In a race down inclined plane why does a cube reach bottom first?

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Discussion Overview

The discussion revolves around the dynamics of objects racing down an inclined plane, specifically comparing a cube and a solid cylinder. It explores the factors influencing their speeds, including energy conversion, moment of inertia, and the effects of friction on their motion.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants suggest that the cube reaches the bottom first because it slides down a frictionless surface, while the cylinder rolls down a surface with friction, converting some of its energy into rotational kinetic energy.
  • There is a proposal that the outcome of the race depends on the moment of inertia of the objects, with the one having the smaller moment of inertia winning.
  • One participant notes that the moment of inertia for a solid cylinder is 1/2*M*R^2, while for a hoop it is M*R^2, implying that the solid cylinder would go down faster.
  • Another participant questions whether a cube with a greater moment of inertia than the cylinder would reach the bottom second.
  • There is a light-hearted suggestion that cubes do not roll well, and alternative shapes like tetrahedrons or icosahedrons could be considered for smoother motion.

Areas of Agreement / Disagreement

Participants generally agree on the role of moment of inertia and energy conversion in determining the outcome of the race. However, there are competing views regarding the implications of different shapes and their rolling behavior, and the discussion remains unresolved on whether the moment of inertia of the cube affects its performance relative to the cylinder.

Contextual Notes

The discussion assumes a frictionless surface for the cube and a frictional surface for the cylinder, which may not reflect real-world conditions. Additionally, the implications of mass and radius on the outcome are noted but not fully resolved.

positron
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In a race down inclined plane why does a cube reach bottom first? The other object is a solid cylinder. The cylinder rolls without slipping, and the cube slides. The cylinder has radius R, and a cube has radius R. Does this depend on the mass of the objects? Is it because since the cube doesn't slide, none of its energy is converted into rotational KE as happens in the case of the cylinder?
 
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positron said:
Is it because since the cube doesn't slide, none of its energy is converted into rotational KE as happens in the case of the cylinder?
That's right. Of course you have to "cheat" a bit and assume that the cube slides down a frictionless surface, while the cylinder rolls down a surface with friction. (Otherwise the cylinder would just slide down also.)

For fun: Solid cylinder versus hoop--which wins that race? Does it depend on mass? On radius? (Figure it out.)
 
Doc Al said:
That's right. Of course you have to "cheat" a bit and assume that the cube slides down a frictionless surface, while the cylinder rolls down a surface with friction. (Otherwise the cylinder would just slide down also.)
For fun: Solid cylinder versus hoop--which wins that race? Does it depend on mass? On radius? (Figure it out.)

It be the one with the smaller moment of inertia. I for a solid cylinder of the same radius and mass as the hoops is larger, so it would go down faster. I for the solid cylinder is 1/2*M*R^2 and I for the hoops is just M*R^2. If the moment of inertia of the cube were greater than the cylinder, would it reach the bottom second?
 
positron said:
It be the one with the smaller moment of inertia. I for a solid cylinder of the same radius and mass as the hoops is larger, so it would go down faster. I for the solid cylinder is 1/2*M*R^2 and I for the hoops is just M*R^2.
Right. The one with the smallest rotational inertia per unit mass would win. (Note: It doesn't depend on mass or radius as long as the object rolls without slipping.)
If the moment of inertia of the cube were greater than the cylinder, would it reach the bottom second?
Cubes don't roll very well. :wink:
 
Doc Al said:
Cubes don't roll very well. :wink:

You should use a tetraheder ; it eliminates one bump :biggrin:
 
vanesch said:
You should use a tetraheder ; it eliminates one bump :biggrin:

How about an icosahedron so the bumps are smaller? :wink:
 

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