The one division of main scale of Vernier callipers reads 1mm and 10 divisions of Vernier scale is equal to the 9 division on main scale. When the two jaws of the instrument touch each other, the zero of the Vernier lies to the right of zero of the main scale and its fourth division coincides with a main scale division. When a spherical bob is tightly placed between the two jaws, the zero of the Vernier scale lies in between 4.1 cm and 4.2 cm and 6th Vernier division coincides with a main scale division. The diameter of the bob will be ___ × 10–2 cm.
Correct Answer: 412
Approach Solution - 1
The Vernier caliper problem involves understanding and calculating measurements using Vernier principles. Let's solve this step by step.
- Least Count Calculation: The least count is the difference between one main scale division (MSD) and one Vernier scale division (VSD).
- Given: 1 MSD = 1 mm; 10 VSD = 9 MSDs.
- Hence, 1 VSD = \(\frac{9}{10} \, \text{mm}\).
- Least Count = 1 MSD - 1 VSD = 1 mm - 0.9 mm = 0.1 mm.
- Zero Error Determination: When the jaws touch, the fourth division of the Vernier coincides with a main scale division. This indicates a zero error.
- Since the fourth Vernier division coincides, zero error = \(-4 \times \text{Least Count} = -4 \times 0.1 \, \text{mm} = -0.4 \, \text{mm}\).
- Measurement of the Diameter: When the bob is placed, the zero of the Vernier lies between 4.1 cm and 4.2 cm, and the sixth Vernier division coincides with a main scale division.
- Main Scale Reading is between 4.1 cm and 4.2 cm, or between 41 mm and 42 mm.
- Vernier Scale Reading = \(6 \times \text{Least Count} = 6 \times 0.1 \, \text{mm} = 0.6 \, \text{mm}\).
- Reading in mm = \((41 + 0.6 - 0.4) \, \text{mm} = 41.2 \, \text{mm}\).
- Verification: Measurement = 41.2 mm ≤ 42 mm (confirmed range).
- Conversion to Result Format: Convert mm to cm: \(41.2 \, \text{mm} = 4.12 \, \text{cm}\).
The diameter of the bob = \(4.12 \times 10^{-2} \, \text{cm}\).
Approach Solution -2
The correct answer is 412
1 MSD = 1 mm
10 VSD = 9 MSD
LC \(=\frac{1}{10} mm\)
\(0+4(\frac{1}{10}) mm=0.4 mm\)
Reading \(=41+6(\frac{1}{10})\)
= 41 + 0.6
= 41.6 mm
True reading = 41.2 mm
= 412 × 10–2 cm
Therefore , diameter of the bob will be 412 × 10–2 cm.
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Concepts Used:
Units and Measurement
Unit:
A unit of a physical quantity is an arbitrarily chosen standard that is broadly acknowledged by the society and in terms of which other quantities of similar nature may be measured.
Measurement:
The process of measurement is basically a comparison process. To measure a physical quantity, we have to find out how many times a standard amount of that physical quantity is present in the quantity being measured. The number thus obtained is known as the magnitude and the standard chosen is called the unit of the physical quantity.
Read More: Fundamental and Derived Units of Measurement
System of Units:
- CGS system
- FPS system
- MKS system
- SI units
Types of Units:
Fundamental Units -
The units defined for the fundamental quantities are called fundamental units.
Derived Units -
The units of all other physical quantities which are derived from the fundamental units are called the derived units.