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Consider a wheel upright like this:

enter image description here

Is the force required to move the wheel by pushing it in the direction that its tire is facing the same magnitude as the force required to move it by pushing at its side?

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No they are not the same.

If you push the wheel to roll without skidding there is no surface friction, there is some other significantly smaller forces opposing the rolling such as the deformation of the tire and friction at the axel bearing.

But if you push the wheel on its side trying to force it to slide sideways, initially the static friction will resist any motion and will increase equally to the force you're pushing with until it reaches the threshold of motion $ \ f_{max\ static} = \mu_{s}N \ $ , with N being the vertical load on the wheel.

Then it will drop down and stay constant at the level of kinetic friction $ \ f_{k \ friction} = \mu_{k}N $.

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  • $\begingroup$ Literally thought of this right before you answered. Thank you! $\endgroup$ – B V Apr 15 at 6:52
  • $\begingroup$ What is the force required to roll a wheel in the ideal world? $\endgroup$ – B V Apr 15 at 20:37
  • $\begingroup$ In world with no friction the wheel will roll forever if it was rolling, if not it take just the initial small amount of work w=mv^2/2+ I omega^2/2. $\endgroup$ – kamran Apr 15 at 22:40

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