Question regarding centripetal force and tangential force

Question

I was asked to think of a situation where a car driver only feels centripetal acceleration, but exerts no tangential acceleration.

The first thing that came to mind was orbit, where the satellite does not need to speed up but be caught by the gravity of Earth, and keeps orbiting around it.

So, my question is this.

I tried to consider a case where a car is driving towards a loop with no friction so that it will not need any tangential acceleration, but the centripetal acceleration is not constant due to gravity.

One of my friends told me that a banked road can do the same thing as orbit, but I did not understand his explanation at all. I cannot imagine a car moving in circles on a banked road without accelerating tangentially.

Can someone explain me this kinematics ?

Answer 1

You don't even need a banked road. The analogy of your orbit example would simply be a car moving at a constant speed in a circle (for simplicity we, for example, restrict to the case without skidding). In that case, there is no tangential force, and the force keeping the car in a circle is the force of static friction. We can see that this is possible in real life as follows. Let $m$ be the mass of the car, then a free body diagram shows that the normal force on the car has magnitude \begin{align} N = mg. \end{align} It follows that the force of static friction on the tires satisfies \begin{align} F_\mathrm s\leq \mu_\mathrm s N = \mu_\mathrm s mg \end{align} In other words, the force of static friction can have any value between $0$ and the product of the coefficient of static friction and the weight of the car.

Now, suppose that the car wants to drive in a circle of radius $r$. In this case, it will experience an acceleration with magnitude $v^2/r$ in the radial direction, and it will require a corresponding force \begin{align} F = \frac{mv^2}{r} \end{align} to do so. Since the force of static friction will be supplying this force, we require $F_\mathrm s = F$ which tell us that the car can move at a constant speed $v$ in a circle of radius $r$ without skidding provided \begin{align} \frac{v^2}{r}\leq \mu_s g. \end{align} It's easy to make this inequality satisfied; you just need to make sure that the circle has large radius, or the car has small speed, or some appropriate combination of the two.