Question

In an unusual vernier, 9 vernier scale divisions coincide with 8 main scale division, then what is the least count of the vernier? (A) \((8 / 9) \mathrm{mm}\) (B) \((1 / 9) \mathrm{mm}\) (C) \((1 / 17) \mathrm{mm}\) (D) \((1 / 8) \mathrm{mm}\)

Short answer

The least count of the unusual vernier scale is \(\frac{1}{9}\) mm, which corresponds to option (B).

Step-by-step solution

Determine the values of the main scale and vernier scale divisions

We know that 9 vernier scale divisions coincide with 8 main scale divisions. Let's assume the length of one main scale division is M, and one vernier scale division is V. Since 8 main scale divisions are equivalent to 9 vernier scale divisions, we can write the equation: \(8M = 9V\)

Find the least count (LC) of the vernier scale

The least count of a vernier scale is defined as the difference between the values of one main scale division (M) and one vernier scale division (V). Least Count (LC) = M - V We will now use the equation from step 1 to express V in terms of M: \(V = \frac{8}{9}M\)

Substitute the value of V in the least count formula

Now that we have found the value of V in terms of M, we can substitute it back into the least count formula: LC = M - \(\frac{8}{9}M\)

Calculate the least count

To calculate the least count, we need to solve for LC: LC = \(\frac{9}{9}M - \frac{8}{9}M\) LC = \(\frac{1}{9}M\) Since the main scale is given in millimeters (mm), the least count will also be in millimeters. LC = \(\frac{1}{9}\) mm The least count of the vernier scale is \(\frac{1}{9}\) mm, which corresponds to option (B).

Key concepts

Main Scale Division

The main scale on a Vernier caliper is the fixed, linear scale. Each division or mark on this scale represents a specific measurement, most often in millimeters. The main scale provides the baseline measurement with which the Vernier scale interacts.

Imagine a ruler with individual lines spaced apart. These lines, or divisions, create a way to measure length. On a Vernier caliper, the main scale typically mimics this concept. Its individual divisions define the primary measurement unit. That's why it plays a crucial role in helping determine the overall reading on the calipers.

• Main scale as fixed standard: Each division on the main scale signifies a fixed unit of measurement.
• Consistency: Regardless of the object being measured, the main scale divisions maintain their size and value.

The main scale serves as the foundation, offering a clear, long baseline against which measurements can be added or adjusted with the help of the Vernier scale.

Vernier Scale Division

The Vernier scale is an auxiliary sliding scale placed next to the main scale on a Vernier caliper. Its divisions are slightly different than those on the main scale. By having different sizes, the Vernier divisions help improve measurement precision.

The configuration of the Vernier scale is carefully designed. It allows for more accurate measurements than using just the main scale alone.

• Smaller increments: Often, more Vernier scale divisions fit into the space of a few main scale divisions, increasing precision.
• Sliding scale: The Vernier must move along the object being measured, aligning its lines with those on the main scale.
• Improved accuracy: By observing where the Vernier aligns with the main scale, one can detect smaller differences or fractional readings not obtainable with the main scale alone.

The Vernier scale divisions offer a second, finer determination, ensuring less margin of error in physical measurements.

Least Count Formula

The least count of a Vernier caliper represents its smallest measurable value, determined by how much the Vernier scale divisions differ from those of the main scale. This part of the instrument is vital because it dictates the caliper's measurement precision.

The formula to find the least count is:\[ \text{Least Count (LC)} = M - V \]This equation shows the difference between the main scale division (M) and the Vernier scale division (V). Understanding this concept can help in practical measurement tasks, ensuring clarity in readings.

• Measurement edge: The least count describes how finely the caliper can measure, or the smallest value discernible by the instrument.
• Mathematical representation: Subtract the value of one Vernier scale division from one main scale division to find the least count.
• Real-world application: On a practical level, knowing the least count aids in determining how accurately one can read or trust the caliper's measurements.

In essence, the least count formula empowers users to grasp the precision level of the tool, guiding them in making meticulous and reliable measurements.