I have the following question which makes no sense to me:
Car is moving at $20 \;\text{m/s}$ speed on the road. Curvature radius of the hill is $R = 100 \;\text{m}$. What is the maximum possible horizontal acceleration at the hill peak, if the friction coefficient is 0.3?
The car is in a circular motion where the circle is on the paper plane. From the question it seems that the car is moving at a constant speed. This means that the acceleration of this circular motion is in the direction of the radius, that is fully vertical at the hill peak. This means there is no horizontal acceleration at all. Therefore the question makes no sense to me.
The solution is given as: $1.8 \;\text{m}\,\text{s}^{-2}$
Best Answer
As the car drives over the hill, the bottom of the tires does not slip. If the car accelerates, the tires push the road backward, and the reaction force pushes the car forward. This is a question about how big can static friction be. For that, you need to know about is the normal force pushing the car onto the road.
Usually that would be the weight of the car. But this car travels in a circular path over the hill. Therefore there is a centripetal force that makes it travel in a circle. Gravity pulls the car down onto the road, and the reaction force of the road keeps the car from penetrating the surface of the road. You should be able to figure out the normal force from that.
Edit - More about static friction
$F = \mu N$ doesn't tell you what the static friction force is. It tells you how big it can be without the tire slipping.
Think of a car parked on a level surface. It doesn't matter if it is parked on ice or asphalt. The car doesn't move. There is no horizontal force trying to accelerate the car.
Now push the parked car. The car on ice moves because there is no friction to prevent it.
If the car is on asphalt, it does not move. The friction of the tires is just strong enough to oppose your force and keep the car from accerating. It is not stronger than your force. It does not make the car move against you.
Static friction is like other reaction forces. Gravity pushes the car onto the road. The road pushes up hard enough to prevent the car from accelerating downward and penetrating the surface.
Reaction forces often have limits to how strong they can be. A meteor digs a crater. It generates forces strong enough to break bonds and dig up the road.
Likewise, static friction has a limit. If you pushed the car with a bulldozer, it would move. The bulldozer generates a force strong enough to break the bonds that hold the tires fixed.
The formula tells you this maximum force. Friction is very complex on a microscopic level. This is not a force calculated from the nature of bonds and such. It is from many measurements of static friction with a wide variety of materials and normal forces. It doesn't always hold. For example things would change if you lubricate the road with some oil. But it is useful for learning about forces and friction. Problems can be created that make you think about them.