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What shape of object will roll down the fastest through the decline piece of wood if travelling with a straight line assuming there are frictionless ramp to stop them from going to the side given the same amount of material with their smoothness being the same? Could it be sphere or a partial sphere with cross section trimmed off? Please also including the distribution of the mass. Is the mass focused in the center given much faster velocity to roll down? What about the 3D shape generate from other curve for example, elliptical or hyperbola, etc...

Please show your math formulation...

In addition, I know that a hollow cylinder will roll down slower a solid-centered cylinder.

interesting video: https://www.youtube.com/watch?v=cB8GNQuyMPc

Interesting thing to learn from this video is that the density doesn't matter, bigger cylinder VS smaller cylinder are the same, even the weight doesn't matter in his experiments...

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  • $\begingroup$ Hollow should roll faster (atleast accelerate), unless your friction becomes an issue. Partial sphere can fall on its side but basically a suitable compromise between stability and possility to hollow it down and lower friction that does not fall over is best. As light as possible as it has least amount of innertia to accelerate but produces enough force to overcome stick friction. OTOH it also climbs worst back up again. $\endgroup$
    – joojaa
    Aug 31 '16 at 20:25
  • $\begingroup$ Not sure I understand your question. You seem to be asking what shape of an item will roll down faster on an inclined plane with normal Earth atmosphere, correct? But do the different shapes also include the distribution of their matter within the outer shape? And their mass? And their smoothness? $\endgroup$
    – Jens
    Aug 31 '16 at 21:01
  • $\begingroup$ Joojaa, hollow is slower. Have a look at the video. World nenowned physics Professeor Lewin from MIT makes a strong case for solid sections. $\endgroup$
    – SlydeRule
    Aug 31 '16 at 21:02
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    $\begingroup$ Just a minor pedantic point that has always irritated me about these brain teaser questions: if the ramp is truly frictionless, the items will not roll. They'll all slide at the same acceleration. (I'm happy to be corrected if anyone thinks I'm wrong though.) $\endgroup$
    – Andy
    Sep 1 '16 at 8:11
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A solid cylinder with a protrusion like a cam if positioned such that the rolling down starts from the inclined tip of the cam will go down faster. The cam should be formed sharp so as not to change the rotational inertia, J, of the mass too much. The cylinder should be placed in an initial tilted imbalance, ready to fall down.
The length of the ramp should be long enough to let the cylinder roll several times. This will average out the wobbling up and down of CG of the roller. This geometry will promote rapid start of rotation and faster roll down.

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Let's assume the ramp does have friction so things will roll, but not so much that it's "sticky." Then you simply need to calculate the torque being applied, which is a function of the radial distribution of mass for each object.
This is a standard set of problems in any first-year mechanics text book, so I won't dive into the math of calculating the angular moments for different shapes.
Just for a quick example: given constant total mass, a (infinitely thin) cylinder has all its mass one radius away from the axis. If you build an object of the same total mass, but with a massless cyclinder of same radius and all mass concentrated along the cylinder's axis, far less torque is required to accelerate it and it'll get going much faster than the first cylinder.

In any case, please take a read thru some introductory physics texts, or peek at good online sites such as physicsguide and hyperphysics .

After that, you might want to re-post to physics.SE

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