Determine the normal pull (kg) for a tape of 20 m long standardized at a pull of 30 kg. The cross-section area of the tape is 0.5 square centimetres, and the weight of the tape per metre is 20 gm.

Discussion - MCQs on Chain Surveying (2) - Normal Pull

Determine the normal pull (kg) for a tape of 20 m long standardized at a pull of 30 kg. The cross-section area of the tape is 0.5 square centimetres, and the weight of the tape per metre is 20 gm. Take the modulus of elasticity for the tape material as 2,100,000 kg per square centimetre.

A. 28

B. 36.8

C. 64

D. 78

Correct Answer: B. 36.8

Note: According to the official commission, in the revised answer key for the exam this question appeared in, none of the options were considered correct. The calculated value is closest to option (b).

📝 Detailed Explanation: Calculating Normal Pull

Normal pull is a specific tension applied to a suspended measuring tape where the positive correction due to pull (stretching) perfectly cancels out the negative correction due to sag. By applying this specific pull, surveyors can eliminate two significant sources of error simultaneously.

Formula and Principle

At normal pull (P_m), the following condition is met:

Pull Correction = Sag Correction

(P_m - P₀) × L / (A × E) = (w × L)² × L / (24 × P_m²)

Given Data:

Length of tape (L) = 20 m
Standard pull (P₀) = 30 kg
Cross-section area (A) = 0.5 cm²
Weight of tape per unit length (w) = 20 gm/m = 0.02 kg/m
Modulus of elasticity (E) = 2,100,000 kg/cm²

Step-by-Step Calculation

First, we insert the given values into the equation. Note that since A and E are both given in cm², their units will cancel, so we don't need to convert them to m².

Step 1: Substitute values
(P_m - 30) × 20 / (0.5 × 2,100,000) = ((0.02 × 20)²) × 20 / (24 × P_m²)

Step 2: Simplify left side
20(P_m - 30) / 1,050,000 = (P_m - 30) / 52,500

Step 3: Simplify right side
(0.4²) × 20 / (24 × P_m²) = 0.16 × 20 / (24 × P_m²) = 3.2 / (24 × P_m²)

Step 4: Equate and cross multiply
(P_m - 30) / 52,500 = 3.2 / (24 × P_m²)
P_m² × (P_m - 30) = (3.2 × 52,500) / 24 = 7,000

Step 5: Final cubic equation
P_m³ - 30P_m² - 7000 = 0

This cubic equation is best solved by trial and error, testing the given options.

Let's test option (b), P_m = 36.8 kg:

(36.8)² × (36.8 - 30) = 1354.24 × 6.8 ≈ 9208.8

This is not exactly 7000, indicating a potential issue in the question's options or data. However, a more precise solution to the cubic equation yields approximately P_m ≈ 35.54 kg. Among the given choices, 36.8 kg is the closest value.