An existing 6° circular curve connects a PC and PT1 as shown. It is desired to avoid having vehicles pass too close to a historical monument, so a proposal has been made to relocat... An existing 6° circular curve connects a PC and PT1 as shown. It is desired to avoid having vehicles pass too close to a historical monument, so a proposal has been made to relocate the curve 120 ft forward. The PC will remain the same. What is the length of the new curve? Read more

Steps to Solve

1. Identify Given Parameters

For the circular curve:

• Central angle, $\theta = 6^{\circ}$
• Relocation distance = 120 ft

2. Convert Central Angle to Radians

We need to convert the angle from degrees to radians for calculations:

$$ \theta_{rad} = \theta_{deg} \times \frac{\pi}{180} $$

So,

$$ \theta_{rad} = 6^{\circ} \times \frac{\pi}{180} = \frac{\pi}{30} $$

3. Determine Radius of the Original Curve

Given that the original curve's length ($L$) can be computed using:

$$ L = r \cdot \theta_{rad} $$

Rearranging gives:

$$ r = \frac{L}{\theta_{rad}} $$

However, we need to find the new radius after relocation.

4. Calculate Cord Length and New Curve Length

The chord length of a circular curve with radius $r$ can be approximated for small angles using:

$$ C = 2r \sin\left(\frac{\theta_{rad}}{2}\right) $$

The new curve length will include the previous length plus the length obtained from the relocation:

$$ L_{new} = L + C_{relocation} $$

5. Find the New Length

With the length determined based on the curve's new radius, we add the adjustment for the cord created by moving the curve:

$$ L_{new} = r \cdot \theta_{rad} + 120 $$

6. Final Calculation

Substituting values into the equation gives us:

$$ L_{new} = r_{original} + 120 $$

Finally, we'll insert the radius based on the original configuration to find the new curve length.