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It is known that static friction prevents sliding motion.

When we are pushing a block, we need to overcome static friction before our block starts sliding across the surface.

However, when we are rolling a wheel, we need static friction for the wheel to move. If we provide too much torque then at the point of contact between wheel and surface, we see that the force applied to the left overcomes Static friction to the right.

This means our wheel skids/slips and does not move.

My confusion is:

  1. In the block case, we need to overcome static friction to move.

  2. In the wheel case, if we overcome static friction, we don't move.

Would greatly appreciate your input on this matter.

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in the wheel case you move the wheel, since you overcome the static friction . or once the wheel moves you could push the car,If you want to move the cars back up you still have to relay on static friction on the front wheels.

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Consider a car with power from the engine applied to the drive wheels. With no friction the wheels keep increasing in angular velocity but the center of mass (CM) of the car does not move since there is no external force on the car. With friction from the ground, the force of friction opposes the increasing angular velocity, and the car moves forward. Friction is an external force on the car and hence the CM moves in the direction of the force of friction, forward for this case.

Friction always opposes the actual or impending motion. This site has numerous answers to this question, search for car acceleration.

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We need to overcome a static friction for two surfaces starting to slide against each other. In both cases, either block or wheel starts to slide against ground, when static friction threshold is reached.

Whether or not sliding thing center of mass will have a linear movement with respect to ground- is a different story dependent on body shape, initial conditions, applied torque amount, etc. For example, even block can spin too,- make good punch into a brick edge resting on a top of mountain and see it rolling down the hill. So, moving linearly or spinning is not a solely property of static friction, but combined state of forces+torques involved and body features itself.

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