Understanding the Mid-Ordinate in Circular Curves

The mid-ordinate (often denoted by M) is a specific geometric element of a circular curve. It represents the distance from the midpoint of the curve's tangent chord (the long chord) to the curve itself, measured along a line perpendicular to the tangent chord from the curve's center.

Given Information

We are provided with the following details for a simple circular curve:

• Radius ($R$) = 560 m
• Deflection Angle ($\Delta$) = 120°

Formula for Mid-Ordinate

The formula used to calculate the mid-ordinate ($M$) of a simple circular curve is:

$$ M = R \times (1 - \cos(\frac{\Delta}{2})) $$

Where:

• $M$ is the Mid-ordinate
• $R$ is the Radius of the curve
• $\Delta$ is the Deflection Angle

Step-by-Step Calculation

Let's calculate the mid-ordinate using the given values:

1. Calculate half of the deflection angle: $$ \frac{\Delta}{2} = \frac{120^\circ}{2} = 60^\circ $$
2. Find the cosine of the half deflection angle: $$ \cos(\frac{\Delta}{2}) = \cos(60^\circ) = 0.5 $$
3. Calculate the term $(1 - \cos(\frac{\Delta}{2}))$: $$ 1 - \cos(60^\circ) = 1 - 0.5 = 0.5 $$
4. Calculate the Mid-ordinate ($M$): $$ M = R \times (1 - \cos(\frac{\Delta}{2})) $$ $$ M = 560 \text{ m} \times 0.5 $$ $$ M = 280 \text{ m} $$

Result Analysis

The calculated value for the mid-ordinate is 280 m. Comparing this result with the given options:

The calculation shows that the mid-ordinate is 280 m, which corresponds to the second option.