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Physics help regarding roller coasters

M

Maggotry

Platinum Member
I won't go into the details, but the subject came up about roller coaster loops. There were 2 arguments:

The first, that the weight of the train doesn't mater. The only considerations are speed, the height of the loop, resistance (wind, track friction), and the gravitational constant.
The second was the same as the first with the addition of weight.

It's been 10+ years since I've had a physics class but I think I remember working an equation like this. If I remember right, weight is unimportant.

So what's the deal? Does weight have any relevance or not? The correct equations would be nice too.
 
If I remember right, as long as the first hill is higher than any other hill or loop, everything else doesn't matter. If any hill or loop after the first hill is taller than the first hill, the rollercoaster is pretty much fvcked
 
I havent taken physics, but I'm pretty sure that weight has something to do with it. Doesnt weight affect your momentum, which carries you through the whole ride?
 
Originally posted by: Maggotry
I won't go into the details, but the subject came up about roller coaster loops. There were 2 arguments:

The first, that the weight of the train doesn't mater. The only considerations are speed, the height of the loop, resistance (wind, track friction), and the gravitational constant.
The second was the same as the first with the addition of weight.

It's been 10+ years since I've had a physics class but I think I remember working an equation like this. If I remember right, weight is unimportant.

So what's the deal? Does weight have any relevance or not? The correct equations would be nice too.
Click to expand...

Do you mean that it doesnt matter in terms of whether the train cant successfully complete the loop or not? Or what? Be a little more specific please.

If you mean in terms of completing the loop, weight isnt a factor.
 
Originally posted by: Mookow
If you mean in terms of completing the loop, weight isnt a factor.
Click to expand...

Yes, that's what I mean. I remember working an equation to determine the maximum height a loop can be based on the train's speed at that point in the track. The faster the train, the taller the loop can be, regardless of weight.

Anyone remember how to work the problem?
Click to expand...
 
Weight (well mass, and the acceleration due to gravity) cancels out in the equation IIRC. Too bad I torched the physics notes after the final 🙂
 
Originally posted by: Maggotry
Originally posted by: Mookow
If you mean in terms of completing the loop, weight isnt a factor.
Click to expand...

Yes, that's what I mean. I remember working an equation to determine the maximum height a loop can be based on the train's speed at that point in the track. The faster the train, the taller the loop can be, regardless of weight.

Anyone remember how to work the problem?
Click to expand...
Click to expand...

Well, look at it from the PE perspective: PE = Mass*Gravity*Height. However, the energy needed to get the CM of the train to the top of the loop from the bottom is = M*G*H (seems similar, doesnt it 😉 ). So, basically, the height that the CM gets to on the first hill (not the height of the hill, as with the standard rollercoaster, where the track goes up, crests, and immediately heads down, the CM passes under the top of the hill), must be greater than the highest point the CM passes through in any succeeding hills/loops, plus it has to have enough leftover height to account for the frictional forces on the train.

As an aside, you could argue that the mass of the train matters as it does affect friction (Ff=(mu)*m*g). However, it does not directly affect it.
 
If there is no friction taken into consideration, mass is inconsequential.

It all has to do with energy (not momentum). You always have the same total energy, between gravitational potential, PE = mgh, and kinetic energy, KE = (1/2)mv^2. When you lose one, you gain the same amount in the other. At that original height, you are all potential. When you touch the ground, your speed is fastest, as it is all kinetic. These cancel out because as your mass increases your potential increases, but it divides out of the kinetic.

If there is friction, mass counts.

This is because as mass increases, the normal force of the track increases equally, becase it is equal to weight. since friction equals the mass times the coefficient of friction, the larger the mass, the more friction. This will take energy away from the car, and make it impossible to get over that last big hill at the end of Space Mountain 😉.

Hope this helps.
 
Originally posted by: Maggotry
I won't go into the details, but the subject came up about roller coaster loops. There were 2 arguments:

The first, that the weight of the train doesn't mater. The only considerations are speed, the height of the loop, resistance (wind, track friction), and the gravitational constant.
The second was the same as the first with the addition of weight.

It's been 10+ years since I've had a physics class but I think I remember working an equation like this. If I remember right, weight is unimportant.

So what's the deal? Does weight have any relevance or not? The correct equations would be nice too.
Click to expand...

Actually, both are wrong. As I posted earlier, mass is somewhat relevant (to the force of friction between the track and the train), and gravity cancels (as long as you treat it as a PE problem, which is easier, anyway).
 
weight is unimportant to what? to keep it from falling? to keep the tracks from getting crushed? it really depends on what you are talking about. the question is really vague.
 
Originally posted by: dighn
weight is unimportant to what? it really depends on what you are talking about. the question is really vague.
Click to expand...

A train is approaching a loop. What determines if the train will successfully complete the loop? I want the mathematical answer. The equations are what I'm looking for.

Edit: I'm not really looking for a "how-to" on roller coaster design, just the above mentioned problem. 🙂
 
Energy wise, a loop is the same as a hump or hill, with the equations mentioned.

If you really want a loop, you have to calculate the current speed of the train through the loop to find centripetal acceleration, compare that to the gravitational acceleration, then find a new force of friction.

Coefficient of drag? You mean air resistance? I don't think there's an equation for that.
 
If I'm reading right, maggotry is trying to get at whether mass matters in having a rollercoaster come into a loop, reach the top, and still follow the loop down and safely continue (as opposed to the Coyote/Roadrunner version where the car reaches the top, stops, then drops straight down). I think weight doesn't matter as it becomes a matter of centripetal force keeping the car against the track to overcome the gravitational pulling down.

Newton's:
F=ma

Centripetal Force:
F_up = mv^2 / r

Gravitational Force:
F_dn = ma

For the car to stay on the track:
F_up >= F_dn
==>
mv^2 / r >= ma
v^2 /r >= a

mass cancels.

edit: if you want to add coefficient of friction, it gets messy as then you'd have to integrate F=ma and find continual velocity, then F_up. Air drag... well, that's fluid dynamics and I'm not. 😛
 
with that you find only the velocity at the top. the car needs enough kinetic energy at the bottom to reach the top

so you have

0.5mv^2 = mg(h = 2r)
v^2 = 4rg
v = 2sqrt(rg) -> atleast this amount of speed needed

more energy is needed because you can't have zero speed at the top
i think a more detailed analysis would be required to find the real min speed.

again this is ignoring friction 😀
 
Originally posted by: tenchim
If I'm reading right, maggotry is trying to get at whether mass matters in having a rollercoaster come into a loop, reach the top, and still follow the loop down and safely continue (as opposed to the Coyote/Roadrunner version where the car reaches the top, stops, then drops straight down). I think weight doesn't matter as it becomes a matter of centripetal force keeping the car against the track to overcome the gravitational pulling down.

Newton's:
F=ma

Centripetal Force:
F_up = mv^2 / r

Gravitational Force:
F_dn = ma

For the car to stay on the track:
F_up >= F_dn
==>
mv^2 / r >= ma
v^2 /r >= a

mass cancels.

edit: if you want to add coefficient of friction, it gets messy as then you'd have to integrate F=ma and find continual velocity, then F_up. Air drag... well, that's fluid dynamics and I'm not. 😛
Click to expand...



THAT'S what I'm looking for! Thanks! 😀
 
To say that mass doesn't matter if you ARE going to account for wind resistance is silly. If the train weighs nothing, for example, it's not gonna fall. If it has negative mass, made of milar and filled with helium, it isn't gonna fall either. It might ever run backwards. But if the train is very light, wind resistance will stop it on a shallow hill. It just ain't gonna run if it doesn't have sufficient mass to pretty much make the air resistance factor irrelevant. At the normal ranges that costers in the real world exist, mass shouldn't matter because the velocity at which things fall is independent of mass, as per Newton. On a frictionless track the only factor of importance would be that the second hill was ever so slightly lower than the starting point and so on for the third and forth respectively. They could be the same heights, actually if you gave the cars a little push to get them over the first hill.
 
Originally posted by: Maggotry
Originally posted by: tenchim
If I'm reading right, maggotry is trying to get at whether mass matters in having a rollercoaster come into a loop, reach the top, and still follow the loop down and safely continue (as opposed to the Coyote/Roadrunner version where the car reaches the top, stops, then drops straight down). I think weight doesn't matter as it becomes a matter of centripetal force keeping the car against the track to overcome the gravitational pulling down.

Newton's:
F=ma

Centripetal Force:
F_up = mv^2 / r

Gravitational Force:
F_dn = ma

For the car to stay on the track:
F_up >= F_dn
==>
mv^2 / r >= ma
v^2 /r >= a

mass cancels.

edit: if you want to add coefficient of friction, it gets messy as then you'd have to integrate F=ma and find continual velocity, then F_up. Air drag... well, that's fluid dynamics and I'm not. 😛
Click to expand...



THAT'S what I'm looking for! Thanks! 😀
Click to expand...

keep in mind though that's the speed at the top of the loop, not the bottom as it approaches the loop.
 
here's another analysis that will land you closer to the answer.

say you got the train coming at the loop at a speed vi. it will spend some kinetic energy to get to the top of the loop (raised GPE) and it has to maintain enough speed at the top to not fall off so you have

vi is the nintial speed
vf is the speed at the top of the loop

-0.5m (vf^2 - vi^2) = mg2r
vi^2 - vf^2 = 4rg

vf needs to be big enough so there's enough centripetal acceleration to account for the gravity so vf^2 = rg

so vi^2 = 5rg

i'm not guaranteeing this is right in fact i'm not sure at all. a more detailed analysis is required. but i know you need at least this amount of speed
 
whoah nelly, all this reading is definitely over my head, I'm way confused now 😕
I'm glad I didn't take physics 'cause I know I would've flunked it. Math isn't my strong point anyhow.

Still an interesting read though 🙂

So if the first loop is not the biggest, then as it approaches other loops it will begin to lose momentum? Is that correct?
 
Peelucky, it isn't that hard. Galaleo said that a light weight falls as fast as a heavy one. Thirty two meters per second per second on earth. Thqt means that a heavy coaster will fall at exactly the same speed as a light one on the exact same track, cause with all identical slopes you are just talking about falling down an inclined plane. Everything falls down that too at the same speed. So the two coasters will fall at the same speed and slow down as the rise exactly the same if you leave out friction. Now since we're talking real world masses of coasters, friction won't have that much effect so mass is not an issue as galaleo said. The whole issue is just a slightly more complecated version of do things of different mass fall at the same rate. Yes, so mass is not an issue.
 
Believe it or not the length of the train will change how large the loop can be. Smaller train, taller loop.

And no it is not because of the way they couple. It has been a while for me as well, but we spent about a week on this in Physics @ ISU.
 
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