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A vernier caliper is a precision measuring tool used in measuring the size of an object with high accuracy. This can measure internal and external dimensions, depths, and step dimensions. The guide explains its parts, step-by-step usage, examples, and mathematical relations along with tables to explain the concepts.
Parts of a Vernier Caliper
- Main Scale : It is a fixed scale marked in millimeters or inches.
- Vernier Scale: A sliding scale with finer graduations that can be used for accurate readings.
- Jaws:
- Outer Jaws: It is used to measure the outer dimensions of an object
- Inner Jaws: This jaw measures the inner dimensions, i.e., diameter of the hole
- Depth Rod: This rod will measure the depth of the holes and recess
- Lock Screw: Locks the vernier scale in place
- Thumb Screw: It helps to move the vernier scale freely
You may also want to know How Projectile Motion Works
Step-by-Step Guide to Use a Vernier Caliper
- 1. Zero Calibration
- Close the external jaws completely.
- Ensure the “0” on the vernier scale aligns perfectly with the “0” on the main scale.
- If there is a misalignment, note the zero error (positive or negative).
- 2. Measuring External Dimensions
- Open the external jaws and place them around the object to be measured (e.g., a cylinder or block).
- Tighten the thumb screw gently so the object is securely held.
- Read the measurement from the main scale and vernier scale.
- 3. Measuring Internal Dimensions
- Open the internal jaws and insert them into the object’s opening (e.g., a pipe).
- Expand the jaws until they touch the internal edges.
- Tighten the thumb screw and record the readings from the main and vernier scales.
- 4. Measuring Depth
- Extend the depth rod by sliding the main scale.
- Insert the depth rod into the hole or recess.
- Read the depth measurement on the scale once the rod reaches the bottom.
- 5. Reading the Measurement
- Main Scale Reading: Note the reading just before the “0” mark on the vernier scale.
- Vernier Scale Reading: Find the line on the vernier scale that aligns perfectly with any line on the main scale.
- Final Measurement: Add the main scale reading and the vernier scale reading. Use the least count for precise calculation.
Key Concepts and Mathematical Relations
- Least Count (LC)
The least count is the smallest measurable value by the instrument. It is calculated as:
![\[LC = \dfrac{\text{Smallest Division on Main Scale}}{\text{Number of Divisions on Vernier Scale}}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-4f479524b615d33fe21612364b8966d3_l3.png "Rendered by QuickLaTeX.com")
Example:
- Main scale division: 1 mm
- Vernier scale divisions: 10
![\[LC = \dfrac{1}{10} = 0.1 \, \text{mm}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-ab6cae5c6c150834aea2cc40b3525be0_l3.png "Rendered by QuickLaTeX.com")
- Zero Error and Correction
- If the “0” of the vernier scale does not coincide with the “0” of the main scale, then there is a zero error.
- Correct the final measurement by adding or subtracting the error.
- Positive Zero Error:
- The vernier “0” lies to the right of the main scale “0”.
- Subtract the error from the measurement.
- Negative Zero Error:
- The vernier “0” lies to the left of the main scale “0”.
- Add the error to your measurement.
Tips for proper Measurement
- Ensure it is zero calibrated before you use it.
- Make use of minimum force necessary while measuring
- Keep the instrument clean and free of dust
- Use the locking screw to fix the reading during measurement
- Take readings multiple times to ensure accuracy
Formulas for Measurements Using Vernier Caliper
When using a vernier caliper, the measurement formula depends on the type of measurement being performed. Below are the general formulas explained step by step:
- Formula for Total Measurement
![\[\text{Final Measurement} = \text{Main Scale Reading} + (\text{Vernier Scale Reading} \times \text{Least Count})\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-c077ebc3f6f5fcc86e6365b95e74a03f_l3.png "Rendered by QuickLaTeX.com")
Where:
- Main Scale Reading (MSR): The reading on the main scale just before the vernier scale’s “0”.
- Vernier Scale Reading (VSR): The division on the vernier scale that aligns exactly with a main scale line.
- Least Count (LC): The smallest value measurable by the caliper.
- Measuring External Dimensions
![\[\text{External Dimension} = \text{MSR} + (\text{VSR} \times \text{LC})\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-f4e2195341f0d2950c4ae254a5c1daef_l3.png "Rendered by QuickLaTeX.com")
Example:
- Main Scale Reading (MSR) = 24 mm
- Vernier Scale Reading (VSR) = 6 (6th division aligns)
- Least Count (LC) = 0.1 mm
![\[\text{External Dimension} = 24 + (6 \times 0.1) = 24 + 0.6 = 24.6 \, \text{mm}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-0edcdc44a69e935644f6f1247d27169a_l3.png "Rendered by QuickLaTeX.com")
- Measuring Internal Dimensions
![\[\text{Internal Dimension} = \text{MSR} + (\text{VSR} \times \text{LC})\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-b6b998e7cade7effc3443af04a902b3b_l3.png "Rendered by QuickLaTeX.com")
Example:
- Main Scale Reading (MSR) = 15 mm
- Vernier Scale Reading (VSR) = 3 (3rd division aligns)
- Least Count (LC) = 0.1 mm
![\[\text{Internal Dimension} = 15 + (3 \times 0.1) = 15 + 0.3 = 15.3 \, \text{mm}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-83c973b6a242cc7e18876d0de413f450_l3.png "Rendered by QuickLaTeX.com")
- Measuring Depth
![\[\text{Depth} = \text{MSR} + (\text{VSR} \times \text{LC})\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-2d708ac7333b715b50356a1bfe546fe4_l3.png "Rendered by QuickLaTeX.com")
Example:
- Main Scale Reading (MSR) = 12 mm
- Vernier Scale Reading (VSR) = 8 (8th division aligns)
- Least Count (LC) = 0.1 mm
![\[\text{Depth} = 12 + (8 \times 0.1) = 12 + 0.8 = 12.8 \, \text{mm}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-71bb9522ec8e4031af718bacac1db867_l3.png "Rendered by QuickLaTeX.com")
- Zero Error Adjustment
If there is a zero error, the formula for the final measurement is adjusted as:
![\[\text{Final Measurement} = (\text{MSR} + \text{VSR} \times \text{LC}) \pm \text{Zero Error}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-afb0124c4c9689e315ae5bb5ed82660b_l3.png "Rendered by QuickLaTeX.com")
- Positive Zero Error: Subtract the error.
- Negative Zero Error: Add the error.
Example:
- Main Scale Reading (MSR) = 20 mm
- Vernier Scale Reading (VSR) = 4 (4th division aligns)
- Least Count (LC) = 0.1 mm
- Zero Error = +0.2 mm (positive)
![\[\text{Final Measurement} = (20 + (4 \times 0.1)) - 0.2 = (20 + 0.4) - 0.2 = 20.2 \, \text{mm}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-eab5dd7fce85c1832a747fe0d12b00c0_l3.png "Rendered by QuickLaTeX.com")
- Least Count Formula
The least count is calculated as:
![\[\text{LC} = \dfrac{\text{Smallest Division on Main Scale}}{\text{Number of Divisions on Vernier Scale}}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-08a1c69a622c771a914e08d669f556e6_l3.png "Rendered by QuickLaTeX.com")
Example:
- Smallest division on main scale = 1 mm
- Number of divisions on vernier scale = 10
![\[\text{LC} = \dfrac{1}{10} = 0.1 \, \text{mm}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-7db8897962386719984763807f8027e5_l3.png "Rendered by QuickLaTeX.com")
Summary of Formulas
| Measurement Type | Formula |
|---|---|
| Final Measurement |  |
| With Zero Error |  |
| Least Count |  |
| External Dimensions | |
| Internal Dimensions | |
| Depth Measurement | |
These formulas ensure precise measurements in various applications.
1 MSD – 1 VSD: Definition and Explanation
In the context of a vernier caliper, 1 MSD (1 Main Scale Division) and 1 VSD (1 Vernier Scale Division) are fundamental to understanding the least count of the instrument. The relationship between these divisions is used to determine the precision of measurements.
1 MSD – 1 VSD Formula
![\[\text{1 MSD - 1 VSD} = \text{Least Count (LC)}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-4c637dee509598c144337a6504e869a1_l3.png "Rendered by QuickLaTeX.com")
Where:
- 1 MSD (Main Scale Division): The smallest unit on the main scale, typically measured in millimeters or centimeters.
- 1 VSD (Vernier Scale Division): The smallest unit on the vernier scale, typically smaller than 1 MSD.
How It Works
The difference between one main scale division and one vernier scale division determines the least count or the minimum measurement the vernier caliper can make.
Derivation of Least Count
- Assume:
- The smallest division on the main scale (1 MSD) = 1 mm
- The number of divisions on the vernier scale = 10
- The total length of the vernier scale = 9 main scale divisions:
- Each vernier division (1 VSD) =
- Difference between 1 MSD and 1 VSD:
![\[\text{1 MSD - 1 VSD} = 1 \, \text{mm} - 0.9 \, \text{mm} = 0.1 \, \text{mm}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-2663eeb503e5ab79379d2525e6538b5b_l3.png "Rendered by QuickLaTeX.com")
Thus, the least count of this example is 0.1 mm.
Example Calculation
If:
- 1 MSD = 0.5 mm
- The vernier scale has 20 divisions spanning 19 main scale divisions:
![\[1 \, \text{VSD} = \dfrac{19 \, \text{MSD}}{20} = \dfrac{19 \times 0.5}{20} = 0.475 \, \text{mm}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-03060281f4752a7db68a45d0650cc818_l3.png "Rendered by QuickLaTeX.com")
Then:
![\[1 \, \text{MSD} - 1 \, \text{VSD} = 0.5 - 0.475 = 0.025 \, \text{mm}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-723fdc400c4cfea6ca09d3e18932b766_l3.png "Rendered by QuickLaTeX.com")
The least count in this case is 0.025 mm.
Key Takeaway
The formula 1 MSD – 1 VSD directly gives the least count of a vernier caliper. This relationship ensures precise measurement by exploiting the small difference between the two scales.
The formula for the Least Count (LC) of the vernier caliper as:
![\[\text{LC} = \frac{\text{Smallest Division on Main Scale}}{\text{Number of Divisions on Vernier Scale}}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-d62e90776d85a6ecd6fcfd4634d2d01b_l3.png "Rendered by QuickLaTeX.com")
Relation Between the Two Formulas
The above formula is the standard method to calculate the least count directly using the scale values. However, 1 MSD – 1 VSD is a derived formula that also gives the least count by using the difference between the value of one main scale division and one vernier scale division.
Linking the Two:
If the main scale division (1 MSD) is 1 mm and the vernier scale has 10 divisions:
![\[\text{LC} = \frac{1 \, \text{mm}}{10} = 0.1 \, \text{mm}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-01cdb3dcefa2e11abf3f9ad81dc81a58_l3.png "Rendered by QuickLaTeX.com")
From the 1 MSD – 1 VSD Formula:
- Each vernier division (1 VSD) is calculated as:
![\[1 \, \text{VSD} = \frac{\text{Total Main Scale Length Spanned}}{\text{Number of Vernier Divisions}} = \frac{1 \, \text{mm} \times 9}{10} = 0.9 \, \text{mm}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-c49502f8c39c644a47f3d7bb4feaa505_l3.png "Rendered by QuickLaTeX.com")
- The least count is:
![\[\text{LC} = 1 \, \text{MSD} - 1 \, \text{VSD} = 1 \, \text{mm} - 0.9 \, \text{mm} = 0.1 \, \text{mm}\]](https://physicsqanda.com/wp-content/ql-cache/quicklatex.com-079dc9adf7c3dc77369f4fcf9bedda99_l3.png "Rendered by QuickLaTeX.com")
Both methods give the same result. The image formula is simpler and is widely used for direct calculations, while the 1 MSD – 1 VSD method gives insight into the mechanical working of the vernier scale.
Examples:
1. Understanding Components
Question: Identify the parts of a vernier caliper and describe their functions.
Answer:
- Main Scale: A fixed scale marked in millimeters or inches.
- Vernier Scale: A sliding scale that allows for precise fractional readings.
- External Jaws: Measure external dimensions like thickness or diameter.
- Internal Jaws: Measure internal dimensions such as hole diameters.
- Depth Rod: Measures depths of holes or recesses.
- Lock Screw: Secures the position of the sliding scale.
- Thumb Screw: Facilitates fine adjustment of the sliding scale.
2. Reading Measurements
Question: A vernier caliper has a main scale reading of 12.5 mm, and the 7th vernier scale division aligns with a main scale division. The least count is 0.1 mm. Calculate the measured length.
Solution:
- Main Scale Reading (MSR): 12.5 mm
- Vernier Scale Reading (VSR): 7 × 0.1 mm = 0.7 mm
- Total Measurement: MSR + VSR = 12.5 mm + 0.7 mm = 13.2 mm
Final Answer: 13.2 mm
3. Calculating Least Count
Question: A vernier caliper has 10 divisions on the vernier scale that coincide with 9 divisions on the main scale. If one main scale division is 1 mm, determine the least count.
Solution:
- Value of 1 Main Scale Division (MSD): 1 mm
- Value of 1 Vernier Scale Division (VSD): (9 MSD) / 10 = 9 mm / 10 = 0.9 mm
- Least Count (LC): MSD – VSD = 1 mm – 0.9 mm = 0.1 mm
Final Answer: 0.1 mm
4. Identifying Zero Error
Question: When the jaws of a vernier caliper are closed, the zero of the vernier scale is 0.02 mm to the right of the zero of the main scale. What is the zero error, and how should it be corrected?
Answer:
- Zero Error: +0.02 mm (positive zero error)
- Correction: Subtract 0.02 mm from all measurements to obtain the true value.
5. Practical Application
Question: Describe how to measure the internal diameter of a pipe using a vernier caliper.
Answer:
- Open Internal Jaws: Slightly open the internal jaws.
- Insert into Pipe: Place the jaws inside the pipe at the desired measurement point.
- Expand Jaws: Gently expand the jaws until they touch the internal surface.
- Lock Position: Use the lock screw to fix the position.
- Read Measurement: Note the main scale reading just before the zero of the vernier scale and add the vernier scale reading multiplied by the least count.
6. Error Analysis
Question: A student measures the thickness of a sheet using a vernier caliper with a least count of 0.02 mm. The main scale reading is 5 mm, and the 3rd vernier division aligns with the main scale. If the zero error is -0.04 mm, what is the correct thickness?
Solution:
- Main Scale Reading (MSR): 5 mm
- Vernier Scale Reading (VSR): 3 × 0.02 mm = 0.06 mm
- Measured Thickness: MSR + VSR = 5 mm + 0.06 mm = 5.06 mm
- Zero Error: -0.04 mm (negative)
- Corrected Thickness: Measured Thickness – Zero Error = 5.06 mm – (-0.04 mm) = 5.06 mm + 0.04 mm = 5.10 mm
Final Answer: 5.10 mm
Conclusion
A vernier caliper is one of the most important tools adopted for precise measurements in science, engineering, and industry. This will be used to know the value of least count and zero error correction, ensuring the result will be on target. As this tool is put into practice with real examples, and the proper maintenance of the instrument will make it a reliable companion in carrying out dimensional analysis.
