Can you explain #9 on pg 273 in the PWN book please?

Sure. Here it is:

Break this into a few steps. First, note that when a wheel rolls without slipping, it travels a distance of one circumference when it makes one complete revolution. Since this wheel has a radius of 12 cm, it would travel 24π cm if it made one complete revolution. Of course, it’s not going to go quite as far.

Now, note that the wheel needs only to turn 30° for the line to be parallel to the ground. (Not obvious? Draw a line parallel to the ground through the wheel’s center.)

Once you turn that first 30°, you only need to turn 90° more to get the line to be perpendicular to the ground. So in total, the wheel will turn 30° + 90° = 120°.

120° is only one third of 360°, so the wheel only makes one third of a complete revolution. Therefore, it only travels one third of its complete circumference of 24π. One third of 24π is 8π, so the answer you’d grid in would be 8.

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