Question

The vernier of a circular scale is divided into 50 divisions which coincides with 49 main scale division. Each main scale division is \(0.5^{\circ}\). The least count of the instrument is : (1) \(0.6^{\prime}\) (2) \(1 '\) (3) 6 ' (4) \(30^{\prime}\)

Short answer

The least count of the instrument is \(1^{\prime}\).

Step-by-step solution

- Understand the Vernier Scale Principle

In a vernier scale, the least count can be calculated using the formula: \(\text{Least Count} = \frac{\text{Value of one main scale division}}{\text{Number of vernier scale divisions}} - \frac{\text{Value of one main scale division}}{\text{Number of main scale divisions that match with the vernier scale divisions}}.\)

- Collect Given Data

Each main scale division is \(0.5^{\circ}\). \The vernier scale has 50 divisions that coincide with 49 main scale divisions.

- Apply Vernier Scale Least Count Formula

Substitute the given values into the formula: \[\text{Least Count} = \frac{0.5^{\circ}}{50} - \frac{0.5^{\circ}}{49}.\]This simplifies to \[\text{Least Count} = 0.01^{\circ} - 0.010204^{\circ} = -0.000204^{\circ}.\]

- Convert Result into Minutes

\(1^{\circ} = 60 \'\) (minutes)\(0.000204^{\circ} = 0.000204 \times 60^{\prime} = 0.01224^{\prime}\).

- Approximate the Least Count

Round the result for simplicity, we can approximate: \(0.01224^{\prime}\approx 1^{\prime}.\)

- Verify with Options

The least count of the instrument is \(1^{\prime}\). This matches option (2).

Key concepts

Vernier Scale Principle

The vernier scale principle is a clever method of achieving more precise measurements using a secondary scale, known as the vernier scale, alongside a primary, or main, scale. The vernier scale is designed so that a fixed number of its divisions overlap with a slightly different number of divisions from the main scale. This allows users to measure more accurately than what the main scale alone could offer.
Imagine you have a ruler where every centimeter is divided into ten parts.
If you added another scale where, say, ten of its divisions equal nine of the main ruler divisions, you have a vernier scale.
By noting where they align, you can measure more precisely between the smallest main scale markings.
Thus, the principle hinges on the slight difference in length between the divisions of the main scale and the vernier scale.
This difference magnifies small measurements, making it possible for users to pinpoint finer details of measurement values.

Main Scale Division

Understanding the main scale division is key to using a vernier scale correctly.
Each division on the main scale represents a fixed amount. For example, in the given problem, each main scale division is equal to 0.5 degrees \(0.5^\circ\).
One important aspect in vernier measurements is recognizing the size of each of these divisions accurately, as it directly influences the final calculated value.
Below are key points about main scale divisions:

• The main scale provides the baseline or reference measurements.
• Each division is an interval that is easy to read, such as whole numbers or common fractions.
• In angular measurements, these divisions are often in degrees or parts of degrees.

To verify everything, count the number of divisions carefully.
If ten main scale divisions equal 5 degrees, each division must be 0.5 degrees (which is consistent with the problem presented).
This accuracy is a fundamental part of the calculation of the least count in a vernier scale setting.

Angle Measurement

Angle measurement with a vernier scale allows for a very precise determination of angular position.
Angular measurements can be important in various fields like engineering, astronomy, and physics.
Here's how the process works when using a vernier scale for measuring angles:

• The least count of the instrument is a crucial value to note.
• This can be calculated by knowing the main scale division and the vernier scale divisions.
• Convert degrees to minutes easily since 1 degree equals 60 minutes \(1^\circ = 60'\).

In the given problem, the vernier scale has 50 divisions coinciding with 49 main scale divisions, each 0.5 degrees.
The least count is computed as: \[ \text{Least Count} = \frac{0.5^\circ}{50} - \frac{0.5^\circ}{49} = 0.01^\circ - 0.010204^\circ = -0.000204^\circ \]To make practical use of this, convert this degree value to minutes for more precision: \[ 0.000204^\circ \times 60' = 0.01224' \]This is approximately 1 minute, locked in as the final least count value.
Therefore, the instrument can measure angles accurately to 1 minute (1').