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

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SUMMARY

In a race down an inclined plane, a cube reaches the bottom before a solid cylinder due to differences in energy conversion. The cube slides down a frictionless surface, while the cylinder rolls down a surface with friction, converting some of its gravitational potential energy into rotational kinetic energy. The moment of inertia plays a crucial role; the solid cylinder has a moment of inertia of 1/2*M*R^2, while a hoop has M*R^2. The object with the smaller moment of inertia per unit mass will always win the race, regardless of mass or radius, as long as it rolls without slipping.

PREREQUISITES
  • Understanding of gravitational potential energy and kinetic energy
  • Familiarity with the concept of moment of inertia
  • Knowledge of rolling motion and friction
  • Basic principles of physics related to inclined planes
NEXT STEPS
  • Research the concept of moment of inertia in different shapes, such as spheres and cylinders
  • Explore the physics of rolling motion and the effects of friction on different surfaces
  • Learn about energy conservation in mechanical systems
  • Investigate the dynamics of various geometric shapes on inclined planes
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Physics students, educators, and anyone interested in understanding the principles of motion and energy conversion in rolling objects.

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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