What Speed Range is Safe for a Car on a Wet, Banked Curve?

[Hydroshock]

Homework Statement

A curve of radius 60m is banked for a design speed of 100km/h. If the coefficient of static friction is 0.3 (wet pavement), at what range of speeds can a car safely make the curve?

v = 100km/hr = 27.8m/s
x = ?
r = 60m
a_r = 12.9m/s²
u = 0.3

Homework Equations

tan(x) = v²/rg
a_r = v²/r

The Attempt at a Solution

The part I'm stuck at is I know I'm missing an equation. The one that mass cancels out to find how the forces can give me the max speed etc. with the coefficient of friction.

tan(x) = (27.8)²/(60)(9.8) = 1.31
x = 52.6°

a_r = (27.8)²/60 = 12.9m/s²

 

[Doc Al]

Don't go equation hunting; instead, draw a free body diagram showing all forces acting (including friction) for the two cases. Then apply Newton's 2nd law.

(An important first step is to solve for the angle of the road--and you've done that. Good.)

 

[Hydroshock]

Alright, well to help myself out on drawing a free body diagram I made a nominal mass of 1000kg (although I do realize it's possible to do without one). so I got $$F_N = (1000)(9.8)sin(52.6) = 12900N$$ which in turn the force is the same as $$F_R$$ which is something I didn't exactly know was meant to happen. Although makes sense as it makes the speed for no friction.

(I'm taking an online course and I find just teaching myself from the book confusing as the work I'm meant to do isn't from the book)

So with $$F_N$$ I found $$F_fr = (.3)(12900) = 3870N$$

From there I added and subtracted the forces to give me max and min for no slipping.

$$F_m_a_x = 16770N = (1000kg)A_r = 16.77m/s^2 = v^2/60m$$
$$F_m_i_n = 9030N = (1000kg)A_r = 9.03m/s^2 = v^2/60m$$

so

$$v_m_a_x = 31.7m/s = 114.1km/hr$$
$$v_m_i_n = 23.3m/s = 83.9km/hr$$

I think I did that right? =)

 

[alphysicist]

Hi Hydroshock,

I believe there are some problems with your calculations.

From your free body diagram for the first case (no friction), which direction is the acceleration in? (Here the normal force will not equal the perpendicular component of the weight.) Which direction is the normal force and weight in? Once you have those, choose components perpendicular and parallel to the acceleration and then use $F_{{\rm net},x}=m a_x$ for the x direction and similarly for the y direction.

The point of the case without friction is to find the angle that the road is banked at, so that you can use that angle for the other two cases.