Understanding the Ratio of Translational and Rotational Kinetic Energy of a Rolling Hoop

Explanation:

Translational Kinetic Energy: The translational kinetic energy (KT) of the hoop is given by the formula: KT = 1/2 * m * v^2, where m is the mass of the hoop and v is its linear speed.

Rotational Kinetic Energy: The rotational kinetic energy (KR) of the hoop is given by the formula: KR = 1/2 * I * ω^2, where I = mr^2, m is the mass of the hoop, r is the radius of the hoop, and ω is the angular speed of the hoop.

To find the ratio KT/KR, we can substitute the formulas for KT and KR into the ratio:

KT/KR = (1/2 * m * v^2) / (1/2 * (mr^2) * (v/r)^2)

Simplifying the equation:

KT/KR = (m * v^2) / ((m * r^2) * (v/r)^2) = (m * v^2) / (m * v^2) = 1

Therefore, the ratio of the hoop's translational kinetic energy to its rotational kinetic energy is equal to 1. This means the translational kinetic energy of the hoop is equal to its rotational kinetic energy.

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