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Problem
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problem
Consider a uniform three-dimensional object of elliptical cross section with mass ``, the semi-major axis and semi-minor axis of the cross-section, `` and ``, respectively. It is placed on a horizontal ground, with stable equilibrium established. A perturbation is applied to the object to make a small pure rolling oscillation on the ground (without slipping). Find the period of this oscillation given by ``, with the acceleration due to gravity as ``. Assume there are no energy losses, and the only energy input comes from the perturbation.
Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.
Plain-text mathematical notation (without MathML)
Consider a uniform three-dimensional object of elliptical cross section with mass `m`, the semi-major axis and semi-minor axis of the cross-section, `A` and `B`, respectively. It is placed on a horizontal ground, with stable equilibrium established. A perturbation is applied to the object to make a small pure rolling oscillation on the ground (without slipping). Find the period of this oscillation given by `ω`, with the acceleration due to gravity as `g`. Assume there are no energy losses, and the only energy input comes from the perturbation. Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.
Original LaTeX notation
Consider a uniform three-dimensional object of elliptical cross section with mass `\(m\)`, the semi-major axis and semi-minor axis of the cross-section, `\(A\)` and `\(B\)`, respectively. It is placed on a horizontal ground, with stable equilibrium established. A perturbation is applied to the object to make a small pure rolling oscillation on the ground (without slipping). Find the period of this oscillation given by `\(\omega\)`, with the acceleration due to gravity as `\(g\)`. Assume there are no energy losses, and the only energy input comes from the perturbation. Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.
subject
physics
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