A certain roller coaster has a circular, vertical loop of radius 10 meters, such that at its top the riders are upside down. How fast does the roller coaster move at this point if the riders feel perfectly weightless when passing the top of the loop?
To determine the speed of the roller coaster at the top of the loop, we need to consider the forces acting on the riders at that point. The key force involved is the force of gravity pulling the riders downward.
When the riders are at the top of the loop, the gravitational force is directed towards the center of the loop (the axis of rotation). The riders feel weightless because the force of gravity is canceled out by a fictitious force called the centrifugal force, which is directed outward and perpendicular to the direction of motion.
At this point, the net force acting on the riders is the centrifugal force. This force is given by the equation:
Fc = mv^2 / r
Where Fc is the centrifugal force, m is the mass of the riders, v is the velocity of the roller coaster, and r is the radius of the loop.
Since the riders feel perfectly weightless, the centrifugal force is equal to zero. Therefore, we can set up the following equation:
0 = mv^2 / r
Rearranging the equation, we can solve for the velocity of the roller coaster:
v^2 = (0 * r) / m
v^2 = 0
v = 0
Hence, at the top of the loop, the roller coaster must momentarily come to a complete stop (velocity of 0) in order for the riders to feel perfectly weightless.