Problem Statement
Find the maximum permissible error in laying off the direction of an offset so that maximum displacement may not exceed 0.025 cm on paper given that length of the offset is 15 m, the scale is 1 cm to 50 cm and the maximum error in length of the offset is 0.5 m.
Step-by-Step Solution
Key Information
- Length of the offset (l) = 15 m
- Scale = 1 cm to 50 cm
- Maximum error in length of the offset = 0.5 m
- Maximum displacement on paper = 0.025 cm
- Goal: Find the maximum permissible angular error (α°)
Step 1: Understand the Sources of Displacement
There are two sources of displacement error:
1. Angular error (direction): l sin α° = 15 sin α°
2. Length error: 0.5 m (given)
Step 2: Calculate the Total Displacement Due to Both Errors
The total displacement can be calculated using the Pythagorean theorem:
Total displacement = √[(15 sin α)² + (0.5)²]
Since we need this in cm on paper, we need to divide by the scale factor:
Displacement on paper = √[(15 sin α)² + (0.5)²] / 50
Step 3: Set Up the Equation Based on Maximum Allowable Displacement
The maximum displacement must not exceed 0.025 cm on paper:
√[(15 sin α)² + (0.5)²] / 50 = 0.025
Step 4: Solve for the Maximum Angular Error (α°)
Multiply both sides by 50:
√[(15 sin α)² + (0.5)²] = 0.025 × 50
√[(15 sin α)² + (0.5)²] = 1.25
Square both sides:
(15 sin α)² + (0.5)² = 1.25²
(15 sin α)² + (0.5)² = 1.5625
Solve for (15 sin α)²:
(15 sin α)² = 1.5625 – 0.25
(15 sin α)² = 1.3125
Therefore:
225 sin² α = 1.3125
sin² α = 1.3125 / 225
sin² α = 0.00583
Taking the square root:
sin α = √0.00583
sin α = 0.07638
Finally, find α in degrees:
α° = sin⁻¹(0.07638)
α° = 4° 23′
Final Result
Conceptual Explanation & Applications
Core Concepts:
- Error Propagation: In surveying, multiple error sources combine to produce a total error effect.
- Scale Factor: How physical measurements translate to drawing representations.
- Angular Precision: The requirement for angular precision increases with the length of measurement.
- Pythagorean Combination: When errors are independent, they combine as the square root of the sum of squares.
Real-World Applications:
- Topographic Surveying: Setting control points requires understanding error tolerances.
- Construction Layout: Building foundations and structural elements need precise offset measurements.
- GIS Data Collection: Understanding error propagation in spatial data collection.
- Map Production: Ensuring that maps maintain acceptable levels of positional accuracy.
- Engineering Design: Translating between field measurements and design documentation.
Why It Works:
The solution considers two independent sources of error – directional (angular) and distance measurement. The angular error causes a displacement proportional to the length of the offset, while the distance error is fixed at 0.5 m. Since these errors are perpendicular to each other, they combine using the Pythagorean theorem.
When drawn on paper at the given scale (1:50), the maximum allowable displacement is 0.025 cm. By solving backward from this constraint, we can determine the maximum angular error that would still keep the total displacement within acceptable limits.
The final result of 4° 23′ represents the tolerance for directional error when laying off a 15 m offset, considering that we already have a 0.5 m potential error in the measured length.
