A disk an a sphere roll off two inclined planes of the same altitude and length. Which of the two objects will get to the bottom of the respective plane first? how does the result depend on the masses and diameters of the disk and the sphere?

When considering the scenario of a disk and a sphere rolling down inclined planes of equal height and length, the outcome hinges on the concepts of rotational motion and moment of inertia. Let's break this down step by step to understand which object reaches the bottom first and how their masses and diameters play a role.

The Basics of Rolling Motion

Both the disk and the sphere experience gravitational force pulling them down the incline. However, as they roll, they also rotate about their axes. This rolling motion involves two types of energy: translational kinetic energy (due to linear motion) and rotational kinetic energy (due to spinning). The total mechanical energy is conserved, meaning that the potential energy at the top of the incline converts into kinetic energy at the bottom.

Moment of Inertia

The moment of inertia is a crucial factor in determining how quickly an object rolls down an incline. It represents how mass is distributed relative to the axis of rotation. For our two objects:

• The moment of inertia for a solid disk about its center is given by \( I_{disk} = \frac{1}{2} m r^2 \).
• The moment of inertia for a solid sphere about its center is \( I_{sphere} = \frac{2}{5} m r^2 \).

Here, \( m \) is the mass and \( r \) is the radius of the object. Notice that the sphere has a smaller moment of inertia compared to the disk for the same mass and radius, which means it can convert more of its potential energy into translational kinetic energy as it rolls down.

Acceleration Down the Incline

The acceleration of each object down the incline can be derived from the equations of motion. For rolling objects, the acceleration \( a \) can be expressed as:

• For the disk: \( a_{disk} = \frac{g \sin(\theta)}{1 + \frac{I_{disk}}{m r^2}} = \frac{g \sin(\theta)}{1 + \frac{1}{2}} = \frac{2g \sin(\theta)}{3} \)
• For the sphere: \( a_{sphere} = \frac{g \sin(\theta)}{1 + \frac{I_{sphere}}{m r^2}} = \frac{g \sin(\theta)}{1 + \frac{2}{5}} = \frac{5g \sin(\theta)}{7} \)

From these equations, we can see that the sphere has a greater acceleration than the disk, meaning it will reach the bottom of the incline first.

Influence of Mass and Diameter

Interestingly, the mass and diameter of the objects do not affect the outcome in terms of which reaches the bottom first. This is because both the mass and the radius cancel out in the equations of motion. What matters is the distribution of mass, which is captured in the moment of inertia. Therefore, regardless of whether the disk and sphere are heavy or light, or large or small, the sphere will always roll down faster than the disk due to its lower moment of inertia.

Final Thoughts

In summary, when a disk and a sphere roll down inclined planes of the same height and length, the sphere will reach the bottom first. This result is primarily due to its lower moment of inertia, allowing it to convert potential energy into translational kinetic energy more efficiently than the disk. The masses and diameters of the objects do not influence this outcome, as they do not affect the relative accelerations of the two shapes.