Question

If the scale arm is not touching the tree stem in caliper measurement, the error in diameter measurement is expressed as one of the following options:

• A. $\pm$ (D tane)/2
• B. $\pm$ (L/D) tane $\times$ 100
• C. $\pm$ D (sece-1)
• D. D tane

Hint:

Think of the geometric relationship: the tilted measurement forms a right-angled triangle where the measured distance is the hypotenuse (\( D \sec e \)) and the true diameter is the base (\( D \)).
Subtracting the base from the hypotenuse gives the error: \( D(\sec e - 1) \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A tree caliper is a tool used to measure the diameter of a tree stem.
It consists of a graduated rule with one fixed arm and one parallel sliding arm.
To obtain an accurate measurement, both arms must be perpendicular to the graduated scale bar and must make firm contact with the tree stem.

Step 2: Detailed Explation:
If the movable arm of the caliper is not perpendicular to the scale arm but is tilted by a small angle \( e \), a measurement error is introduced.
Instead of measuring the true perpendicular diameter \( D \), the tilted arm measures a longer distance along the scale bar.
Using trigonometry, this measured diameter \( D' \) can be expressed as:
\[ D' = D \sec e \]
The resulting error in the diameter measurement is the difference between the measured diameter and the true diameter:
\[ \text{Error} = D' - D = D \sec e - D = D(\sec e - 1) \]
This error is represented with a \( \pm \) sign depending on the direction of the tilt:
\[ \text{Error} = \pm D(\sec e - 1) \]
This formula represents the standard error introduced in caliper measurements when the arms are not parallel.

Step 3: Fil Answer
The correct expression for the measurement error is \( \pm D(\sec e - 1) \).