The objects have both translational and rotational kinetic energy since the objects are rolling.

From the conservation of momentum we have:

$\overline{)\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{m}}{{\mathbf{v}}}^{{\mathbf{2}}}{\mathbf{+}}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{I}}{{\mathbf{\omega}}}^{{\mathbf{2}}}{\mathbf{=}}{\mathbf{m}}{\mathbf{g}}{\mathbf{h}}}$

We solve for h as follows:

$\begin{array}{rcl}\mathbf{h}& \mathbf{=}& \frac{\mathbf{(}{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}\mathbf{m}{\mathbf{v}}^{\mathbf{2}}\mathbf{+}{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}\mathbf{I}{\mathbf{\omega}}^{\mathbf{2}}\mathbf{)}}{\mathbf{m}\mathbf{g}}\end{array}$

It is clear that the maximum height reached by the objects depends on the moment of inertia of the objects.

The five objects of various masses, each denoted m, all have the same radius. They are all rolling at the same speed as they approach a curved incline.

Rank the objects based on the maximum height they reach along the curved incline. Rank from largest to smallest. To rank items as equivalent, overlap them.

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