A vertical angle is measured to a signal mounted on top of a tower. with the transit telescope in direct position and the reading on the circle is +22°32'. If the reading on the circle is +22°38' with the telescope in reversed position, the index error and the index correction, respectively are:
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To find the index error and index correction, we can use the method of reciprocal observations.
Step 1: Calculate the Vertical Angle
First, we need to determine the vertical angle by taking the difference between the two readings on the circle.
Vertical Angle = Reading with Telescope in Direct Position - Reading with Telescope in Reversed Position
= +22°32' - +22°38'
Before performing the subtraction, note that the minutes have to be converted to decimal form. One minute is equal to 1/60 of a degree.
Vertical Angle = 22° + (32/60) - 22° - (38/60)
= 22° + 0.5333 - 22° - 0.6333
= 0° - 0.1
= -0°10'
Step 2: Determine the Index Error
The index error is the apparent difference between the actual zero reading on the circle and the observed zero reading. It can be positive or negative.
In this case, the index error is equal to half the vertical angle because we have reciprocal observations.
Index Error = Vertical Angle / 2
= (-0°10') / 2
= -0°5'
Therefore, the index error is -0°5'.
Step 3: Calculate the Index Correction
The index correction is the algebraic sum of the index error and the value by which the observed reading must be corrected.
Index Correction = Index Error + Observed Vertical Angle
= -0°5' + +22°32'
= -0°5' + 22° + (32/60)
= -0°5' + 22° + 0.5333
= 21° + 0.0333
= 21° + (2/60)
= 21°2'
Therefore, the index correction is +21°2'.
To summarize:
Index error: -0°5'
Index correction: +21°2'
To find the index error and index correction, we need to know the difference between the reading in direct and reversed positions of the telescope.
Given:
Reading in direct position = +22°32'
Reading in reversed position = +22°38'
First, let's determine the difference between the readings:
Reading in reversed position - Reading in direct position
= (+22°38') - (+22°32')
To subtract angles, we need to convert them into a common format. Since both angles are in degrees and minutes, we can directly subtract them:
22°38' - 22°32' = 6'
The difference is 6 minutes.
Now, let's determine the index error and index correction:
1. Index Error:
The index error represents the difference between the actual position of the zero mark on the vertical circle and the indicated zero mark. In this case, the index error is the difference obtained from the previous step, which is 6 minutes (+6').
Therefore, the index error = +6'
2. Index Correction:
The index correction compensates for any systematic errors in the instruments. It is the negative of the index error. Therefore, the index correction is the negative of the index error:
Index correction = -6'
So, the index error is +6' and the index correction is -6'.