A hollow, thin-shelled cylinder of mass rolls along a horizontal surface with a translational speed of v. What percent of its total kinetic energy is translational?

translational=1/2 m v^2

rotational=1/2 I w^2 where w=v/r
rotaltonal=1/2 I v^2/r but I for thin shell cylinder is http://en.wikipedia.org/wiki/List_of_moments_of_inertia
rotational=1/2 mr^2*v^2/r^2=1/2 m v^2

Hmmm. Looks like 1/2 of the energy is translational.

To determine the percent of the total kinetic energy that is translational, we need to know the ratio of the translational kinetic energy to the total kinetic energy.

Let's break it down step by step:

1. The total kinetic energy of an object is given by the formula: K = (1/2)mv², where K is the kinetic energy, m is the mass of the object, and v is its velocity.

2. For a hollow, thin-shelled cylinder, the total kinetic energy includes both rotational and translational kinetic energy. The rotational kinetic energy (K_rot) is given by the formula: K_rot = (1/2)Iω², where I is the moment of inertia of the object and ω is its angular velocity.

3. In this case, we are interested in the translational kinetic energy (K_trans), which is the kinetic energy associated with the overall linear motion of the cylinder.

4. Since the cylinder is rolling without slipping, there is a relationship between its linear speed (v) and angular speed (ω) called the rolling condition: v = ωR, where R is the radius of the cylinder.

5. We can substitute this relationship into the formulas for kinetic energy:
- For translational kinetic energy: K_trans = (1/2)mv²
- For rotational kinetic energy: K_rot = (1/2)I(ω²)

6. If we substitute the relationship v = ωR into the rotational kinetic energy formula, we get: K_rot = (1/2)I((v/R)²) = (1/2)m(v²/R²)

7. Now, let's calculate the percentage of the total kinetic energy that is translational:
- Percent_trans = (K_trans / (K_trans + K_rot)) * 100

8. Substituting the formulas for K_trans and K_rot:
Percent_trans = [(1/2)mv² / ((1/2)mv² + (1/2)m(v²/R²))] * 100

9. Further simplification yields the final formula for the percentage of the total kinetic energy that is translational:
Percent_trans = [(v²) / (v² + (v²/R²))] * 100

Thus, the percentage of the total kinetic energy that is translational can be calculated using this formula.