Question

You are upgrading an amusement park ride by adding a circular loop to an existing roller coaster ride. The first hill for the existing roller coaster is 55 m tall, and you are asked to build, right at the bottom of this hill, the tallest loop possible without having the cars fall out of the track or the passengers fall out of the cars. The roller coaster starts from rest at the top of the hill. What is the maximum radius for the loop?

Answer 1

Answer:

110m

Explanation:

We need first have a couple of equation to give a relation between the maximum force and the centrifugal foce.

We know that the maximum force F=mg, that is the condition for the coster to just stay on the track.

In the other hand we know that the centrifugal force is given by,

$F= \frac{mv^2}{g}$

where r is the radius of the loop and v the velocity of the coaster.

Assume that both equation are equal, then

$\frac{mv^2}{r} = mg$

The velocity of the coaster is given by,

$v= \sqrt{2gh}$

Where h=55m and g=9.8m/s^2

Replacing

$v=\sqrt{2(9.8)(55)}$

$v= 32.833m/s$

The maximus radius of the loop is,

$\frac{mv^2}{r}=mg$

$r=\frac{v^2}{g} = \frac{32.83^2}{9.8}=110m$