Problem 60 - Master Resource Book in JEE Main Physics

Question

A spectrometer gives the following reading when used to measure the angle of a prism. Main scale reading : \(58.5\) degree Vernier scale reading : 09 divisions Given that 1 division on main scale corresponds to \(0.5\) degree. Total divisions on the vernier scale is 30 and match with 29 divisions of the main scale. The angle of the prism from the above data is (a) \(58.59^{\circ}\) (b) \(58.77^{\circ}\) (c) \(58.65^{\circ}\) (d) \(59^{\circ}\)

Step-by-Step Solution

Answer

The angle of the prism is (c) \( 58.65^{\circ} \).

Step 1: Understanding the Vernier scale reading

The vernier scale has 30 divisions, and these align with 29 divisions on the main scale. If we denote the length of one main scale division as \( M \), then \( 29M = 30V \), where \( V \) is a vernier scale division.

Step 2: Calculate the value of one Vernier Scale Division

From \( 29M = 30V \), we find the value of one vernier division: \( V = \frac{29}{30} M \). Since one main scale division \( M = 0.5^{\circ} \), we plug in the values to get \( V = \frac{29}{30} \times 0.5^{\circ} = 0.4833^{\circ} \).

Step 3: Vernier Constant or Least Count Calculation

The least count (LC) is calculated as the difference between one main scale division and one vernier division: \( LC = M - V = 0.5^{\circ} - 0.4833^{\circ} = 0.0167^{\circ} \). This is the smallest reading the vernier scale can measure.

Step 4: Calculate the Vernier Scale Reading Contribution

Each vernier scale division corresponds to \( 0.0167^{\circ} \). So, for a reading of 9 vernier scale divisions, the contribution is \( 9 \times 0.0167^{\circ} = 0.1503^{\circ} \).

Step 5: Find the Total Angle Measured

The total angle is the sum of the main scale reading and the vernier contribution: \( 58.5^{\circ} + 0.1503^{\circ} = 58.6503^{\circ} \). Rounding to two decimal places gives \( 58.65^{\circ} \).

Step 6: Determine the Correct Answer

Comparing \( 58.65^{\circ} \) with the options, the closest match is option (c): \( 58.65^{\circ} \).

Key Concepts

Vernier Caliper

The vernier caliper is a precision instrument that measures internal and external dimensions as well as depths. Although the task here involves spectrometer readings, understanding a vernier caliper is crucial because it works on the same principle of the vernier scale being utilized. A vernier caliper comprises two scales: the main scale, which is fixed, and the vernier scale, which slides parallel to the main scale. When making measurements, the zero of the vernier scale aligns with the zero on the main scale when the device is completely closed. It is widely used in scientific experiments due to its ability to provide highly accurate measurements. By reading where the vernier and main scale lines align, one can understand and derive extremely precise measurements down to fractions of a millimeter or degrees.

Least Count

The least count of a measuring instrument is the smallest difference in dimensions that it can measure. In other words, it represents the precision of the instrument. For a vernier scale, it is particularly significant as it signifies the smallest reading that can be obtained.A simple formula to calculate the least count (LC) is:

• \[LC = \text{Value of one Main Scale Division} - \text{Value of one Vernier Scale Division}\]

Applying this, we understand how much each division represents. For the spectrometer in question, the least count was calculated:

• One main scale division was \(0.5^{\circ}\)
• One vernier scale division ended up being approximately \(0.4833^{\circ}\)
• This gave a least count of \(0.0167^{\circ}\)

With this knowledge, users can comprehend repeated measurements and establish their accuracy.

Spectrometer Readings

Spectrometer readings involve determining the angle of a prism, a common task in physics labs. This process requires both the main scale and the vernier scale to provide an accurate angle measurement.When using a spectrometer, the main scale provides the initial reading. In this example, it was \(58.5^{\circ}\). The vernier scale assists in capturing finer details with an additional smaller measurement, achieved through analyzing multiple divisions.Each vernier division contributes additionally to the reading, calculated as the number of divisions multiplied by the instrument's least count. This was done with 9 divisions contributing \(0.1503^{\circ}\) in the present example.Such accurate readings allow scientists to ensure very fine precision during experiments involving light, color, or other waveforms and angles.

Vernier Scale Division

Each division of a vernier scale forms a fraction of the main scale division. It plays a pivotal role in providing accurate measurements by compensating for any excess or deficit in the main scale reading.The vernier scale in this instance has \(30\) divisions, which correspond perfectly with \(29\) on the main scale. Mathematically, this balances the calculation as each division on the vernier scale represents:

• \[V = \frac{29}{30} \times 0.5^{\circ} = 0.4833^{\circ}\]

This slightly smaller division allows the vernier to decrease the margin of error, thereby capturing more precise measurements by effectively "zooming in" on the angle represented by these scale divisions.

Main Scale Division

The main scale division is the basic unit of measurement on an instrument which provides a coarse, yet crucial, reading. The main scale of a spectrometer typically measures larger increments which are then refined by the vernier scale.In the provided spectrometer example, each main scale division represents \(0.5^{\circ}\), laying the foundational measurement before any more precise readings are taken.It is important to grasp that although the main scale gives the broader outlook of measurement, it requires the fine-tuning of the vernier scale to obtain accurate and meaningful experimental data.