A road curve is to be laid out on a circle. What radius should be used if the track is to change direction by 42° in distance of 44 m? (Assume π = 22/7)
60 m
Calculating Road Curve Radius
This question asks us to determine the radius of a circular road curve. We are given the distance along the curve (arc length) and the total change in direction (angle). To solve this, we need to use the relationship between arc length, radius, and the angle subtended at the center of the circle.
Understanding Arc Length and Angle
The distance along a circular arc is given by the formula:
\[ s = r \theta \]Where:
- \(s\) is the arc length (distance along the curve).
- \(r\) is the radius of the circle.
- \(\theta\) is the angle subtended by the arc at the center, measured in radians.
The question provides the angle in degrees, so the first step is to convert this angle into radians.
Converting Angle from Degrees to Radians
The relationship between degrees and radians is \(180^\circ = \pi\) radians.
To convert an angle from degrees to radians, we use the formula:
\[ \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180^\circ} \]Given angle = \(42^\circ\). We are also given to assume \(\pi = \frac{22}{7}\).
So, the angle in radians is:
\[ \theta = 42^\circ \times \frac{\frac{22}{7}}{180^\circ} \] \[ \theta = 42 \times \frac{22}{7 \times 180} \]Let's simplify the expression:
\[ \theta = \frac{42 \times 22}{7 \times 180} \]Divide 42 by 7:
\[ \theta = \frac{6 \times 22}{180} \]Multiply 6 by 22:
\[ \theta = \frac{132}{180} \]Simplify the fraction \(\frac{132}{180}\). We can divide both numerator and denominator by their greatest common divisor, which is 12.
\[ \theta = \frac{132 \div 12}{180 \div 12} \] \[ \theta = \frac{11}{15} \text{ radians} \]So, the angle of change in direction in radians is \(\frac{11}{15}\).
Calculating the Radius of the Road Curve
Now we can use the arc length formula \(s = r\theta\) and solve for the radius \(r\). We are given the distance \(s = 44\) m and we calculated \(\theta = \frac{11}{15}\) radians.
\[ 44 \text{ m} = r \times \frac{11}{15} \]To find \(r\), rearrange the formula:
\[ r = \frac{44}{\frac{11}{15}} \] \[ r = 44 \times \frac{15}{11} \]Divide 44 by 11:
\[ r = 4 \times 15 \] \[ r = 60 \text{ m} \]Therefore, the radius that should be used for the road curve is 60 m.
| Given Information | Value |
|---|---|
| Change in direction (\(\theta\)) | \(42^\circ\) |
| Distance along track (arc length \(s\)) | 44 m |
| Assumed \(\pi\) value | \(22/7\) |
Summary of Steps
- Identify the given values: arc length \(s = 44\) m, angle \(\theta = 42^\circ\).
- Recognize the need to use the arc length formula \(s = r\theta\) where \(\theta\) is in radians.
- Convert the angle from degrees to radians using the formula \(\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180}\), using \(\pi = \frac{22}{7}\).
- Substitute the converted angle and the arc length into the formula \(s = r\theta\).
- Solve the equation for the radius \(r\).
Following these steps, we found the radius \(r\) to be 60 m.
Revision Table: Road Curve Calculations
| Concept | Formula | Notes |
|---|---|---|
| Arc Length | \(s = r\theta\) | \(\theta\) must be in radians |
| Degrees to Radians | \(\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180^\circ}\) | Conversion factor |
| Radians to Degrees | \(\theta_{degrees} = \theta_{radians} \times \frac{180^\circ}{\pi}\) | Conversion factor |
Additional Information: Circular Curves in Engineering
Circular curves are fundamental elements in civil engineering design, particularly for roads, railways, and pipelines. They are used to provide a gradual transition between straight sections (tangents).
- The radius of a road curve is a crucial design parameter. A larger radius means a gentler curve, which allows for higher speeds and improves safety and comfort.
- The arc length is the actual distance traveled along the curve.
- The central angle (the angle \(\theta\) used in calculations) represents the total change in the direction of the path over the length of the curve.
- Engineers use these calculations to stake out the curve during construction, calculate earthwork volumes, and design superelevation (banking of the curve) and sight distances.
Understanding the relationship \(s = r\theta\) is essential for many practical applications involving circular motion and geometry.
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