My physics teacher today used a car driving over a circular hill as an example of uniform circular motion, and he said that the car always accelerates towards the center because the normal force is off to the side and less than the force of gravity. I understand that, observationally, some kind of acceleration must occur or else the car would just fly off the hill. However, at the top of the hill, how can the normal force and the force of gravity be unbalanced? What makes them be unbalanced? I really hope someone will at least know what I’m talking about - this is going to keep me up at night! (If someone can answer this one, I’ve got another too!)
It sounds to me like you’re confusing acceleration and velocity. If so, you’re not the first person to do this. At the top of the hill, the car is accelerating downward, but it is not moving downward (yet). The same could be said for a projectile at the top of its ballistic path. Objects can only be in this state (acceleration but no motion) for an instant, because any acceleration means that the velocity is changing, and if it’s zero now, it will be nonzero soon.
Does that make any sense?
As you should already know, an object moving in uniform circular motion experiences centripital acceleration, a[sub]c[/sub] = v[sup]2[/sup]/r.
If the car were traveling on flat ground, it would not be accelerating towards the earth. For objects in static equilibrium, the sum of all forces must be zero, so mg would have to be balanced by an equal and opposite normal force F[sub]N[/sub].
However, if you drive the car over the crest of the hill, part of that acceleration due to gravity is used to change the velocity vector, and only a fraction of g contributes to the normal force acting on the wheels; F[sub]N[/sub] = m(g - v[sup]2[/sup]/r).
If one travelled fast enough over the crest of the hill, you would fly off. The border between the two regimes is where there is absolutely NO normal force on the car going up the hill.
Isn’t it true that unless v is less than the sqrt[grcos(theta)] the car will fly off?
A FBD of the car indicates that the normal force due to gravity must remain larger than the centripetal acceleration force or the car will fly off. This is true for any point on the circular hill. I may have overlooked something. Think about a car on a flat surface traveling in a circle. The minute the centripetal acceleration force is too large for the friction on the tires, the car slides out of the circle. Should be the kinda the same deal. But I have been wrong many times! 
Once the cars tires leave the circle, the car starts slowing down–but if the car makes it all the way to 180 degrees, it’s all downhill from there, unless the tires are sticky.
Thanks for the responses 
… I get the difference between velocity and acceleration, and I get why the car has to be accelerating. But why, when it reaches the top of the hill, do the forces not momentarily balance? What causes the normal force to be lessened? I know why it’s less, but not how. If that makes any sense. KeithT … I think you might be on track with what I’m looking for. Could you explain the last thing you said more?
Shimmery, the lessening of the normal force is essentially due to the speed of the car at the apex of the hill. The faster the car goes, the more centripetal acceleration is required to keep the car on the hill. Since the only two forces we are considering are the weight of the car and the normal force on the car, and because the weight of the car is constant, the normal force must necessarily decrease as the speed of the car increases. (Note that the centripetal force at the apex of the hill, which is a vector directed “down,” is equal to the weight of the car minus the normal force.)
However, the normal force can only decrease to zero. It cannot be negative. The maximum centripetal force possible, then, is equal to the weight of the car. If the car goes any faster than the speed corresponding to zero normal force, it no longer travels in a circle. Instead, it goes airborne in a Dukes of Hazard type scenario.
Hope this helps somewhat.
Hm.
The normal force is always perpandicular to the plane where the object is traveling, but it will only be balanced by the force of gravity if that plane is also tangental to the surface of the earth. So, the two forces only balance at the crest of the hill.
Hm, I might just be completely confused.
Maybe a little confused, Urban.
The two forces don’t have to balance at the crest of the hill. It depends on the car’s speed. (Actually, if the car has any speed then there will be a centripetal acceleration and so the normal force must always be less than the gravitational force) As mentioned before, if the car is going just the right speed, the normal force will equal zero, the centripetal force will be just the gravitational force, and the driver will feel a floating free-fall sensation as he crests the hill.
Right, the normal force, in general, is exactly enough to keep an object on the surface. With a flat surface, this means that the normal force will be such that the acceleration tangential to the surface is zero. But with a curved surface, it’s not so simple. The opposite case as the car on the hill is also true: a marble rolling in a bowl will have a greater normal force than gravitational force when it reaches the bottom.
Achernar, when you put it that way, wow! I get it! However, the normal force sort of bothers me. It seems like a force that physicists made up because they decided there had to be a force to balance gravity, and it’s always just large enough to make their problems work out correctly. Is this true? Does the normal force actually come from somewhere, and I just haven’t learned about it yet?
The normal force is actually electromagnetic in origin (it’s due to charged particles in the atoms interacting) but because both objects have no net charge, the force falls off very quickly with distance. So if the normal force were too big, then the objects would fly apart, but once they’re separated by even a tiny amount, they’re not being pushed apart anymore, so that doesn’t happen.
Sorry to bump this but I wanted to say thanks for the explanation, Chronos. I’m relieved to know its a real force.
(Also sorry I took so long to respond - I got my wisdom teeth out last Friday 
and I’ve been recovering.