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6. A roller coaster car in a vertical loop requires a force toward the center of the loop. At the top of the loop, the car requires a force toward the center of the loop, which is straight down. This force can be supplied by the force of gravity and by the normal force of the track.

a) In the limiting case, where the car is traveling as fast as possible, how much force will be required to keep it moving in the circle? Since the force of gravity is mg and doesn't change its value, what produces the very large force?
b) In the limiting case, where the car is moving as slowly as it can while still moving in a circle, the force of gravity is the only force acting on the car. Describe what would happen if the car moved faster than this. Where would the extra force come from? If the car moved slower than this, what would happen? (Hint: It would need a smaller force, but the force of gravity can't get smaller.) Would the car be able to travel in the circle?
c) Describe how the construction of a roller coaster track in a vertical loop is impacted by the speed of the roller coaster.

Part C: Is There an Equation?

1. The success of physics in describing the world is due to the discovery that mathematics can describe events precisely, accurately, and concisely. The equation for circular motion is

	Fc =
	where Fc is the centripetal force,
	m	is the mass,
	v	is the velocity, and
	R	is the radius.

This equation accurately describes your observations. (You can see how the equation is derived in the Physics Talk section.)

The centripetal force, F_c , on the left side of the equation is the force required to move something in a circle. It is always directed toward the center of the circle. (Reminder: when something moves in a circle a force is required. Remember the toy car with the string attached. The string always supplied a force toward the center of the circle and the car moved in a circle.)