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A balsa-wood sphere is a perfect example of a uniform solid sphere. Balsa wood is known for its lightweight properties, which makes it an interesting material for physics problems. When we examine the moment of inertia of a balsa-wood sphere, it's essential to recall that it is generally calculated about its diameter. Moment of inertia depends heavily on both mass distribution and geometry.

For a uniform solid sphere, like our balsa-wood sphere, the formula to calculate the moment of inertia about a diameter is:

• \( I_{\text{balsa}} = \frac{2}{5}MR^2 \)

where \( M \) is the mass and \( R \) is the radius.
These spheres have mass uniformly spread throughout their volume, which makes the moment of inertia calculation more straightforward compared to objects with less uniform mass distribution, such as a hollow sphere.

To find an axis where this balsa-wood sphere shares the same moment of inertia as another object, like a hollow lead sphere, we often need to use advanced techniques such as the parallel axis theorem. This theorem allows us to find the moment of inertia about different axes, that are not through the center of the sphere.

A hollow lead sphere differs significantly from a solid sphere in terms of mass distribution. Lead is much denser, and a hollow sphere only has mass on its outer shell, which changes how its moment of inertia is calculated.

For a thin-walled, hollow sphere, the mass is situated away from the axis of rotation in the center. This unique distribution gives the hollow sphere a distinct formula for its moment of inertia:

• \( I_{\text{lead}} = \frac{2}{3}MR^2 \)

where \( M \) is the mass of the sphere and \( R \) is the radius.
The moment of inertia is generally larger for hollow objects compared to uniform ones of the same mass and radius because more mass is situated further from the rotational axis.

In practical terms, understanding this difference is crucial for comparing the rotational properties of different objects, such as the uniform balsa-wood sphere and the hollow lead sphere. This understanding allows us to solve problems like identifying an axis for the solid sphere where its moment of inertia matches that of the hollow sphere.

The "Parallel Axis Theorem" is a helpful tool in physics that allows us to calculate the moment of inertia of an object about an axis parallel to an axis through its center of mass. It tells us how to adjust the moment of inertia when the rotation isn't about the axis at the object's center of mass.

The theorem formula is:

• \( I = I_{\text{cm}} + Md^2 \)

where \( I_{\text{cm}} \) is the moment of inertia at the center of mass, \( M \) is the mass, and \( d \) is the distance between the two axes.
When using this theorem in practice, we calculate the moment of inertia about the center of mass and then add \( Md^2 \) to account for the new axis.

This theorem is particularly useful in exercises like the one at hand, where the goal is to find an equivalent moment of inertia for different objects with distinct axes. For instance, the balsa-wood sphere, which normally has its moment of inertia calculated over its diameter, needs adjustment to match the moment of inertia of the hollow lead sphere, which was given for the diameter.
Using the parallel axis theorem allows us to find the specific axis for the balsa sphere that matches the moment of inertia of the lead sphere, thus solving the problem through mathematical insight and understanding of physical properties.