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If you had two bodies with the same weight but one having mass concentrated more in the center, while the other had most mass concentrated on the outside, but both had the same center of mass and gravity, and an equal force was applied on the same spot on both, would the torque induced be different?

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The definition of torque is force times lever arm. As you state it, force and lever arm are identical, so is the torque.

There are differences. Because the moment of inertia differs, the applied torque will cause a different angular velocity (if the force is not in line with the centre of mass).

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