The sleeper density of a BG track is (n+6). If the length of one BG rail is 13 m then find the number of sleepers per 1.024 km of track.
1497
Understanding Sleeper Density in Railway Tracks
Sleeper density is an important parameter in railway track construction. It refers to the number of sleepers used per rail length. This density determines the spacing between sleepers and affects the overall stability and load distribution of the track.
The question provides the sleeper density for a Broad Gauge (BG) track as \((n+6)\). Here, \(n\) represents the length of the rail in meters. The length of one BG rail is given as 13 meters.
Calculating Sleeper Density
Given that the length of one BG rail (\(n\)) is 13 meters, we can find the sleeper density using the formula \((n+6)\):
\[ \text{Sleeper Density} = n + 6 \] \[ \text{Sleeper Density} = 13 + 6 = 19 \text{ sleepers per rail length} \]This means for every 13-meter length of rail, there are 19 sleepers.
Calculating Number of Sleepers per Kilometre
We need to find the number of sleepers for a track length of 1.024 km. First, let's convert the track length from kilometers to meters.
We know that 1 kilometer = 1000 meters.
\[ \text{Track Length} = 1.024 \text{ km} = 1.024 \times 1000 \text{ meters} = 1024 \text{ meters} \]Now, we need to determine how many rail lengths of 13 meters are present in the total track length of 1024 meters.
\[ \text{Number of Rail Lengths} = \frac{\text{Total Track Length}}{\text{Length of one rail}} \] \[ \text{Number of Rail Lengths} = \frac{1024 \text{ m}}{13 \text{ m/rail}} \approx 78.769 \]Since each rail length requires 19 sleepers (as calculated by the sleeper density), the total number of sleepers for the 1024-meter track length is the number of rail lengths multiplied by the sleeper density per rail length.
\[ \text{Total Number of Sleepers} = \text{Number of Rail Lengths} \times \text{Sleeper Density} \] \[ \text{Total Number of Sleepers} = \frac{1024}{13} \times 19 \] \[ \text{Total Number of Sleepers} \approx 78.769 \times 19 \] \[ \text{Total Number of Sleepers} \approx 1496.62 \]Since the number of sleepers must be a whole number, we look at the given options. The value 1496.62 is very close to 1497. In practical scenarios, the number of sleepers is rounded up to the nearest whole number to ensure sufficient support.
Therefore, the number of sleepers per 1.024 km of track is approximately 1497.
Summary of Calculation Steps
- Identify the given rail length \(n\).
- Calculate the sleeper density using \(n+6\).
- Convert the total track length to meters.
- Calculate the number of rail lengths in the total track length.
- Multiply the number of rail lengths by the sleeper density to get the total number of sleepers.
Calculation:
| Parameter | Value |
|---|---|
| Rail Length (\(n\)) | 13 m |
| Sleeper Density (\(n+6\)) | \(13+6=19\) sleepers/rail length |
| Track Length | 1.024 km = 1024 m |
| Number of Rail Lengths | \(1024 / 13 \approx 78.769\) |
| Total Sleepers | \(78.769 \times 19 \approx 1496.62\) |
Rounding 1496.62 to the nearest whole number gives 1497.
Important Questions from Track Design
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