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?
V^2/R = g. Solve for V
To find the speed of the roller coaster at the top of the loop, we can use the concept of centripetal force. At the top of the loop, the riders will feel perfectly weightless because the normal force from the track balances their weight. This means that the net force acting on the riders is only due to the centripetal force.
When the riders are at the top of the loop, the net force acting on them is the difference between the gravitational force and the normal force:
Net force = mg - N
where m is the mass of the riders, g is the acceleration due to gravity, and N is the normal force.
The centripetal force is given by:
Centripetal force = (mass) * (velocity)^2 / (radius)
In this case, the radius of the loop is 10 meters.
At the top of the loop, the radius and the velocity are in the same direction, so we have:
mg - N = (mass) * (velocity)^2 / (radius)
Since the riders feel weightless, the normal force N is equal to zero.
Therefore, we have:
mg = (mass) * (velocity)^2 / (radius)
Rearranging the equation, we get:
velocity = √(g * radius)
Plugging in the values, with g being approximately 9.8 m/s^2 and the radius being 10 meters, we can calculate the velocity:
velocity = √(9.8 * 10) = √(98) = 9.9 m/s
So, the roller coaster needs to move at approximately 9.9 m/s at the top of the loop for the riders to feel weightless.