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so let's suppose there is a wheel, which rotates thus moving on a flat plane, without any slopes, and the gravitational pull throughout the plane is the same as well. The height of the center of mass of the wheel does not change as well. So my question is does the wheel do work against gravity?

Personally, I believe it does because when you take the contact point of the wheel to the plane, its height changes as the wheel rotates, thus changing the potential energy of the point while rotating. However, I couldn't be sure if my hypothesis was correct by researching in various websites and couldn't conduct any simulations or experiments. So I would love to hear your opinions on the topic.

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The centre of mass stays at the same height so there is no displacement of the centre of mass along the direction of the gravitational force, thus work done is zero.

when you take the contact point of the wheel to the plane, its height changes as the wheel rotates, thus changing the potential energy of the point while rotating

If the wheel is symmetric then there is a the part of the wheel which was in contact with the ground moving upwards but there is a point which is diametrically opposite to that point which moves downwards by exactly the same amount.
So overall the wheel does no work against gravity.

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So my question is does the wheel do work against gravity?

Yes, the work done is related to the distance moved and weight of the wheel. So, work has been done against gravity.

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