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I'm doing a marble roller coaster project for my physics class at school. The first part of our roller coaster involves a marble falling downwards into a curve that will drop then go up 12 inches. What is the optimal shape of the curve so that the ball preserves the most momentum possible (so we can minimize the height that we drop it from)?

  • Maximum initial height of 36 inches (although I'm trying to minimize that)
  • Initial speed from drop of 0
  • Time taken is irrelevant
  • We're assuming a level surface
  • PVC insulator is pretty rough but will still allow a steel ball to roll down it
  • The PVC pipe is pliable (since we cut it in half) so that we can pretty much shape it in whatever way we want
  • The steel ball is around 7/16 inch diameter

Thanks Edit: Feel free to close the question as Norbert has already sufficiently explained it both in his comments and in my answer (a restatement of what he said)

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Andrew Hu
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    I'm voting to close this question as off-topic because of insufficient prior research. (The second hit on googling "optimal curve fastest" works.) – Norbert Schuch Jan 02 '16 at 21:59
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    If you actually want to build this, you most likely want a compromise between the optimal curve and sth. you can actually build. – Norbert Schuch Jan 02 '16 at 22:00
  • oh my bad my search of "marble roller coaster optimal curve" didn't work out as well. Thanks – Andrew Hu Jan 02 '16 at 22:00
  • sth doesn't show up on google. Whats that? – Andrew Hu Jan 02 '16 at 22:01
  • "Roller coaster" is probably not a good search term. -- In any case, if you only care about minimal initial height the shape is completely irrelevant due to conservation of energy, unless you include friction (which is tricky, and for which you haven't provided enough information). Brachistochrones are about the fastest way to get there. – Norbert Schuch Jan 02 '16 at 22:02
  • OK Thanks, I'm using PVC pipe insulator, so friction is an issue. Let me see if I can find something that takes friction into account – Andrew Hu Jan 02 '16 at 22:03
  • If you have a concrete scenario, provide as many details as possible, and be precise about your figure of merit (initial height? speed?). – Norbert Schuch Jan 02 '16 at 22:04
  • Maximum initial height of 36 inches (although I'm trying to minimize that), Initial speed from drop of 0, time taken is irrelevant, we're assuming a level surface, PVC insulator is pretty rough but will still allow a steel ball to roll down it, and the PVC pipe is pliable (since we cut it in half) so that we can pretty much shape it in whatever way we want. The steel ball is around 7/16 inch diameter – Andrew Hu Jan 02 '16 at 22:09
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    Please edit these informations into the question, rather than just leaving comments. But with the data you describe the shape is basically irrelevant as long as you don't have sharp bends -- that should be rather obvious intuitively, shouldn't it? – Norbert Schuch Jan 02 '16 at 22:12
  • Intuitively, yes, the shape seems irrelevant, but me being a newbie to physics (this is a high school course), I was wondering if there was another side of the problem that I didn't know about. Thanks – Andrew Hu Jan 02 '16 at 22:14
  • FYI: "sth" = something, a horrendous abbreviation IMO. – Kyle Kanos Jan 02 '16 at 22:44
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    @KyleKanos More horrendous than FYI or IMO? – Norbert Schuch Jan 02 '16 at 22:45
  • @NorbertSchuch: Yes because those are easily understood and actually convey something. – Kyle Kanos Jan 02 '16 at 22:46
  • I thought sth was some physics abbreviation meaning something like Standard Height – Andrew Hu Jan 02 '16 at 22:48
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    @KyleKanos Sth. also means sth. :-) And FYI and IMO don't convey much, because obviously your comments are for our information and reflect your opinion. ;-) – Norbert Schuch Jan 02 '16 at 22:48
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    @NorbertSchuch: Technically speaking, FYI & IMO are acronyms while sth is an abbreviation, so the two I used cannot, by definition, be compared to abbreviations, apples & oranges kind of thing – Kyle Kanos Jan 03 '16 at 00:20
  • I think that my answer shows that there is quite a bit of interesting physics in this question and I vote to re-open. This is not at all about the brachistochrone curve, as the goal is not "least time" but "lowest height". That makes it about conservation of energy, and brings out aspects of rolling vs sliding friction. – Floris Jan 03 '16 at 20:37

2 Answers2

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You want to minimize energy lost by the marble since you want the lowest height that still allows you to climb 12 inches.

A marble starting from rest will lose energy due to slipping before rolling - this means the initial slope should not be too steep; after that you want the curvature to be gradual but the more rapidly you curve back up, the shorter the total distance - which ought to minimize the total energy lost due to friction.

A bit of math: we can calculate the maximum slope allowed for a ball to roll down the slope without sliding. If the coefficient of friction is $\mu$, then the maximum slope is given by

$$\tan\theta = \frac72 \mu$$

for a solid sphere - the derivation of this can be found in an earlier answer I wrote, which also includes some interesting calculations on the amount of power dissipated due to sliding. If you have a sphere of arbitrary moment of inertia $I = k m r^2$, then the general equation is

$$\tan\theta < \frac{\mu}{1+k}$$

The derivation of this can be found at this link - although my definition of $k$ is different than the one in that link, the math should be easy to follow.

Since the stated goal here is to lose the least amount of energy (so you can start the marble from the lowest height) it seems obvious to me that you want the initial slope to be just less than the critical slope: this will minimize the distance travelled which is favorable to minimize rolling friction (but I am pretty sure sliding friction would be much more significant).

Note that the rising slope (the 12" climb at the end) also needs to be less than the critical slope - otherwise the ball will not be able to convert its rotational energy back into potential energy (it will slow down without stopping spinning).

Incidentally, if you wanted a setup that was both "fast" and "lose little energy", you might find that dropping the marble vertically off a (slightly angled) steel anvil so it bounces back to the required height might be the most efficient mechanism - although that will depend on the coefficient of restitution of the marble-anvil system... worth a little side experiment.

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Floris
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  • I never thought that slipping would cause it to lose energy. I guess that does make sense, thanks! I'll experiment to find out the best slope to put it at. This was the answer I needed in the end. – Andrew Hu Jan 03 '16 at 01:29
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Thanks to @Norbert Schuch for this answer

While time taken for the ball to travel the curve can be minimized through the use of a Brachistochrone curve, the shape of the curve doesn't really affect the upwards distance that the ball travess when you drop it from a certain height (unless there are oddities like sharp bends in the "track")

Andrew Hu
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  • Not sure brachistochrone curve works for a rolling ball (vs sliding bead)... Especially in the presence of friction. It is derived for an idealized point mass with no friction. With friction, the best path is likely shorter. – Floris Jan 03 '16 at 01:29
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    @Floris: The brachistochrone curve should work fine for a rolling ball without friction. A rolling ball as behaving exactly the same way as a sliding ball except for a constant factor given by the moment of inertia. – Peter Shor Jan 03 '16 at 06:35
  • @PeterShor yes that makes sense. But friction definitely changes things, favoring a shorter path. – Floris Jan 03 '16 at 14:28