Potential energy and climbing upwards

At the end of last week, seeing a good(ish) weather forecast, we had a short break away in Wellington, taking the train both directions. Despite the lovely weather (Wellington isn’t ALWAYS blowing a gale) we didn’t get to see the volcanoes en-route, in either direction – they were shrouded in cloud as is often the case. The train journey from Auckland (or Otorohanga in our case) to Wellington is much hyped, and indeed it is very scenic in patches, and even spectacular in the occasional place (when the cloud lifts).  But it’s not the most spectacular main-line train trip I’ve ever been on – e.g. the St Gotthard line in Switzerland beats it hands down. In my opinion, anyway.

One interesting feature on the line is the Raurimu spiral – a very loopy section of track where the line climbs about 200 meters up onto the top of the volcanic plateau in about 5 km. That’s 5 km in a straight line – making an average gradient of about 1/25, (i.e. 1 metre up for every 25 m along), which, although fine on foot or a breeze in a car, is too much for a train to handle. Consequently the line does twice this distance, making a more manageable gradient of 1/50, achieving this through horseshoe bends and one complete loop, where the line crosses over itself. (Though note, a looped line is NOT unique to the Raurimu spiral, unlike the commentary implied, I know for sure, having travelled on it,  there are plenty of them on the St Gotthard line in Switzerland for example)

With ascending hills, its fairly obvious that, no matter how you do it, if you want to get from A to B you have to gain a specific amount of height. If B sits 200 metres above A, you need to climb 200 metres – whether you do this gradually or quickly, it is inescapable that you have to do it. That means moving 200 metres against a gravitational force. From a physics perspective, we can refer to this force as ‘conservative’, meaning that the amount of movement against it, from going from B to A, is the same no matter how you get there.

For example, you could climb a ladder that’s 200 m high, straight upwards. In which case you move 200 m, and every step of it is in the exact opposite direction to the gravitational force. Or, you could walk up a 1/50 incline for 10 km (i.e. follow the train track). In which case you move 10000 m, but the gravitational force is almost at right-angles to your movement. For every 50 m you move horizontally, you only move 1 m against the force. By the time you do your 10 km walk, you’ve moved 200 m against the force, as you would climbing the ladder. Hopefully that’s obvious.

In physics, gravity isn’t the only force we consider. Take the electric force. If we have an electric field, a charged particle will feel it, and experience a force. The electric force is also conservative. That means if we move a charged particle from A to B, it will move the same distance against the force, no matter which route it goes in. The amount of potential energy it gains is the same, no matter the route. This is just like the gravity example. In gravitational fields (the earth) we consider lines of equal height as being ‘contours’ on a map – in an electric field the analogous lines are called ‘equipotentials’, meaning they are lines or surfaces of equal potential energy.

However, not all forces we come across are conservative. Consider a loop from A to B back to A again. Sometimes, when we do a loop, the total distance we move against the force is zero. That’s the case in the gravity case – if we raise up 200 m then fall 200 m we end up at the same height we started with. We moved against the gravitational field on the way up (hard!), and with it (easy!) on the way down. But with the frictional force, for example, it doesn’t work like that. No matter how you move, the frictional force acts exactly against you. If you push a box from A to B and back to A, it doesn’t matter what route you take, friction acts against you all the time. Sometimes it might be more, sometimes less, depending on what surface you are on, but all the time it is against you. It’s not a conservative force, and so it doesn’t have equipotentials, or contours.

Of course, that doesn’t mean friction is necessarily a bad thing – the train wouldn’t be going anywhere if the wheels slid on the track. Conservative forces, however, are particularly neat from a physics perspective, and from a maths point of view the ‘equipotentials’, or ‘contours’, make them easy to handle. Working out how much energy the train uses to overcome the force of gravity on the ascent of the Raurimu spiral is easy – working how much energy the train uses to overcome friction or air resistance enroute is hard.

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