Static vs Kinetic Friction on an Unbanked Ramp

[mancity]

Homework Statement

A car with a mass of 1000 kg is moving on an un-banked ramp with a radius of 100 m. What is the maximum speed the car can move without skidding if the coefficients of static and kinetic friction are 0.80 and 0.60 respectively?

Relevant Equations

mgμ=mv^2/r

I used kinetic friction and did mgμ_k=mv^2/r. However, the solution is mgμ_s=mv^2/r. I am confused on why we consider static friction and not kinetic friction, thanks!

 

[haruspex]

I am confused on why we consider static friction and not kinetic friction, thanks!

It is a common misunderstanding.
Friction is about relative motion of surfaces in contact. Kinetic friction occurs when there is such relative motion; static occurs when there is no such relative motion, only the potential for it.
If a wheel is not skidding ("rolling contact") then there is no relative motion. The part of the wheel touching a road has, instantaneously, zero velocity.

 

[nasu]

Here the friction has double role. Besides enabling the rolling without slipping, which applies even when the car moves along a straight road, the friction provides the centripetal force for the circular motion. "Skidding" here refers to the case when the friction is not enough to provide the centripetal force for the car to move in a circle of the given radius with the given velocity. In this case the car moves outwards from the center. You are looking for the situation when this does not happen and the car does not move along the radial direction and the component of the friction acting along the radial direction provides the centripetal force. No motion along the radial direction means that the radial component of friction is static.
There is a tangential component of friction that ensures rolling without slipping. This is also static, as described by @haruspex.

 

[haruspex]

There is a tangential component of friction that ensures rolling without slipping.

Only if there is tangential acceleration. This is important since any tangential frictional force contributes to the total frictional force, thereby "using up" some of the available $\mu_sN$ and reducing the max speed.