Question

Roller coaster loops are rarely perfectly circular. Instead, they are tightly curved at the top, where the cars are moving more slowly, and they have a gentler curve at the bottom, where the cars are moving much faster. Explain why the loops are designed this way.

Answer 1

Answer:

$a_{c}=v^{2}/R$

The radius of curvature changes so that centripetal acceleration is similar along the entire roller coaster.

Explanation:

We know that the centripetal acceleration is directly proportional to the tangential velocity and inversely proportional to the radius of curvature:

$a_{c}=v^{2}/R$

By energy conservation (and common sense), we know that the speed at the top of the roller coaster is smaller. Therefore if the roller coaster has similar accelerations (therefore also similar normal forces) at the top and at the bottom, it is necessary that the difference in speed be compensated with the radius of curvature, i.e. smaller radius at the top than at the bottom.