A 30 cm chain was found to be 20 cm too long after changing the distance of 2000 m. It was found to be 28 cm too long at the end of the day's work after changing a total distance of 3400 m. Find the true distance of the chain was corrected before the commencement of work.





The given problem is related to the correction of a survey chain. The chain was found to be too long, and it was corrected by changing the distance of the chain. The correction was done twice, and the chain was found to be too long by 20 cm and 28 cm, respectively, after changing the distance. We need to find the true distance of the chain before the commencement of work.



- Correction due to temperature and tension
- Measurement error
- Linear Interpolation



Let us assume that the original length of the chain is "L" cm. Due to temperature and tension, the measured length of the chain is more than the actual length. Hence, the measured length of the chain after correction is given by

Measured length = Actual length + Correction

The correction is given by

Correction = (Coefficient of linear expansion) x (Change in temperature) x Length of the chain + (Tension in the chain) / Cross-sectional area

Let us assume that the coefficient of linear expansion is "a" cm/cm/degree Celsius, the change in temperature is "ΔT" degree Celsius, the tension in the chain is "T" N, and the cross-sectional area of the chain is "A" sq. cm. Then, the correction can be calculated as

Correction = a x ΔT x L + T / A



Let us assume that the correction due to temperature and tension for the given chain is "C" cm. Then, the measured length of the chain after the correction is given by

Measured length = L + C

At the first instance, the distance of the chain was changed by 2000 m. Hence, the correction due to temperature and tension would have changed. Let us assume that the correction due to temperature and tension after changing the distance by 2000 m is "C1" cm. Then, the measured length of the chain after changing the distance of 2000 m is given by

Measured length = L + C1

The difference between the measured length of the chain before and after changing the distance is equal to the error in measurement. Hence, the error in measurement due to changing the distance of 2000 m is given by

Error1 = (L + C1) - (L + C)

Error1 = C1 - C

Similarly, at the end of the day's work, the distance of the chain was changed by a total of 3400 m. Let us assume that the correction due to temperature and tension after changing the distance by 3400 m is "C2" cm. Then, the measured length of the chain after changing the distance of 3400 m is given by

Measured length = L + C2

The error in measurement due to changing the distance of 3400 m is given by

Error2 =