Conceptual Question on acceleration and circular motion

[RoboNerd]

Homework Statement

If a roller coaster car enters the circular-loop portion of the ride and navigates it successfully, then the net force on the car at its topmost point is straight down. Then why does not the car fall down?

So I am having issues understanding my textbook's solution to the conceptual problem above.

The textbook's solution is as following:
"Remember that force tells an object how to accelerate. If the car had zero velocity at this point, then it would certainly fall straight down, but the car has a non-zero velocity to the left at this point. The fact that acceleration is downward means that at the next moment, vector v will point down to the left at a slight angle, ensuring that the car remains on a circular path, in contact with the rack. The minimum centripetal acceleration of the car at the top of the track would be equal to the acceleration of gravity, g = 9.8 m/s^2. If centripetal acceleration were less than g, then the car would fall off its circular path."

Homework Equations

F[c][/SUB]=m*a[c][/SUB]

The Attempt at a Solution

I understand the portion of the solution where they say that the acceleration would cause the direction of the leftward velocity vector to move downwards. However, I do not understand why they say that the minimum centripetal acceleration of the car at the top of the track would be equal to the acceleration of gravity and if acceleration of gravity would be greater than centripetal acceleration, the car would fall off.

Could someone please help me understand this?

Thanks in advance!

 

[Doc Al]

Try this: Imagine that the car is speeding around the loop (way faster than the minimum speed). At the top of the loop, what forces act on the car?

 

[RoboNerd]

The normal force of the loop and the force of gravity.

Yes, I know that.

 

[Doc Al]

The normal force of the loop and the force of gravity.

Right, and those forces add to produce the centripetal force.

What happens to those forces as the speed of the car is reduced?

 

[RoboNerd]

Right, and those forces add to produce the centripetal force.

What happens to those forces as the speed of the car is reduced?

Gravity remains constant. Normal force decreases

 

[Doc Al]

Gravity remains constant. Normal force decreases

Right. As the speed decreases, the needed centripetal force decreases. When the normal force equals zero, that's the minimum speed. Go slower and the actual force (gravity) is greater than that needed to keep the car in circular motion. It gets pulled off the track.

 

[RoboNerd]

Right. As the speed decreases, the needed centripetal force decreases. When the normal force equals zero, that's the minimum speed. Go slower and the actual force (gravity) is greater than that needed to keep the car in circular motion. It gets pulled off the track.

The mass terms in centripetal force and gravity force cancel... so the accelerations matter. Right. Thanks!