To balance a 29.8 kg automobile tire and wheel, a mechanic must place a 58.2 g lead weight 25.0 cm from the center of the wheel. When the wheel is balanced, its center of mass is exactly at the center of the wheel. How far from the center of the wheel was its center of mass before the lead weight was added?

To solve this problem, we need to understand the concept of balancing a wheel.

Balancing a wheel involves adjusting the weight distribution so that the center of mass of the wheel coincides with the center of the wheel itself. This ensures that the wheel rotates smoothly without vibrations.

In this case, we have a tire and wheel weighing 29.8 kg and a lead weight of 58.2 g (or 0.0582 kg) placed 25.0 cm from the center of the wheel.

To find the distance from the center of the wheel to its center of mass before the lead weight was added, we need to use the principle of moments.

The principle of moments states that in equilibrium, the sum of the clockwise moments about any point must be equal to the sum of the counterclockwise moments about that same point.

Therefore, we can set up the following equation:

Clockwise moments = Counterclockwise moments

For the clockwise moments, we have the weight of the tire and wheel acting at their center of mass, which is unknown (let's call it x) from the center of the wheel:

Clockwise moment = (29.8 kg) * x

For the counterclockwise moments, we have the lead weight acting at a distance of 25.0 cm (or 0.25 m) from the center of the wheel:

Counterclockwise moment = (0.0582 kg) * (0.25 m)

Setting these two moments equal to each other, we can solve for x:

(29.8 kg) * x = (0.0582 kg) * (0.25 m)

x = (0.0582 kg * 0.25 m) / 29.8 kg

x = 0.0004855 m

Therefore, the center of mass of the wheel before the lead weight was added was approximately 0.0004855 meters (or 0.4855 mm) from the center of the wheel.