Car turning with friction: maximum speed for a car rounding a level curve.

In this problem, we find the maximum speed for a car rounding a level turn.

To analyze a car turning with friction, we begin with a complete force diagram. We view the turning car from overhead and rear views, and we observe that the car accelerates toward the center of curvature. This means a horizontal force must push toward the center of curvature, and that's the static friction force. To complete the force diagram, we include the weight of the car, mg, and the normal force, which is also mg because there are no other forces tampering with the vertical direction.

Next, we apply Newton's second law F_net=ma in the horizontal direction, defining the positive direction to be "toward the center of curvature" to keep the math as simple as possible. The net horizontal force is just the static friction force pointing to the center of curvature, and the observed acceleration is the centripetal acceleration a=v^2/r.

Now we assume the static friction force is maximized, because we are looking for the maximum speed just before the tires start to slip. Subbing in the formula for maximum static friction force and canceling mass out of the equation, we are able to solve for the maximum speed for a car rounding a level curve: v_max=sqrt(mu_s*g*r).