Apparently I had no idea how to interpret percent grade numbers, because I would've thought a 110% grade was impossible or meant going directly vertical or something.
But actually the internet tells me it represents rise/run as a percent, so a 110% grade is slightly more than a 45 degree angle. The more you know!
It’s related to a ratio. A ~6 degree slope would have you dropping one meter for every ten meters you move forward: 10%. 45 degrees means one meter for every meter forward: 100%. Keep plotting that out and it approaches infinity as you reach 90 degrees (or just ask, “for every meter forward, how many up/down?” if it is 90 degrees.)
The problem is that "forward" is ambiguous. You can measure it relative to your direction of travel, or perpendicular to the gravitational vector. For small gradients these are more or less the same, but the steeper the slope the more these two measures diverge.
> In general, beginner slopes (green circle) are between 6% and 25%. Intermediate slopes (blue square) are between 25% and 40%. Difficult slopes (black diamond) are 40% and up. ...
If you used the difference in height over the actual distance traveled instead of the horizontal distance, then 100 % would indeed be vertical. I.e. instead of using sin(a)/cos(a) you could just use sin(a) and for small angles the values are close but the later one approaches 1 while the former one diverges as you approach 90°. I would probably have picked the later one but the former one is probably somewhat easier to work with in practice, e.g. for construction work.
The other variant was actually meant to read sin(a)/1 to highlight the ratio in both cases but I abandoned that halfway through because I thought it makes the hypotenuse look special while it really is not in the sin(a)/cos(a) variant where the more natural description would be tan(a)/1 but that kind of ruins the simple association with the side lengths of a right triangle. I had another idea how to formulate it that would not favor one of the variants but as far as I can tell the necessary distinction between the two catheti that exists in German does not exist in English and so I left it as it is.
That's right. It's simple to remember once you get the hang of it. 100% grade: every foot forward is one foot of climb, or every meter forward is one meter of rise. 10% grade: ten feet forward, one foot up, or ten meters forward, one meter up. 110% grade: one foot forward, 1.1 feet up, or one meter forward, 1.1 meters up. 6% grades are steep enough to earn warning signs, if you pay attention to signage in the mountains.
Sonora Pass, for example, has sections of 26% grade. That sounds immensely steep (and is, for driving), but it sounds more reasonable when you say "every 50 feet forward, you've climbed 13 feet." Solve the triangle, and that's 14.574°. If you've been over the Sonora Pass, you know that it feels like you're in a rocket climbing vertically at a couple points, and you might be surprised to learn that's only 14.6° of grade.
22nd Street in SF is over 31%, IIRC, and is one of the steepest streets I can think of.
I went to Germany this year and I made a point of planning a visit to a funicular. To my great disappointment, the funicular we found in Freiburg am Breisgau used a counterweight instead of another car. Now I wonder if it was really a funicular at all.
There's one EU directive [1] that concerns "funicular railways and other installations with vehicles mounted on wheels or on other suspension devices where traction is provided by one or more cables", which "shall not apply to [...] lifts". Those are addressed in [2] which "shall apply to lifts permanently serving buildings and constructions and intended for the transport of [persons and goods] [...] serving specific levels, having a carrier moving along guides which are rigid and inclined at an angle of more than 15 degrees to the horizontal", but "shall not apply to [...] funicular railways".
Specifically, they're called snow sheds[0] as they're angled to capture and hold snow deposits, serving as a physical obstacle to accumulated snow that could then later slide. The big risk there would be a slab avalanche,[1] possibly triggered by the funicular's own vibrations. Because the snow sheds are directly supporting any heavy deposits of snow, they isolate the slab from a weaker snowpack that'd otherwise be underneath it.
I went with my family before they shuttered the old funicular. The old funicular was expensive to maintain and prone to interruption by falling rocks. So they decided to scrap the old funicular and build a new one. However not everything worked out to plan. After all there was a two-year delay. They underestimated the difficulties the steep grades created. For example they had to tether all workers to ropes. Tools falling down turned into dangerous projectiles. Tunnel boring machines had to use a climbing contraption to bore literally upwards. An so on.
By the way would have Elon Musk helped by combining his Boring Company and SpaceX to bore upwards?
I rode the cog railway up Pike's Peak a few years ago.
If you EVER get a chance to ride the train documented here, or any other such train, BY ALL MEANS take it. The views are just spectacular. And, on a train, you're not distracted by driving, so you can watch the view the whole time.
There used to be a funicular nearly adjacent to the starting point of the Cog Railway that would have given the Swiss one a run for its money, as the Manitou Incline hit 68° in some points. The Manitou Incline shuttered in the 90s after a rockfall damaged the tracks. It's a popular endurance trail now, with people 'running' up the old incline track to the top of Mount Manitou.
I'd like to second your post though. I took the Cog Railway up to the summit of Pikes Peak right before Christmas a few years ago and we were blessed with having to stop as a herd of Rocky Mountain big-horn sheep moved across the path. There is a protected lambing area on the backside of the mountain which of course I've never visited but to see these animals up close at 11,000 feet was spectacular.
Plus you can buy fresh donuts and coffee up at the summit. A nice way to visit 14,000 feet if you're not inclined to the wonderful but strenuous Barr Trail.
Trams are expensive because of all the pipes and cables under the street that have to be moved. Also a funicular train is driverless and unpowered, more like big elevator car than a train.
You also have to account for dealing with traffic and pedestrian crossings, lights, planning etc. The Dublin tram was a new line across existing streets. This funicular probably doesn't cross any roads.
Also things take longer and cost more in Ireland. Case in point, near my house the local council had to dig up a road and lay some pipe to channel a stream underneath. This took two months. In the UK they put a train tunnel under a motorway in 72 hours
Intamin offers comparable systems.[1] Intamin is primarily a roller coaster manufacturer. As a sideline they offer the "Mountain People Mover", which uses roller coaster track and wheel systems. It's electric drive, not cable drive, so it's a people mover, not a funicular. The track can be much more curved, in all axes, than with a cable driven system.
Intamin offers the tilting cabin option, but their existing installations don't use it.
At some inclination, cable-driven systems are considered elevators. The George Washington Masonic Temple has cable-hauled elevators which are 82.5 degrees from horizontal. This is a 759% slope. They're tilted in two axes, because the building is a narrowing tower with big meeting rooms on several floors, and the elevators don't break up the space.
For the California/bay area folks there is a tram from Silver Lake to Lake Agnew over in the Eastern Sierra (near june lake) that has a max of about a 105% grade. Sadly, you can't ride it but you can hike up parts and around the station at the top. Even trying to climb up is quite a challenge. Its quite amazing. Built in 1915 to support construction/operation of the Lake Agnew and Gem Lake dams. I suggest going up in the fall when the aspen are changing color.
63 comments
[ 2.6 ms ] story [ 130 ms ] threadBut actually the internet tells me it represents rise/run as a percent, so a 110% grade is slightly more than a 45 degree angle. The more you know!
100*tan(slope) = grade%
slope = arctan(grade%/100)
so 110% is about 47.7 degrees, and tan(90 degrees) is infinitely large.
E.g. "this ski slope has a gradient of 90%!!!"
http://www.foxnews.com/travel/2017/01/30/5-most-dangerous-sk...
> In general, beginner slopes (green circle) are between 6% and 25%. Intermediate slopes (blue square) are between 25% and 40%. Difficult slopes (black diamond) are 40% and up. ...
https://en.wikipedia.org/wiki/Piste
Also generally with ski slopes the quoted gradient is some tiny portion of the run. Top to bottom the actual gradient is much less.
A 10% grade over a horizontal distance of X will gain exactly twice as much height as a 5% grade over the same distance.
Also called tan(a) ;)
Sonora Pass, for example, has sections of 26% grade. That sounds immensely steep (and is, for driving), but it sounds more reasonable when you say "every 50 feet forward, you've climbed 13 feet." Solve the triangle, and that's 14.574°. If you've been over the Sonora Pass, you know that it feels like you're in a rocket climbing vertically at a couple points, and you might be surprised to learn that's only 14.6° of grade.
22nd Street in SF is over 31%, IIRC, and is one of the steepest streets I can think of.
It seems the distinction is that funiculars operate in balanced pairs, while elevators are independent units.
https://en.wikipedia.org/wiki/New_Town_Hall_(Hanover)#Histor...
[1] http://eur-lex.europa.eu/legal-content/EN/TXT/HTML/?uri=CELE...
[2] http://eur-lex.europa.eu/LexUriServ/LexUriServ.do?uri=OJ:L:2...
because of course there is.
https://m.youtube.com/watch?v=C5YoI5F3xEw
Is that to try to catch the train if it falls off the tracks?
Edit: s/avalanche protection/snow/g
> https://en.wikipedia.org/wiki/Snow_fence
[0] https://en.wikipedia.org/wiki/Snow_shed
[1] https://www.meted.ucar.edu/afwa/avalanche/navmenu.php?tab=1&...
By the way would have Elon Musk helped by combining his Boring Company and SpaceX to bore upwards?
If you EVER get a chance to ride the train documented here, or any other such train, BY ALL MEANS take it. The views are just spectacular. And, on a train, you're not distracted by driving, so you can watch the view the whole time.
I'd like to second your post though. I took the Cog Railway up to the summit of Pikes Peak right before Christmas a few years ago and we were blessed with having to stop as a herd of Rocky Mountain big-horn sheep moved across the path. There is a protected lambing area on the backside of the mountain which of course I've never visited but to see these animals up close at 11,000 feet was spectacular.
Plus you can buy fresh donuts and coffee up at the summit. A nice way to visit 14,000 feet if you're not inclined to the wonderful but strenuous Barr Trail.
https://www.salzburg.info/en/travel-info/arrival-traffic/cab...
<del>I can't find any info on an exact percentage but I'm surprised another railway beat it as it really is very steep</del>
https://www.youtube.com/watch?v=1m507EEHp34
EDIT: 62% according to wikipedia (once I found the right name for this railway). That's less than half the one linked in this post. Impressive!
https://en.m.wikipedia.org/wiki/Katoomba_Scenic_World#Railwa...
In comparision, new Dublin tram line - COTS cars, 6 km line for 368 mln Euro. Or Detroit Qline, $180 mln. for 5.3 km line.
Also things take longer and cost more in Ireland. Case in point, near my house the local council had to dig up a road and lay some pipe to channel a stream underneath. This took two months. In the UK they put a train tunnel under a motorway in 72 hours
Intamin offers the tilting cabin option, but their existing installations don't use it.
At some inclination, cable-driven systems are considered elevators. The George Washington Masonic Temple has cable-hauled elevators which are 82.5 degrees from horizontal. This is a 759% slope. They're tilted in two axes, because the building is a narrowing tower with big meeting rooms on several floors, and the elevators don't break up the space.
[1] http://www.intamintransportation.com/transportation/Mountain...