The Physics of Rolling Objects Explained

When it comes to rolling objects on an incline, the hoop will go the farthest compared to a solid ball or a solid disk.

Let's break down the physics behind this phenomenon. The kinetic energy of each object plays a crucial role in determining how high they can roll up the incline.

Solid Ball:

The kinetic energy of a solid ball is calculated using the formula 1/2 MV² + 1/2 I ω². For a sphere, the moment of inertia is 2/5 MR².

So, the kinetic energy of a solid ball can be represented as 1/2 MV² + 1/5 MR² ω² = 0.7 MV².

Solid Disk:

For a solid disk, the kinetic energy formula is 1/2 MV² + 1/2 I ω², where the moment of inertia for a disk is 1/2 MR².

The kinetic energy of a solid disk is then 1/2 MV² + 1/4 MR² ω² = 0.75 MV².

Hoop:

Lastly, for a hoop, the kinetic energy formula is 1/2 MV² + 1/2 I ω², where the moment of inertia for a hoop is MR².

The kinetic energy of a hoop simplifies to MV², which is the largest among the three objects.

Therefore, the hoop has the highest kinetic energy, allowing it to roll the farthest up the incline. On the other hand, the solid ball will reach the lowest height due to having the least kinetic energy among the three objects.