Barron's 2nd Edition DT Q31

Barron's Book

31) We need to find the minimum speed at the top of a roller coaster’s loop. For this problem, it will be crucial to use centripetal force. Let’s look at the forces acting on the roller coaster at the top of the loop.

Where F[g] is force of gravity

The only force acting on the object is the gravitational force. Now we need to equate the centripetal force to the sum of the vertical forces. However, since at the top point, the centripetal force points downward, to the center, I am going to consider the downward direction as positive.

Where F[c] is centripetal force, F[g] is force of gravity, m is mass, v is speed, r is radius, and g is acceleration due to gravity