To solve for the angle $\alpha$ defined by $\angle PAC$ in the given belt-pulley system, we can approach the problem using trigonometry and the geometry of the setup. Here's a step-by-step explanation:

Given:

• Radii $AP = 10$ units and $CQ = 5$ units.
• The distance $AC = 20$ units.
• Points $P$ and $Q$ are where the perpendicular tangents from the pulleys meet the belt.

Steps to solve for $\alpha$:

1. Identify the Relevant Triangle:

   • $\angle PAC$ forms a right triangle $\triangle PAC$ with $AP$ as one leg, $AC$ as the hypotenuse, and $CP$ (not explicitly mentioned but can be inferred as part of the system) as the other leg.

2. Apply the Cosine Law (Alternatively, use right triangle trigonometry):

   • Since we know $AC$, $AP$, and $CQ$, we can use the cosine rule, but given that $P$ and $Q$ are perpendicular to the belt, a simpler approach is using right triangle trigonometry: $\cos(\alpha) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{AP}{AC}$ So, $\cos(\alpha) = \frac{10}{20} = 0.5$ $\alpha = \cos^{-1}(0.5) = 60^\circ$

Conclusion:

The angle $\alpha$ defined by $\angle PAC$ is $\boxed{60^\circ}$.

Would you like more details on any step, or do you have any questions? Here are five questions related to this problem:

1. What are the conditions for using the cosine rule instead of basic trigonometry in triangle problems?
2. How would the angle change if the distance $AC$ were different?
3. Why is the relationship between the pulley radii and distances important in determining angles in pulley systems?
4. Can the sine rule be applied here, and if so, how?
5. How can this concept be extended to understand angles in more complex mechanical systems?

Tip: When working with angles in geometry problems, always check if the problem can be simplified using basic trigonometric functions before applying more complex rules like the law of cosines.