What is the minimum speed that a roller coaster must have so that the people do not fall out at the top of the circle in an upside down loop? Take the radius to be 8 meters.

their weight has to equal centripetal force.

mg=mv^2/r solve for v.

To determine the minimum speed that a roller coaster must have so that the people do not fall out at the top of the circle in an upside-down loop, we need to consider the balance between the gravitational force and the normal force acting on the riders.

At the top of the loop, the gravitational force acting on the riders is directed downward towards the center of the loop (in this case, towards the ground). The normal force, on the other hand, is directed towards the center of the loop but acts in an upward direction.

At the instant when the riders are at the top of the loop, the net force acting on them must be pointing towards the center of the loop in order to keep them moving in a circular path. This means that the net force is equivalent to the centripetal force.

The centripetal force is given by the equation:

Fc = m * v^2 / r

where Fc is the centripetal force, m is the mass of the riders, v is the velocity, and r is the radius of the loop.

To prevent the riders from falling out, the net force must be equal to or greater than the gravitational force acting on them. Therefore, we can set up the following equation:

m * v^2 / r ≥ m * g

where g is the acceleration due to gravity.

Considering that g is approximately 9.8 m/s^2, we substitute this value into the equation:

v^2 / r ≥ g

Now, solving for the minimum velocity (v) needed, we multiply both sides of the inequality by r:

v^2 ≥ g * r

Taking the square root of both sides of the inequality gives:

v ≥ √(g * r)

Substituting the given values of g = 9.8 m/s^2 and r = 8 meters, we can calculate the minimum velocity:

v ≥ √(9.8 * 8)
v ≥ √78.4
v ≥ 8.84 m/s

Therefore, the minimum speed that a roller coaster must have to prevent people from falling out at the top of the circle in an upside-down loop, with a radius of 8 meters, is approximately 8.84 m/s.