4.3. Example 2.3

Determine the ordinates of the points on a circular curve having a long chord of 100 m and a versed sine of 5 m. The ordinates are to be measured from the long chord at an interval of 10 m.

In this example, we are tasked with finding the ordinates for points on a circular curve. The long chord provided measures 100 meters, which is the straight line distance between the two ends of the curve. The versed sine, defined as the vertical distance from the arc to the chord, is given as 5 meters. We need to calculate the height of the arc above the chord at specific intervals (10 meters apart) along the curve.

Think of a section of a hilly road with a bridge. The long chord is like the straight line across the flat area between two hills where the bridge might sit. The versed sine indicates how high the bridge is above this flat area. To understand how far exactly above the straight path the bridge is at multiple points along its length, we measure these heights at regular intervals, just as this example details.

Length of long chord (L) = 100 m, Versed sine (O₀) = 5 m, Interval = 10 m.

To compute the ordinates, we first confirm that the long chord is 100 meters long, while the versed sine measures the height difference from the actual curve to this long chord as 5 meters. These measurements are essential for calculating how the curve translates vertically at various intervals.

Imagine stretching a rope (the long chord) between two points on a curvy path. The height of the rope above the highest point of the path (versed sine) gives a sense of how steep and curvy that path is. In our analysis, we'll measure that height incrementally to see how the curvature affects the journey.

The versined or mid-ordinate O₀ = R - √[R² - (L/2)²].

We use the provided formula to calculate the radius (R) of the curve using the length of the chord (L) and its relation to the versed sine. By substituting our known values into the formula, we’ll derive an expression that helps us find the radius needed to maintain that curve accurate to the geographical layout.

Consider how a skateboard ramp is designed. The distance across (the long chord) and its height (versed sine) are used to calculate how rounded that ramp should be to ensure a smooth ride. Similarly, we are designing our curve based on these knowns to ensure it can be navigated safely and comfortably.

Calculating ordinates at intervals, O = √[R² - (x)²] – (R - O₀).

To determine the ordinates at each 10-meter interval, we apply the previous formula to calculate each height at those specified distances. The terms in the equation help us determine how high the arc is as we progress along the curve, indicating the vertical offsets from the long chord.

Think about a roller coaster that takes twists and turns. Each segment of the ride can be seen as an interval where the height from the ground (representing our ordinate) is crucial to understand how steep or flat the ride will be. By calculating these at each part of the ride, engineers ensure safety and enjoyment for the riders.

--