to go around a banked curve without sliding down the incline.
If vcar < vmin, the car will slide down the incline
In terms of the angle:
 v2 
   s
gr
tan     2
 v 
1  s 
 gr 