Find the coefficient of static friction given design velocity and radius of a banked curve. Help please!?design velocity = 20 m/s r=100m actual velocity of vehicle = 25 m/s i'm so confused... |
Find the coefficient of static friction given design velocity and radius of a banked curve. Help please!?design velocity = 20 m/s r=100m actual velocity of vehicle = 25 m/s i'm so confused... |
Let:
a be the angle of banking,
g be the acceleration due to gravity,
v0 be the design velocity,
v1 be the actual velocity,
r be the radius of the curve,
u be the minimum coefficient of static friction needed for velocity v1,
b be the angle of friction such that tan(b) = u.
With no friction (for which ideal banking is designed):
tan(a) = v0^2 / (rg) ...(1)
With friction:
tan(a + b) = v1^2 / (rg) ...(2)
From (1):
tan(a) = v0^2 / (rg)
= 20^2 / (100 * 9.81)
= 0.4077
a = 22.18 deg.
From (2):
tan(a + b) = v1^2 / (rg)
= 25^2 / (100 * 9.81)
= 0.6371
a + b = 32.50 deg.
b = 32.50 - 22.18
= 10.32 deg.
u = tan(b)
= 0.182.
The theory of banked curves without friction is explained here:
http://www.batesville.k12.in.us/physics/...
The explanation with friction is on the next page:
http://www.batesville.k12.in.us/physics/...
The formula derived with friction is:
[ tan(a) + u) ] / [ 1 - u tan(a) ] = v^2 / (rg).
Noting that if u = tan(b) the left hand side becomes the expansion of tan(a + b) gives the formula I have in (2) above.