If the nth division of main scale coincides with (n+1)th division of vernier scale, find the least count of the vernier. Given one main scale divisions equal to 'a' units

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In the context of precision measurements, the vernier caliper is an instrument that allows for more precise measurements than a standard ruler or measuring tape. It consists of a main scale and a vernier scale that slides along it. The least count of the vernier is the smallest measurement that can be obtained with it and is crucial for providing the precision that vernier calipers are known for.

Property	Description
Main Scale	* Shows n full divisions. * Each division length is known as the Main Scale Division (MSD).
Vernier Scale	* Slides along the main scale. * Has a fixed number of divisions (VSD). * Each division is slightly shorter than an MSD.
Alignment	The nth division of the main scale coincides with the (n-1)th division of the vernier scale.
Meaning	There’s a slight overlap between the two scales. The vernier scale “lags behind” by one division compared to the main scale.
Least Count (LC)	The smallest distance the vernier caliper can measure.
LC Formula	LC = MSD – (Total Length of Vernier Scale) / (Number of Vernier Scale Divisions)
Special Case	In this scenario, the Total Length of Vernier Scale is (n-1) * MSD.
Adjusted LC Formula	LC = MSD (1 – (n-1) / VSD)

Contents hide

1 Explanation:

2 Calculation Steps:

3 Final Answer:

Explanation:

The least count of a vernier caliper is the difference between one main scale division and one vernier scale division. It represents the smallest value that can be measured with the instrument. When the nth division of the main scale coincides with the (n+1)th division of the vernier scale, it indicates that n divisions on the main scale are equal to n+1 divisions on the vernier scale.

Calculation Steps:

Identify the Given Values: One main scale division is equal to ‘a’ units.
Understand the Coincidence: The fact that the nth division of the main scale coincides with the (n+1)th division of the vernier scale means that ‘n’ main scale divisions cover the same length as ‘n+1’ vernier scale divisions.
Express the Total Length: The total length of ‘n’ main scale divisions is ‘na’ units.
Set Up an Equation: Since ‘n’ main scale divisions are equal to ‘n+1’ vernier scale divisions, you can set up the equation:
$na = (n+1)V$
where ‘V’ is the length of one vernier scale division.
Solve for V: Rearrange the equation to find the value of ‘V’:
$V = \dfrac{na}{n+1}$
Calculate the Least Count: The least count is the difference between the length of one main scale division and one vernier scale division, which can be calculated as:
$\text{Least Count} = a - V$

Final Answer:

The least count of the vernier is calculated by taking the difference between the length of one main scale division (‘a’ units) and one vernier scale division (‘V’). It is expressed as:
$\text{Least Count} = a - \dfrac{na}{n+1}$