A roller coaster has a loop the loop with a loop radius of 14.5 m. What is the minimum speed that the cart must have so that unconstrained riders do not fall out when the cart is upside down at the top of the loop? What condition on the riders is met at this speed?

To find the minimum speed that the cart must have at the top of the loop, we can apply the centripetal force requirement.

At the top of the loop, the riders are upside down, so the net force acting on them must be directed towards the center of the loop to provide the necessary centripetal force.

The forces acting on the riders are the gravitational force (mg) and the normal force (N). The gravitational force is directed downwards, while the normal force is directed upwards.

To determine the condition on the riders at this speed, we want the net force (centripetal force) to be equal to the gravitational force, meaning that the normal force becomes zero.

The net force is given by the equation:

Fc = m * ac

Where Fc is the centripetal force, m is the mass of the riders, and ac is the centripetal acceleration.

The centripetal force is provided by the normal force N, so we have:

Fc = N

Setting the gravitational force equal to the centripetal force:

mg = N

Now, we need to calculate the centripetal acceleration ac. The centripetal acceleration is given by:

ac = v^2 / R

Where v is the velocity of the cart at the top of the loop, and R is the radius of the loop.

Substituting ac and N into the equation, we get:

mg = N = m * v^2 / R

Solving for v:

v^2 = g * R

v = sqrt(g * R)

where g is the acceleration due to gravity (approximately 9.8 m/s^2).

Now, we can substitute the given value of R:

v = sqrt(9.8 m/s^2 * 14.5 m)

v = sqrt(142.1 m^2/s^2)

v ≈ 11.92 m/s

Therefore, the minimum speed that the cart must have at the top of the loop is approximately 11.92 m/s. At this speed, the riders will experience zero normal force, meaning they will be weightless.