# Simulating linear momentum/KE through angular momentum/KE

I have undertaken a project where I need to design a brake system for a miniature locomotive. To test the effective of the brakes I have decided to create a rig where the brake system is attached to a flywheel which is spun to a certain angular speed in order to simulate the momentum of the locomotive as if it was on a track. I don't have a track available which is why I am doing this.

The mass of the locomotive is 800KG and assumed to be travelling at a const velocity of 4.2 m/s therefore will possess a linear momentum of 3,360 kg.m/s and a linear kinetic energy of 7065J.

I would like to simulate this momentum in the form of a flywheel, in order to determine its dimensions I need to know how much angular momentum the flywheel needs.

I am not sure how I can correctly mathematically/physically be able to "convert" linear momentum into angular momentum. I know that these two quantities are independent which is why I am asking this question.

Where I have ended up so far:

The energy in an object has two components: linear K.E. (El) and rotation K.E.(Er) Et = El + Er

El = 0.5*m*(v^2) , Er = 0.5*I*w^2, Et = total energy where, 'v' is linear velocity, 'w' is angular speed and 'I' is moment if inertia.

If I assume the locomotive only has linear component: 7065J and I assume that the flywheel only has rotation component - can I assume it to have 7065J of energy.

I question myself whether a linear joule is the same as rotational joule.

Any comment is helpful - i can work on from a hint

• This sounds like you need to apply dimensional analysis : dynamic similarity, kinematic similarity etc. – Solar Mike Nov 8 '17 at 10:22

## 2 Answers

Your fundamental assumptions are correct, but I would question the practicality of the system that you are suggesting for testing.

Even with a heavy flywheel at a large diameter, you would need to spin it to a very high rotational speed in order to store the 7065J of energy that you are trying to mimic. Unfortunately brake-force is not independent of the surface speeds, and other important effects such as overheating/longevity of the system may not be accurately modeled in this case.

I would recommend dimensional analysis to allow you to model the system at a smaller scale (see https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem and https://en.wikipedia.org/wiki/Similitude_(model))

To answer your question "whether a linear joule is the same as rotational joule"

They are both forms of energy that are considered equal when using the conservation of energy methods - which I believe is what you're trying to do.