A small uniform disk and a small uniform sphere are released simultaneously at the top of a high inclined plane, and they roll down without slipping. Which one will reach the bottom first?

the sphere

Explanation:

From the given information,

A free flow body diagrammatic expression  for the small uniform disk and a small uniform sphere which are released simultaneously at the top of a high inclined plane  can be seen in the image attached below.

From the diagram;

The Normal force mgsinθ - Friction force F  = mass m × acceleration a

Meanwhile; the frictional force

where

 in a rolling motion

Then;

∴

The Normal force mgsinθ - F  =  m ×  a     can be re-written as:

making a the subject of the formula, we have:

Similarly;

I = mk²  in which k is the radius of gyration

∴

replacing I = mk² into the above equation , we have:

where;

the uniform disk  

the uniform  sphere

∴

We can now see that the uniform sphere is greater than the disk as such the sphere will reach the bottom first.