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Keep on rolling

HS Thomas
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Joined: May 15, 2002
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Jack and David have designed two wheels. The tread on both (the middle O) is the same size and weight.
The hub on each, though different in size and shape, weighs exactly the same, 10 kg. If the boys release the wheel at the same time which wheel will reach at the bottom of a ramp first.You have to give reasons why.


regards
[ November 09, 2003: Message edited by: HS Thomas ]
Jim Yingst
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Ummm... so is -O- intended to indicate a small-hubbed wheel, while oOo indicates a wheel with larger hubs? The treads couldn't be exactly the same then, since they have to contact the hubs, right? But they can have the same mass, and the same outside tread diameter. OK.
Presumeably the wheels are both released from the top of the ramp, at the same time, to see which rolls down fastest.
I'll leave this unaswered for now. Will check in later...


"I'm not back." - Bill Harding, Twister
HS Thomas
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Joined: May 15, 2002
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Jim, you are right.
A wheel has one hub each. The first hub is a solid cylinder passing through the centre of the wheel. The second hub is a hollow cylinder
that passes through the wheel near the outer rim. They both weigh exactly the same 10 kg , though different in size AND shape.
Is that a better description ?
- the first hub.

The inner circle would be the hub on the second wheel with the middle area being all wheel + the area from the hub to the outer rim.
Think of the sorts of wheel you could get on Noddy's car.....
The inner circle weighs the same as the cylinder hub in the first diagram.
(I think it is the first time I've attempted to draw an ASCII diagram and failed! )
regards
[ November 10, 2003: Message edited by: HS Thomas ]
Arjun Shastry
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Joined: Mar 13, 2003
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I think both wheels will reach at the same time.Initial velocity is zero for both.For a ramp with angle of inclination 'alpha',as weight is same for both and so their mass,accleration will be m*g*cos(alpha)m/sec2.


MH
HS Thomas
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Cap , that's what I thought too. But that's not the answer given.
regards
Arjun Shastry
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Joined: Mar 13, 2003
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Oh, I forgot ,they are not sliding blocks but rolling objects.I need to open mechanics book
HS Thomas
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Mechanics ? What fly wheels ? Are / Were you an engineer by profession ?
Mechanics may provide the solution. The important thing is the hub.
regards
[ November 10, 2003: Message edited by: HS Thomas ]
David O'Meara
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Joined: Mar 06, 2001
Posts: 13459

Angular momentum is my answer
Bert Bates
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Man am I rusty, rusty, rusty... :roll:
I think the smaller hubbed wheel's "spinning enertia" (I made that term up :roll: ), will be smaller than the larger hubbed wheel's. For instance if both wheels were spinning at the same RPM, it would take more work to stop the large hubbed wheel. So if it takes more work to stop it, it'll take more work to start it. In this case, the "starting the wheel rolling" work is done by gravity, same amount for each wheel, so the small hubbed wheel will reach the ground first, and I think it'll also be going faster.
Not only do I feel rusty, I feel way out on a limb


Spot false dilemmas now, ask me how!
(If you're not on the edge, you're taking up too much room.)
HS Thomas
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Bert, you are on the right track.
For "spinning enertia" think "rotational inertia".
So what is your answer ?
Dave's "angular momentum" sounds right , too!
regards
[ November 11, 2003: Message edited by: HS Thomas ]
HS Thomas
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Originally posted by Bert Bates:
so the small hubbed wheel will reach the ground first, and I think it'll also be going faster.

Bert, sorry, I mised your answer on the initial reading.

I'll hold off giving the full answer I have and see if anyone else comes with a different explaination.
regards
[ November 11, 2003: Message edited by: HS Thomas ]
Jim Yingst
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Bert's answer looks good to me - "spinning inertia" is generally called "moment of inertia", at least in the US.
Let's expand this a bit. Assume the following objects are all released from the top of an inclined ramp at the same time. Assume there's no air resistance and each object rolls straight down the incline with no slippage. What order do the objects arrive at the bottom?
  • Solid steel cylinder, 10 cm diameter, 10 cm long
  • Solid steel cylinder, 10 cm diameter, 1 cm long
  • Solid plastic cylinder, 10 cm diameter, 10 cm long
  • Steel pipe segment, 10 cm outer diameter, 9 cm inner diameter, 10 cm long
  • Plastic pipe segment, 20 cm outer diameter, 18 cm inner diameter, 1 m long
  • Plastic pipe segment, 20 cm outer diameter, 19 cm inner diameter, 1 m long
  • Solid steel sphere, 10 cm diameter
  • Solid plastic sphere, 10 cm diamteter
  • Solid steel sphere, 1 cm diameter


  • Let's say that plastic has a density 1/10 that of steel. (Dunno, I just made that up because I'm too lazy to look it up.) Assume that the conditions are such that each object must travel the same total distance down the ramp - larger objects don't get an advantage because their leading edge arrives earlier. One way to do this is to make sure that the leading edges are all aligned at start, and then look at arrival times of leading edges. Or, align all the centers of mass initially, and then meaure arrival times of certers of mass. Either way - just don't mix the two methods up.
    [ November 12, 2003: Message edited by: Jim Yingst ]
    HS Thomas
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    Answer to the original question:

    The centre weighted wheel will have less resistance to spinning since it's main mass is more tightly packed and doesn't have far to spin.
    It's rotational inertia is less than the rim weighted wheel and so gets moving faster than the rim-weighted wheel. This ties with Bert's explaination - (ignoring the humming and hawing ).

    Of the last two examples Jim gave above:
    Solid plastic sphere, 10 cm diamteter
    Solid steel sphere, 1 cm diameter
    Assuming these weigh the same, by the same token the steel sphere is more tightly packed and takes less time to complete a spin (spins faster), so it should reach level ground first, IMHO.
    The other examples appear in the right order of reaching level ground first, comparing like shapes with like. I may be wrong.Actually , 9,8,7 would be my answer for the last three.The steel sphere 10 cm diameter would have more resistance to spinning.

    regards
    [ November 12, 2003: Message edited by: HS Thomas ]
    Jim Yingst
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    Solid plastic sphere, 10 cm diamteter
    Solid steel sphere, 1 cm diameter
    Assuming these weigh the same

    If the plastic sphere has 10 times the diameter of the steel sphere, it has 1000 times the volume. It also has 1/10 the density, so it ends up with 100 times the mass and weight.
    The steel sphere 10 cm diameter would have more resistance to spinning.
    Wouldn't it also have more weight causing it to spin? Which effect is more significant?
    HS Thomas
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    I didn't go into calculations re: the weight. You can have really light plastic and heavy steel. I was imagining they both weighed the same.
    Wouldn't it also have more weight causing it to spin? Which effect is more significant?

    Jim, I think you are saying - More weight means more spin means more speed. What about the distribution of that weight ? Or does that help the spin in the case of a sphere ? Why don't cars have wheels that are spheres then ? ( Could cut down on fuel !)
    Oh, Ok weight being the determining factor, 7,8,9 would be the orders of arrival.
    regards
    [ November 12, 2003: Message edited by: HS Thomas ]
    Jim Yingst
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    Jim, I think you are saying - More weight means more spin means more speed.
    Yes. That is, I'm not saying that the net result is more speed; I'm just pointing out that there's an argument which would suggest that bigger --> more weight --> more speed. Just as there's an argument that bigger --> more resistance to spin --> less speed. The question is, which of these is more significant? Or do they balance? I'm being deliberately vague about the answer, for now.
    What about the distribution of that weight ?
    It's very important.
    Or does that help the spin in the case of a sphere ?
    Yup.
    Why don't cars have wheels that are spheres then ? ( Could cut down on fuel !)
    The two biggest reasons that come to mind are
  • Cars need to go uphill as well as downhill, and uphill, weight counts against you.
  • Spheres would require more material than conventional wheels, and material costs money
  • HS Thomas
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    I'll think on this a while. Or leave this for the more dynamics minded to answer.

    From movies/news reels , after a single volcanic explosion , assuming rocks are thrust out at the same force, I seem to remember little rocks travelling faster than big ones. Hmmm, not always, sometimes
    big ones overtake little ones ! Different speeds, I guess.
    regards
    [ November 12, 2003: Message edited by: HS Thomas ]
    HS Thomas
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    1.The bigger the mass, the longer the moments of inertia.
    2.The greater the angular speed of rotation (spin), the more/less energy is required. ( I can't decide which)
    3.The objects at a height have a gravitational potential energy,
    Mgh which gets converted to kinetic energy but the mechanical energy will be conserved(net effect is 0).
    net mechanical energy = net kinetic energy + net gravitational potential energy = 0
    If the distance travelled is kept constant, I think it depends on the height at which the objects start rolling from. Objects travel faster from greater heights.(Or whichever height is needed to overcome 1 and 2 , perhaps).
    Shape matters ; a disk may travel faster than a solid sphere of the same mass and radius even at great heights. The only explanation I can think of for that is the distribution of weight along it's angle of rotation.

    regards
    [ November 13, 2003: Message edited by: HS Thomas ]
    HS Thomas
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    So my answer is equally vague: It depends.
    regards
    Jim Yingst
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    Well, I know y'all have been just dying to know m answer to this, so here it is: it turns out that, just like for falling objects (with no air resistance), total weight is irrelevant. Also (and perhaps more surprisingly) the diameter is irrelevant. The bigger the sphere or cylinder or pipe, the more moment of intertia it has. But a bigger object also needs less angular velocity to cover distance by rolling - basically, a big wheel needs fewer revolutions to roll down a hill. These effects ultimately balance, so that a 1 cm sphere and a 10 cm sphere and a 1 m sphere all roll down a hill at the same speed. The only remaining factor is the distribution. Objects which keep most of theiw mass close to the axis of rotation roll faster than objects which keep most of it away from the axis. So, from fastest to slowest:
    7, 8, 9: All spheres tie for first place.
    1, 2, 3: All solid cylinders tie for second.
    4,5: Pipes are slower.
    6: The pipe (relatively) thinnest walls is last.
    [ November 17, 2003: Message edited by: Jim Yingst ]
    HS Thomas
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    Jim, you mean shape doesn't matter. Oh , you meant weight.
    Where would flat disks fit in , in your examples?
    Disks will go faster than the rest.
    According to the above postulates, small disks will go fastest.
    Which ,right now , I can't imagine is correct.
    regards
    [ November 17, 2003: Message edited by: HS Thomas ]
    Jim Yingst
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    Where would flat disks fit in , in your examples?
    They're the same as solid cylinders. The length of the cylinder doesn't matter. Imagine you've got two identical solid cylinders. If you roll them down the hill at the same time, they'd roll at the same rate, since they're identical, right? This is true ieven if they start out right next to and coaxial with each other. They'd remain coaxial all they way down. So now, what difference would it make if we glued the two cylinders together? None. The combined cylinder would roll down at the same speed either individual cylinder would have. A cylinder of length x rolls at teh same speed as an otherwise wquivalent cylinder of length 2x. The same arguments could just as well apply to lengths of 10x or, reversing things, .5x or .1x. The length of the cylinder is irrelevant.
    Disks will go faster than the rest.
    According to the above postulates, small disks will go fastest.

    I don't see where that came from. All uniform solid disks and cylinders roll at the same speed.
     
     
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