Question

You just got a job as an engineer at Disneyland and you are placed in charge
with designing a new rollercoaster. As an initial model, to get some idea of the needs of this
project, you will consider your moving cars of mass m as point-like objects. At the start, the
cars of your rollercoaster are pushed by a huge spring (spring constant k) on a frictionless
surface. After they are released from the spring, the rollercoaster cars move a distance D on
a horizontal surface with a friction coefficient μ. At the end of the horizontal surface, the
rollercoaster cars enter a huge frictionless loop of radius R. What should be the initial
compression of the spring Dx to be sure that the cars are able to complete the loop?

Answer 1

Let initial compression be x

Initial energy stored by the spring = 1/2 kx^2

Workdone by friction= fD=μmgD

Energy contained in the rollercoaster just before reaching the loop= 1/2 kx^2 - μmgD

Now that much energy should be sufficient to make the vehicle reach the top of the loop with velocity at the top 0 and then inertia will make the rollercoaster move forward and complete the loop.

Hence using energy conservation:-

1/2 kx^2 + μmgD= mg(2R)

Hence x= √{2mg(2R-μD)/k}