A 30 meter steel tape weighing 0.040 kg/m is constantly supported at each
quarter-length and at its end points, and is used to measure a line AB with a steady
pull of 5.5 kg. If the measured length of AB is 785.245 m, determine the following:
a. Correction due to sag between supports and for the whole tape length
b. Total sag correction for the whole length measured
c. Correct length of the line AB

To solve this problem, we will first calculate the sag correction for each quarter-length section of the tape, then apply this correction over the entire length of the line AB. Finally, we'll determine the corrected length of the line AB.

### Given Data:
- **Length of steel tape** = \( L = 30 \) meters
- **Weight of tape per unit length** = \( w = 0.040 \) kg/m
- **Steady pull** = \( P = 5.5 \) kg
- **Measured length of AB** = \( L_{AB} = 785.245 \) meters
- **Gravitational acceleration** = \( g = 9.81 \) m/s²

### Step 1: Calculate the weight of the tape per unit length in Newtons
The weight per unit length \( w \) in Newtons is given by:
\[
w = 0.040 \times 9.81 = 0.3924 \, \text{N/m}
\]

### Step 2: Sag correction for one quarter-length section
The sag correction for one span of the tape (suspended between two supports) can be calculated using the formula:
\[
\Delta S = \frac{w^2 L^3}{24 P^2}
\]
Here, \( L_q \) is the quarter-length of the tape, so:
\[
L_q = \frac{30}{4} = 7.5 \, \text{meters}
\]

Now substituting the values:
\[
\Delta S_q = \frac{(0.3924)^2 \times (7.5)^3}{24 \times (5.5 \times 9.81)^2}
\]

### Step 3: Total sag correction for the whole tape length
The total sag correction for the entire length of the tape is obtained by summing up the sag corrections for all four spans. Since the sag correction per span is the same, the total sag correction \( \Delta S_t \) is:
\[
\Delta S_t = 4 \times \Delta S_q
\]

### Step 4: Calculate the total sag correction over the measured length of AB
The total number of tape lengths required to measure AB is:
\[
N = \frac{785.245}{30} \approx 26.1748
\]

Now, the total sag correction over the entire length of AB is:
\[
\Delta S_{AB} = N \times \Delta S_t
\]

### Step 5: Correct length of the line AB
Finally, the corrected length of the line AB is given by:
\[
L_{AB_{corrected}} = L_{AB} + \Delta S_{AB}
\]

Let's compute these values step by step.

### Step 2: Sag Correction for one quarter-length section
\[
\Delta S_q = \frac{(0.3924)^2 \times (7.5)^3}{24 \times (5.5 \times 9.81)^2}
\]
\[
\Delta S_q \approx \frac{0.1539 \times 421.875}{24 \times 2888.3025}
\]
\[
\Delta S_q \approx \frac{64.9584}{69319.26} \approx 0.000937 \, \text{meters}
\]

### Step 3: Total Sag Correction for the Whole Tape Length
\[
\Delta S_t = 4 \times 0.000937 \approx 0.003748 \, \text{meters}
\]

### Step 4: Total Sag Correction Over the Measured Length of AB
\[
\Delta S_{AB} = 26.1748 \times 0.003748 \approx 0.0981 \, \text{meters}
\]

### Step 5: Correct Length of the Line AB
\[
L_{AB_{corrected}} = 785.245 + 0.0981 \approx 785.3431 \, \text{meters}
\]

### Summary:
a. **Sag correction for the whole tape length**: \( \Delta S_t \approx 0.003748 \) meters  
b. **Total sag correction for the whole length measured**: \( \Delta S_{AB} \approx 0.0981 \) meters  
c. **Corrected length of the line AB**: \( L_{AB_{corrected}} \approx 785.3431 \) meters

Would you like more details or have any questions about the calculations? 

Here are some related questions:
1. How does increasing the number of supports affect the sag correction?
2. What other factors can influence the accuracy of distance measurements using a steel tape?
3. Why is it important to correct for sag when measuring long distances?
4. How would using a tape made of a lighter material affect the sag correction?
5. How would environmental factors, such as temperature, affect the measured length?

**Tip:** Always consider both sag and tension when measuring long distances with tapes to ensure accuracy.

Learn how to calculate sag correction for a 30 meter steel tape used to measure a line AB under steady pull. Includes detailed steps and formulas used in mechanics and statics. Suitable for advanced college-level students and professionals.

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