True Ne06847, but that all depends on what zero is. There are contexts in which division by zero can be considered as defined. For example, division by zero in the extended complex plane is defined to be a quantity known as complex infinity. However zero does not have a multiplicative inverse under any circumstances. Limits involving division by a real quantity which approaches zero may in fact be well defined depending on application of the defined limits of the mathematical argument.
Theoretical mathematics has no application in the garden railroad. Since nobody here will even attempt a 90 degree slope or greater than 50% gradient, then nobody will even approach the idea of a complex infinity so division by zero is a red herring to the argument as our defined limits are from 0 gradient to a maximum useable of maybe 6 or 8%.
To sum up:
Sticks are free, string cost a dollar, a yard (meter) stick two dollars, and a day in the garden laying track is priceless.
I was (maybe poorly) trying to show that kimbrit’s method has a large margin of error (50% give or take) to figure out the actual grade of an incline; as well as the error in her information regarding that a grade of 100% is 90º (to it’s “flat” plane of reference).
This is the model world Dan and in my garden I certainly will not be using the maths I’ve been shown how to use, nor stand in a corner with the pointy hat on because I do it another way, rightly or wrongly. The truth of the matter is I never work out a % grade or a degree of grade because in my garden it doesn’t matter, I work out a smooth gradient - as said. I will always maintain though that a gradient must be between 0 and 90 degrees, I know because I’ve walked up them in my hiking days. In my climbing days I’ve also climbed reverse gradients - when 90 degrees is passed and it overhangs. I am now at home with a beer and I am going to incline myself in my favourite chair and grade tonights TV with points out of 10. I will be dividing nothing by zero because that is a pointless exercise, except perhaps when Gail serves me my fish, chips and mushy peas and as I eat it, degree by degree, I will appreciate that said meal is diminishing, bit by bit, towards zero and in the end I will not be able to divide it anymore. Such is life, by the way I’m not a her and show us some pics of your railway.
Festive cheers,
Kim
PS whatever T-Jack was on about, I probably agree.
The question is, how to measure the gradient of a curve.
The answer is pretty straightforward. I assume you already know how to measure the gradient of straight track based on calculating rise / run, using your stepping block and level. So go ahead and calculate the rise by that method, from the beginning of the spot on the curve you want to measure to the end.
Now, divide that number by the total distance around the curve. You can calculate this in one of two ways (as well as a number of more complicated ways):
If you’re using preformed curved track sections of a known radius: multiply the radius by 6.3 (2pi) to get the diameter of the circle. Divide by the number of segments needed to make a circle (usually 12 or 16). That’s the length of each curved track segment. For example, 10’ diameter (5’ radius) track has a diameter of approximately 31.5. Aristo-Craft 5’ radius track comes 12 segments to a circle. Therefore each segment is approximately 31.5 inches (31.5 feet / 12 = 31.5 inches) long.
If not, get a piece of string. Fasten one side to a sleeper where you want to start measuring, and stretch it along the curved side of one of the rails until you reach the end. Mark the end, stretch the string out straight and measure it. (You can also do something similar with a soft measuring tape instead if you prefer). If that’s still not precise enough for you, measure each rail and take an average between them to get the distance measured down the center of the track.
You now have the vertical distance and the horizontal distance. Divide one by the other and you have your gradient.
I give up… ifyou fail to see why I stated you are wrong in your method of determining gradient, then there is no point for going on further. I never said one way was “more right” than the other for modeling, but that your method will provide a gradient that is (roughly) 50% steeper than what was intended - as well as you are not actually measuring the gradient of a given line/angle (amount of rise for a given run), but what part/percentage of 90 degrees that a given line/angle is.
In my post, I had hoped that I could offer better examples of gradient, but it apperas that my originally well intended posts have degraded into some form of childish argument over relatively simple mathemati
Oh Dan, never give up, enjoy the forum for what it is and have some fun. Gradient angles in the garden really don’t matter - except in Australia!! I’m 55 years young, have trains in the garden, drink good beer 'til it flows out of my ears, enjoy spirited conversation with my mates, watch as much football as I can, but most of all, I always have fun. I do not doubt your maths, I think you are at college and do them all of the time, just let us old guys have our bit of fun, well this one anyway.
I firmly agree with Kim on that point, Dan. Don’t get discouraged because of a little debate, it’s only a forum and what’s more we are arguing about TOY trains for Pete’s Sake!
Your math is on the mark, there are no errors in your position, it’s just that it really, truly doesn’t matter in the garden if the grade is + or - a few points just as long as it looks good and the engines can handle them. On my Dixie D SL I did a quick calculation on a scrap paper for what I wanted the maximum grade to be, then went outside with a 2x4 and a tape measure and made it so.
The only math you need is a tape measure. A flexible one like for linen work. Since we know one end of the grade is at zero, there’s half the equation right there. Let’s say for arguement’s sake the high end of the grade will be 2". Ok, pretend it’s a hilly logging line. What’d you say? About a meter long? What’s that…40" give or take, right? Since we know grade is the same regardless of scale, we automatically know that a 1% grade equals a 1 inch rise in 100 inches, right? For measurement’s sake, let’s say 96" or eight feet. Since we know the simple math 96 divided by 2 is 48, we got that licked. So, if you’re curve was 48 inches long, that would be easy. It would be about a 4% grade. Since it’s about eight inches shy, we know the grade will be slightly more, probably in the 4.3-4.5% range, including the four inches we knocked off to get to 96". Make sense? I hate equations, but I bet I’m close off the top of my head.
All well and good; HOWEVER, the question here is that if you have a spirit level equipped with an adjustable vial and you set that vial to give 1 degree of rise, how do you calculate the resulting percentage of grade that the given 1 degree of rise will give.
As far as the curveature is concerned it would appear that a length measurement along the centerline of the track gauge is the consensus for distance.
perhaps I should stick in the MRR forums for a bit then… or hang around here more to figure out how well G scale can take grades… HO wouldn’t do so great*… and curves make it worse… i think to the effect of .5% for “broad” curves.
*from both experience and what I’ve read - they lose ~50% pulling capacity for each % of grade or so. Dunno if this holds true with G (or if it really even matters because trains aren’t 50-something cars long)
To all my good mates that want to divide anything by nought i say phooey. This is digging in the dirt in your backyard mate, and when it is over 30 deg C a cool beer is more important.
I really think i am pretty right, i am convinced that over 1 m track and this metres position and shape changes every time i move on to another small section (276 mm) of track; if i drop 2 cm i have a 2% gradient and i also have about a 2.2deg gradient. Over 4m i have in fact dropped 8 cm, i have proved that now ith my water level.
If anyone disagrees tell me what you think; or any test i can do to get a more realistic figure.
Yes, please hang around here. I feel you have not only a lot to learn, but a lot to give as well. WELCOME TO THE REAL WORLD OF MODEL RAILROADING!!! Quick primer, HO trains are toys and are not real models. Real trains run outside in the dirt, so do we. Real trains get rained on, so do we. There are no critical measurments, no critical scale (unless that is what you want to do), and some garden trains do pull 50-200 cars up and down grades with no problems, in the snow.
Trust me, once you taste Garden Scale, there is never going back to puny little gurlie man toy HO scale. Buy a shovel, you’ll be needing it.
Ian;
Sounds like you started a regular feast here. I’m afraid I am using a far too simple method. Took a piece of string, tied a knot in one end measured and tied knots every quarter meter for 3 meters. I’ll bend the track and have it somewhat in place, put the string between the rails and clamp it to the inside rail with a “alligator paper clip”, pull it tight against the inside rail and at each knot I know how much higher the “lift” will be depending on the grade I’m after. No science, no calculator required here. Each knot represents a quarter cm lift for each percentage of grade I want. After reading some of these posts I’m beginning to think I’m not having enough fun! That’s all right, I’d rather play in the dirt than push calculator buttons.
Tom mate, i read your post several times and i like it but i can’t quite see what you are getting out of what you did. I am even astute enough to work out that you did not use a metric measure at all but 40" and every 10’ and a quarter of a cm is not actually that is it? its 1/10"
That brings up a very good point, Tom. There are a million and one ways to do whatever it is that needs to be done in the garden. No one way is right or wrong. Whatever works for you is the right way.
Guys like Dan are probably getting a culture shock on this forum. I’ve found that it works like this:
an idea or question is brought forward
discussion on its merits
alternate methods or ideas are discussed
Ian poo-poos the whole thing
The Australians begin drinking cold beer
The English begin drinking warm beer (because their refrigerators are made by Lucas Electric)
Capt Bob invests in something
I bring up a bunch of useless technicalities because I spent 25K on a college degree and I’m going to use it one way or another!
Everybody goes outside and plays with some trains
Repeat step 1 through 9
Nobody is under any obligation to pay the least bit of attention to what anybody else has to say, take what you can use and leave the rest. I’ve learned a lot, adapted some ideas and moved in directions I would never have thought of myself. This is in fact MUCH different than the HO forum. I like it this way. I spent many years being stressed over a job that requires 100% accuracy as the only acceptable standard, now I want to be a bit more casual. Maybe it’s good, maybe bad, I don’t know, but I like the format the way it is.
Jack once more i agree with you but i must take you aside and have a man to man talk to you; what time is it there; i know what time it is here and you are up to late or up to early. Its nearly 1 pm here wed 13 th and i think you are 14 hours behind us, so add 10 hours and it is 10 pm last night, right?
Yes, please hang around here. I feel you have not only a lot to learn, but a lot to give as well. WELCOME TO THE REAL WORLD OF MODEL RAILROADING!!! Quick primer, HO trains are toys and are not real models. Real trains run outside in the dirt, so do we. Real trains get rained on, so do we. There are no critical measurments, no critical scale (unless that is what you want to do), and some garden trains do pull 50-200 cars up and down grades with no problems, in the snow.
Trust me, once you taste Garden Scale, there is never going back to puny little gurlie man toy HO scale. Buy a shovel, yo
Model Railroad scales aren’t mutually exclusive, you know… I’ve got HO, N, G (In that order because it can be pronounced.)
Anyway, about the grade math and stuff. If you want to (cold rainy day?) check your method against the actual math, you can figure out how far off you are.
There is another way to do grade measurements, one that doesn’t require a calculator… just a level and tape measure. Place the level on the point you want to measure to, level it, and drop the tape from the other end of the level (making a right angle) and note the distance from the level and the distance to the ground from the tape measure.
(Sure, Puckdropper expects us to go through Geometry and Trigonometry to calculate the degree on the railroad and NOW he tells us an easier way. – If I had thought of it earlier, I would have told you earlier.)