A curve with a 130m radius on a level road is banked at the correct angle for a speed of 20 M/S?

Question

If an automobile rounds this curve at 30 M/S , what is the minimum coefficient of static friction between tires and road needed to prevent skidding?
Express your answer using two significant figures.

Answer 1

The centripetal acceleration is given by:

Ac = V^2 / R

If the road is banked at the correct angle T for a speed of V, then when the vehicle goes at velocity V, there is no force parallel to the plane of the road:

Let N be the force normal to the road and
let W be the weight of the vehicle

Then the vertical component of N has to equal W, and
The horizontal component has to equal the needed centripetal force.

W = Mg

where M is the mass of the vehicle and g is the acceleration due to gravity at the surface of the Earth (about 9.8 m/s^2)

The centripetal force is M times the centripetal acceleration.

So the horizontal and vertical components of the normal force N are:

Nv = Mg
Nh = M(V^2)/R

But we also know that

Nv = N cos T
Nh = N sin T

So tan T = Nh/Nv = (V^2)/(Rg)

We know V (20 m/s), R (130m), and g (9.8 m/s^2), so we can compute tan T and hence T, and then sin T and cos T

As the velocity increases to 30 m/s, the gravitational force remains the same, but the centripetal force has to increase. As long as the car does not skid, we have an additional force operating - the force Fs due to static friction.

Fs operates parallel to the road so it has both vertical and horizontal components. The horizontal component goes in toward the center of the curve and the vertical component goes down.

So we have the equations:

W + Fv = Nv

M(V^2)/R = Nh + Fh

And the connection between N and Fs is given by:

Fs = N Cs

where Cs is the coefficient of static friction.

We don't really care about the forces, just the accelerations, etc so we can divide out the mass or just assume any mass we want - say 1.

So now we have three equations with three unknowns: N, Fs, and Cs.

We solve the first two for N and Fs (recalling the relationship between the components and the magnitudes of the vectors revolve around the sine and cosine of the angle T.)

Then Cs is just Fs/N