Car on a Curve
This next problem is a bit easier than the last one, but it’s still good practice.
A car of mass m travels around a flat curve that has a radius of curvature r . What is the necessary
coefficient of friction such that the car can round the curve with a velocity v ?Before we draw our free-body diagram, we should consider how friction is acting in this case. Imagine
this: what would it be like to round a corner quickly while driving on ice? You would slide off the road,
right? Another way to put that is to say that without friction, you would be unable to make the turn.
Friction provides the centripetal force necessary to round the corner. Bingo! The force of friction must
point in toward the center of the curve.
We can now write some equations and solve for μ , the coefficient of friction.
The net force in the horizontal direction is F (^) f , which can be set equal to mass times (centripetal)
acceleration.
We also know that F (^) f = μ (^) N . So,
Furthermore, we know that the car is in vertical equilibrium—it is neither flying off the road nor being
pushed through it—so F (^) N = mg .
Solving for μ we have