Question 236: Units , Dimensions and Error Analysis

Question

The diameter of a spherical bob is measured using a vernier callipers. 9 divisions of the main scale, in the vernier callipers, are equal to 10 divisions of vernier scale. One main scale division is 1 mm. The main scale reading is 10 mm and 8th division of vernier scale was found to coincide exactly with one of the main scale division. If the given vernier callipers has positive zero error of 0.04 cm, then the radius of the bob is ___________ $$	imes$$ 10$$-$$2 cm.

Accepted Answer

To understand the measurement of the spherical bob's diameter using Vernier calipers, let's break it down step by step:
Vernier Scale Calculation:
We are given that 9 divisions on the main scale (MSD) are equivalent to 10 divisions on the Vernier scale (VSD).
One main scale division is 1 mm. Therefore, 9 MSD = 9 mm.
Since 9 mm = 10 VSD, one VSD equals $ \frac{9}{10} = 0.9 \, 	ext{mm} $.
Least Count (LC) of the Vernier Callipers:
The least count is the smallest measurable value and is calculated as:
$      	ext{LC} = 1 \, 	ext{MSD} - 1 \, 	ext{VSD} = 1 \, 	ext{mm} - 0.9 \, 	ext{mm} = 0.1 \, 	ext{mm}      $
Calculate the Initial Reading:
The main scale reading (MSR) is 10 mm.
The 8th division of the Vernier scale coincides with the main scale division, meaning the Vernier scale reading (VSR) is 8.
Total reading before zero error correction = MSR + VSR ÃÂ LC:
$      10 \, 	ext{mm} + 8 	imes 0.1 \, 	ext{mm} = 10.8 \, 	ext{mm}      $
Adjust for the Zero Error:
The Vernier calipers have a positive zero error of 0.04 cm (or 0.4 mm, as 1 cm = 10 mm).
To get the correct measurement, subtract the zero error:
$      10.8 \, 	ext{mm} - 0.4 \, 	ext{mm} = 10.4 \, 	ext{mm}      $
Compute the Radius:
The diameter calculated is 10.4 mm. To find the radius, divide by 2:
$      	ext{radius} = \frac{10.4 \, 	ext{mm}}{2} = 5.2 \, 	ext{mm}      $
Convert the radius from millimeters to centimeters:
$      5.2 \, 	ext{mm} = 0.52 \, 	ext{cm} = 52 	imes 10^{-2} \, 	ext{cm}      $
The radius of the spherical bob is $ 52 	imes 10^{-2} $ cm.

Units , Dimensions and Error Analysis

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