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

In summary, the speed at which a cube and cylinder reach the bottom of an inclined plane depends on the mass and radius of the objects. The cube slides down while the cylinder rolls without slipping. The object with the smaller moment of inertia per unit mass will reach the bottom first. The type and amount of friction also play a role in the race.
  • #1
positron
21
0
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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  • #2
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.)
 
  • #3
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?
 
  • #4
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:
 
  • #5
Doc Al said:
Cubes don't roll very well. :wink:

You should use a tetraheder ; it eliminates one bump :biggrin:
 
  • #6
vanesch said:
You should use a tetraheder ; it eliminates one bump :biggrin:

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

1. Why does a cube reach the bottom first in a race down an inclined plane?

The cube reaches the bottom first because it has a smaller surface area compared to other objects, resulting in less air resistance and friction. This allows it to move down the inclined plane at a faster rate.

2. Does the mass of the cube affect its speed down the inclined plane?

Yes, the mass of the cube does affect its speed down the inclined plane. Objects with a larger mass have a greater inertia, meaning they require more force to move. Therefore, a heavier cube would move slower down the inclined plane compared to a lighter cube.

3. How does the angle of the inclined plane affect the race?

The angle of the inclined plane affects the race by changing the force of gravity acting on the objects. A steeper angle will result in a greater force pulling the objects down, causing them to reach the bottom faster. However, if the angle is too steep, the objects may slide or roll off the inclined plane before reaching the bottom.

4. Are there any other factors that could affect the race down an inclined plane?

Yes, there are other factors that could affect the race down an inclined plane. These include the type of surface the objects are sliding on, the shape and size of the objects, and any external forces acting on the objects such as wind or friction from the air.

5. Is the race down an inclined plane affected by the laws of motion?

Yes, the race down an inclined plane is affected by the laws of motion. The first law of motion states that an object will remain at rest or in motion at a constant speed in a straight line unless acted upon by an external force. The second law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The third law states that for every action, there is an equal and opposite reaction. All of these laws play a role in determining the speed and movement of objects down an inclined plane.

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