Saturday, October 25, 2008

Steep Hill

I have been trying to learn about gradients and how a gradient is expressed. A check online tells me there is no Gradients for Dummies.

Things got a bit frightening when I began to try to analyse this para.

There is a fourth method in which slope may be expressed: the rise is divided by the hypotenuse (the slope length). This is not a usual way to measure slope but it is useful when one only knows the slope length and not the horizontal run. This follows the sine function rather than the tangent function and this method diverges from the "rise over run" method as angles start getting larger (see small-angle formula )

I vaguely recall that I failed both Pure and Applied Algebra....or was it Trigonometry or Calculus? I think I passed Simple Arithmetic. Amazingly, I have managed through life without ever having to use anything I learnt in these subjects. I did not even keep my logarithm table book and slide rule and I have no recollection of what I actually did with them in class. Probably poked other boys in their groins with the slide rule and set fire to the log book with a Bunsen Burner.

I have learnt that a gradient can be expressed as degrees. This sounds so simple and obvious and I am not sure why there are other methods. Just visualise a protractor and where say ten degrees would be, and you get a mental picture of how steep a hill is. But when talking about railways and tramways, they use a different method.

I heard that the Blue Mountains railway was very steep, but since the time when it was first built, the slope has been decreased. (What? They chopped the top off the mountain?)

Rail gradients are expressed as a gradient of say, 1 in 20, to use Wikipedia's example. So, for every 20 metres, you would rise 1 metre. I don't get an instant mental picture but it is very vivid if drawn to scale on a piece of paper. I am led to believe that it is quite steep, but drawn on paper, it does not look steep.

Remember my post about the Balmain counterweight dummy, the system which assisted the Balmain tram up and down from the Darling Street Wharf? That was said to be about a 1 in 8 gradient.

I just drew that inclination on a piece of paper and it does not look at all steep to me.

And by writing this post, I think I have worked out why gradients never look steep to me when drawn on paper. In the above para where I mention Wikipedia, I altered something for we Australians' benefits. I converted the measurements to metric. What it actually said was 'For example, a slope that has a rise of 5 feet for every 100 feet of run would have a slope ratio of 1 in 20.'

Am I on the right track here? It the ratio method won't translate to metric? I have no idea how to represent feet on on a sheet of paper.

All too hard and I have my sock drawer to tidy. I will just accept that the Blue Mountains train line is steep.


  1. I don't know why, but reading this post all I could think was 'the kneebone is connected to thighbone'.

    Does that help?...

    ...Nah, didn't think it would.

  2. Could be apt Victor, if walking up a steep gradient.

  3. Andrew it doesn't matter if it is in metric, imperial or any other system, so long as you use the same units for both the rise and the run.Because it is a ratio, the units cancel out. 5 feet over 100 feet is the same as 5 metres over 100 metres, is the same as 5 twigs over 100 twigs (so long as you use the same twig for both measurements): the slope ratio is 1 in 20.

  4. Anonymous4:58 pm

    Oh how I would love to differentiate every slope I came across.

  5. I have a much simpler method of assessing gradients.

    4th gear: Flat road ahead.
    3rd gear: Lean forward in your seat a bit.
    2nd gear: Let's hope you don't meet a flock of sheep coming the other way.
    1st gear: Time for a new car.
    Reverse: You're buggered. Find a different route.

  6. What Hughes said.

    Gradient is what my knees won't do coming downhill.

  7. altissima is correct - gradient as far as I know has always been expressed as a ratio and regardless of the unit used, it will always come up. Such as models (the miniature kind, not the starvers). I work with 1:35 scale. This is also used for mapping as well. 1 of anything will equal 10,000 of the same unit on the ground, etc.

  8. Ok Altissima and Rob. I am sure you are correct, but then you draw a 1 in 8 gradient on paper, perhaps the steepest hill that a tram will travel up, and tell me if it looks steep to you? Something is not right.

    Reuben, see if you can find out what ORQ stands for. I may or may not be one, but......I know many who are and I have a lot foreign born friends.

    Given you don't drive Brian, your insight is amazing.

    Jahteh, I only learnt when we visited that place where the special rocks are, near Sunbury, can't recall the name, that it harder walking down a steep hill than up it when you get to a certain age.

  9. Andrew,

    I do drive...or at least I used to. I just can't afford a car nowadays. Besides, I'm doing my bit for the cadging lifts off other people.

  10. I'd always been told gradient was measured in pubs, for example if you are walking up a hill and can make it to the top without needing to stop for a beer then the hill is not steep, otherwise if a pub stop is required at the halfway point for an energising ale then you're proabably better off driving or taking the tram.

  11. And the world is a safer place Brian.

    The logic works for me Kezza.

  12. Yeah I agree Andrew, on paper it doesn't look right. However, here are a couple of very rough mind exercises to imagine gradient in the real world...
    Try imagining say a wheelchair ramp of about 1m rise. It would probably be at least 8 metres long (often doubled back on itself). Now imagine having to push yourself up in the wheelchair -pretty hard work.
    Another one -the westgate bridge is approx 1250m long from one approach ramp to the centre. The height is 58m above water, but lets say 45m because the ramps don't start in the water. 45/1250 = about 1/27. If you draw that on paper it looks really flat, but the westgate is bloody steep - i know: I've cycled over it! This is a very rough calculation, as the westgate isn't a regular gradient all along, but it's enough to give you an idea.
    If all that doesn't make sense, Kezza's theory is probably just as useful!

  13. It does make sense Altissima. Clearly we see things in situ differently to how they are on paper. I just need to adjust my thinking,