The Order of Arrival: Sphere, Disk, Hoop

Moments of Inertia Calculation

Let's start by calculating the moment of inertia of each object released on the inclined ramp.

Sphere: \(I_{sphere} = \frac{2}{5} m R^2\)

Disk: \(I_{disk} = \frac{1}{2} m R^2\)

Hoop: \(I_{hoop} = m R^2\)

Energy Conservation and Kinetic Energy

By applying the conservation of energy, we can determine their final velocities at the bottom of the ramp. The initial potential energy is converted into kinetic energy as they roll down.

For the sphere:

\(\frac{1}{5} mv^2 + \frac{1}{2} mv^2 = mgh\)

\(v_{sphere} = \sqrt{\frac{10gh}{7}}\)

For the disk:

\(\frac{3}{4} mv^2 = mgh\)

\(v_{disk} = \sqrt{\frac{4gh}{3}}\)

For the hoop:

\(mv^2 = mgh\)

\(v_{hoop}= \sqrt{gh}\)

Explanation of Results

The moment of inertia determines how quickly an object can accelerate while rolling. The object with the smallest moment of inertia reaches the bottom first due to its ability to accelerate faster.

The solid sphere arrives first, followed by the disk, and finally, the hoop reaches the bottom last due to its larger moment of inertia.

Final Conclusion

In conclusion, the order in which a sphere, disk, and hoop reach the bottom of an incline when released simultaneously from the same height is dictated by their moments of inertia. The solid sphere, with the smallest moment of inertia, arrives first, followed by the disk, and finally, the hoop, with the largest moment of inertia, arrives last.