Calculate the length (m) of tangent of a 5-degree curve, if the deflection angle is 60 degree.
198.6
Calculating Tangent Length for a Circular Curve
This question asks us to determine the length of the tangent for a circular curve, given its degree and the total deflection angle.
Here's what we are given:
- Degree of the curve, \(D = 5^{\circ}\). The degree of curve defines the sharpness of the curve and is related to its radius. A common definition relates the degree of curve to the radius such that \(R = \frac{1718.9}{D}\) for the radius in meters, based on the arc definition where the central angle subtended by a 100ft arc is D degrees, and then converted to meters. Alternatively, it could be based on a metric standard. For this problem, assuming the relationship \(R = \frac{1718.9}{D}\) yields the radius in meters seems appropriate to arrive at the provided options.
- Deflection angle (or intersection angle), \(\Delta = 60^{\circ}\). This is the total angle turned by the curve.
We need to calculate the Tangent Length (T) of the curve.
Steps to Calculate Tangent Length
To find the tangent length, we first need to calculate the radius (R) of the curve using the given degree of curve (D).
Step 1: Calculate the Radius (R)
Using the relationship \(R = \frac{1718.9}{D}\), where R will be in meters:
\[ R = \frac{1718.9}{5^{\circ}} \]
\[ R = 343.78 \text{ m} \]
So, the radius of the 5-degree curve is approximately 343.78 meters.
Step 2: Calculate the Tangent Length (T)
The tangent length (T) of a simple circular curve is calculated using the formula:
\[ T = R \tan\left(\frac{\Delta}{2}\right) \]
We have the radius \(R = 343.78\) m and the deflection angle \(\Delta = 60^{\circ}\). The half deflection angle is \(\frac{\Delta}{2} = \frac{60^{\circ}}{2} = 30^{\circ}\).
Now, plug these values into the formula:
\[ T = 343.78 \times \tan(30^{\circ}) \]
The value of \(\tan(30^{\circ})\) is approximately 0.57735.
\[ T \approx 343.78 \times 0.57735 \]
\[ T \approx 198.59 \text{ m} \]
Rounding to one decimal place, the tangent length is approximately 198.6 m.
This calculated value matches one of the given options.
Summary of Calculations
| Parameter | Symbol | Value |
|---|---|---|
| Degree of Curve | D | \(5^{\circ}\) |
| Deflection Angle | \(\Delta\) | \(60^{\circ}\) |
| Calculated Radius | R | \(343.78\) m |
| Calculated Tangent Length | T | \(198.59\) m |
The calculated tangent length is approximately 198.6 meters.
Revision Table: Simple Curve Elements
| Element | Symbol | Formula | Description |
|---|---|---|---|
| Radius | R | Varies with definition of D (e.g., \(R=\frac{1718.9}{D}\) m) | Radius of the circular arc |
| Tangent Length | T | \(R \tan(\Delta/2)\) | Distance from Tangent Point to Point of Intersection |
| Length of Curve | L | \(R \Delta \frac{\pi}{180}\) (Δ in degrees) or \(R \Delta\) (Δ in radians) | Length along the arc from BC to EC |
| Long Chord | LC | \(2R \sin(\Delta/2)\) | Straight line distance from BC to EC |
| External Distance | E | \(R(\sec(\Delta/2) - 1)\) or \(R(\frac{1}{\cos(\Delta/2)} - 1)\) | Distance from the Point of Intersection to the midpoint of the curve |
| Mid-ordinate | M | \(R(1 - \cos(\Delta/2))\) | Distance from the midpoint of the long chord to the midpoint of the curve |
Where BC is Beginning of Curve, EC is End of Curve, and PI is Point of Intersection.
Additional Information: Degree of Curve Definitions
The term "degree of curve" has different definitions, which can affect the calculation of the radius (R). It's important to know which definition is being used in a particular context.
- Arc Definition: The degree of curve (D) is the central angle subtended by an arc of a standard length (e.g., 100 ft or 20m or 30m).
- For a 100 ft arc: \(R = \frac{100 \times 180}{\pi D} \approx \frac{5729.58}{D}\) feet.
- For a 20 m arc: \(R = \frac{20 \times 180}{\pi D} \approx \frac{1145.92}{D}\) meters.
- For a 30 m arc: \(R = \frac{30 \times 180}{\pi D} \approx \frac{1718.87}{D}\) meters. (The value 1718.9 used in the calculation above is consistent with this definition if the length unit is meters).
- Chord Definition: The degree of curve (D) is the central angle subtended by a standard chord length (e.g., 100 ft or 20m or 30m).
- For a 100 ft chord: \(R = \frac{50}{\sin(D/2)}\) feet.
- For a 20 m chord: \(R = \frac{10}{\sin(D/2)}\) meters.
- For a 30 m chord: \(R = \frac{15}{\sin(D/2)}\) meters.
The problem implicitly used the arc definition relating D to R via \(R \approx \frac{1718.9}{D}\) meters to arrive at the given options.
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