Mine Development & Surveying

Linear Measurement: Errors & Corrections

Standardisation, temperature, slope and sag corrections — turning a raw tape reading into a true horizontal distance.

PART 1

Topic Breakdown & Traps

The Engineering Principle

A steel tape rarely measures the true horizontal distance directly: it may be the wrong length (standardisation), it expands or contracts with temperature, it is pulled with a tension different from its standard, it sags between supports, and it may be laid along a slope rather than the horizontal. Each effect is a small, *signed* correction added to the measured length. Knowing the sign of each correction — and that some (slope, sag) are always subtractive — is half the battle.

The Core Formula Matrix

Standardisation (wrong tape length): true length

= L_{m e a s} \times \frac{L ^{'}}{L}

, where

L^{'}

= actual tape length,

L

= nominal length.

Temperature:

C_{t} = α (T_{m} - T_{0}) L

(sign follows the temperature difference).

Slope (always subtractive):

C_{s l o p e} = - \frac{h ^{2}}{2 L}

, with

h

= height difference over slope length

L

.

Sag (always subtractive, per span):

C_{s a g} = - \frac{w ^{2} L ^{3}}{24 P ^{2}}

w

= weight per unit length,

P

= applied tension.

The ‘IIT Traps’

⚠
Slope and sag are always negative. They shorten the horizontal/true length; adding them with a plus sign is a guaranteed error.
⚠
A tape too long ⇒ measured distance is too short. So the standardisation correction is *positive* when $L^{'} > L$ — the opposite of many students' first instinct.
⚠
**Use consistent units for $α$ .** The coefficient of thermal expansion is per °C (≈ $11.2 \times 1 0^{- 6}$ for steel); keep length in metres so the correction comes out in metres.

PART 2

Progressive 3-Tier Question Suite

Q1BASIC1 Mark · MCQ

100 m

steel tape (

α = 11.2 \times 1 0^{- 6} /^{\circ} C

) standardised at

2 0^{\circ} C

is used at

3 5^{\circ} C

. The temperature correction is:

Q2MEDIUM2 Marks · NAT

A length of

50 m

is measured along a slope with a height difference of

2 m

between its ends. The slope correction (magnitude) to be subtracted is ______ m. (Round off to three decimal places.)

Q3HARD2 Marks · NAT

A line is measured as

1500 m

with a

30 m

tape that is later found to be

0.02 m

too long. The corrected (true) length of the line is ______ m. (Round off to two decimal places.)