A crossed belt connects a 24-centimeter pulley
(x = 12-cm radius)
on an electric motor with a 48-centimeter pulley
(y = 24-cm radius)
on a saw arbor, as shown in the figure below. The electric motor runs at 1,900 revolutions per minute.

Let L be the total length of the belt. Write L as a function of 𝜑, where 𝜑 is measured in radians. 

To find the length of the crossed belt connecting two pulleys, we'll derive the formula based on their radii and the angle 𝜑, measured in radians.

### Problem Breakdown:
- **Pulley 1 (Motor Pulley)**: Radius = 12 cm
- **Pulley 2 (Saw Pulley)**: Radius = 24 cm
- **Revolutions per minute of the motor**: 1,900 rpm (this is additional information that isn't directly needed for the belt length but is useful for rotational speed relations).

### Step-by-Step Solution:

1. **Identify Key Points:**
   - For a crossed belt, the belt wraps around each pulley and crosses between them.
   - The belt length, \( L \), consists of the arcs around each pulley and the straight segments where the belt crosses between the pulleys.

2. **Define 𝜑:**
   - 𝜑 is the angle formed by the belt at each pulley, measured in radians.

3. **Arc Lengths:**
   - The length of the arc on the smaller pulley (Pulley 1) is \( r_1 \times \phi = 12 \times \phi \).
   - The length of the arc on the larger pulley (Pulley 2) is \( r_2 \times \phi = 24 \times \phi \).

4. **Straight Segment Length:**
   - Let’s denote the angle from the point where the belt leaves one pulley to the point where it touches the other pulley as 𝜃. Since this is a crossed belt, the angle is \( \pi - 2\phi \).

5. **Length of the Crossed Segments:**
   - For crossed belts, the straight segment length between two points can be found using geometry:  
     \[
     L_{\text{straight}} = 2 \times \sqrt{(r_1 + r_2)^2 - (r_1 - r_2)^2 \sin^2(\phi)}
     \]

6. **Total Belt Length \( L \):**
   - Combine the arc lengths and the straight segments to get the total length:
   \[
   L = 12\phi + 24\phi + 2 \sqrt{(12 + 24)^2 - (24 - 12)^2 \sin^2(\phi)}
   \]

   Simplifying the equation:
   \[
   L = 36\phi + 2 \sqrt{36^2 - 12^2 \sin^2(\phi)}
   \]

   Further simplified:
   \[
   L = 36\phi + 2 \sqrt{1296 - 144 \sin^2(\phi)}
   \]

Thus, the total length \( L \) as a function of 𝜑 is:

\[
\boxed{L(\phi) = 36\phi + 2 \sqrt{1296 - 144 \sin^2(\phi)}}
\]

Would you like more details on the derivation or have any specific questions?

### Related Questions:
1. How does the belt length change if the radii of the pulleys are different?
2. What is the impact of changing the angle 𝜑 on the total belt length?
3. How does the belt length formula differ for a crossed vs. an open belt system?
4. Can the revolutions per minute (rpm) of the motor affect the belt length calculation?
5. How would you calculate the belt tension given the length and the forces acting on it?

**Tip:** Always double-check the units when dealing with radius and arc lengths to maintain consistency in calculations.

A crossed belt connects a 24-centimeter pulley (x = 12-cm radius) on an electric motor with a 48-centimeter pulley (y = 24-cm radius) on a saw arbor. The electric motor runs at 1,900 revolutions per minute. Let L be the total length of the belt. Write L as a function of 𝜑, where 𝜑 is measured in radians.

Calculate the total length of a crossed belt connecting two pulleys with radii of 12 cm and 24 cm. The problem involves key trigonometric and geometric concepts, including arc length and straight segment calculations. Suitable for students in Grades 10-12.

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