# How do roller coasters use the law of conservation?

Jun 15, 2014

A good example is when a train is lifted to a height so that it has sufficient GPE to complete a loop the loop.

In the above case the GPE must equal the GPE the train requires to reach the top of the loop, plus the required kinetic energy at the top. If friction losses (at the wheels, axels and due to drag) are not neglected – which they cannot be in a real situation – then the work done against friction must be added onto the required initial GPE.

If friction is not neglected then experimental data will most likely be necessary for calculating how much work will be done against friction. But purely theoretical methods can be used to find the minimum height from which a loop can be completed.

So for a loop who radius, $R$, and starting at a height of $h$:
${E}_{{P}_{1}} = {E}_{{P}_{2}} + {E}_{{K}_{2}}$
From circular motion, the minimum possible speed at the top of the loop is:
${v}^{2} = g R$

${E}_{{P}_{1}} = m g h$. ${E}_{{P}_{2}} = m g \left(2 R\right)$. ${E}_{{K}_{2}} = \frac{1}{2} m {v}^{2} = \frac{1}{2} m \left(g R\right)$

⇒mgh=mg(2R) + 1/2m(gR) ⇒ h=(2R) + 1/2R
⇒h = 5/2R

That height is the minimum height that a train can start from in order to successfully complete a loop. As noted above in a real situation a height > h would be required to account for friction losses.