Problem 65 - 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 \(=58.5^{\circ}\) Vernier scale reading \(=09\) division Given that 1 division on main scale corresponding to \(0.5^{\circ}\). Total division on the vernier scale is 30 and match with 29 divisions of the main scale. The angle of the prism from the above data \(\quad\) [AIEEE 2012] (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 \(58.65^{\circ}\) (option c).

Step 1: Understand the Vernier Constant

The vernier constant is found by dividing the difference in a single division between the main scale and vernier scale. Since 30 vernier scale divisions match with 29 main scale divisions, the least count or vernier constant (VC) is calculated as follows:\[\text{VC} = \left(\text{Main Scale Division} - \frac{1}{30} \right) = 0.5^{\circ} - \frac{0.5^{\circ}}{30}\]Calculate this value to find the VC.

Step 2: Calculate the Vernier Constant

Given that a single main scale division is \(0.5^{\circ}\), calculate the vernier constant:\[\text{VC} = 0.5^{\circ} - \frac{0.5^{\circ}}{30} = 0.5^{\circ} - 0.0167^{\circ} \approx 0.0167^{\circ}\]

Step 3: Find Vernier Scale Reading Contribution

The contribution from vernier scale reading can be calculated by multiplying the number of vernier divisions by the vernier constant:\[\text{Vernier Reading} = 09 \times 0.0167^{\circ} = 0.1503^{\circ}\]

Step 4: Calculate Total Angle

Add the main scale reading and the vernier scale contribution to find the total angle:\[\text{Total Angle} = 58.5^{\circ} + 0.1503^{\circ} = 58.6503^{\circ}\]Round this value to two decimal places.

Step 5: Final Answer

Round the total angle to two decimal places gives:\[58.65^{\circ}\]Therefore, the angle of the prism is \(58.65^{\circ}\).

Key Concepts

Least Count

The least count of a measurement instrument like the Vernier caliper is the smallest measurement that can be accurately read by it. This concept is central to obtaining precise measurements and helps in minimizing errors in experimental observations. In the given exercise, 30 divisions on the Vernier scale correspond to 29 divisions on the main scale, which each represents 0.5\(^{\circ}\). Using this information, we calculate the least count, which is also known as the Vernier constant, as:

• One main scale division = 0.5\(^{\circ}\)
• Vernier constant (\(VC\)) = \(\frac{1}{30}\) of a main scale division = \(0.5^{\circ} - \frac{0.5^{\circ}}{30}\)

This gives us a least count of approximately \(0.0167^{\circ}\). This value signifies how small a value can be measured using this setup, ensuring precise readings.

Angle of Prism

Understanding how to measure the angle of a prism with precise tools like a spectrometer involves careful computation of multiple readings. To determine the angle of the prism, first calculate the contributions of both the main scale and Vernier scale readings. For the given exercise:

• Main scale reading = \(58.5^{\circ}\)
• Vernier scale reading = 9 divisions
• Each Vernier division = 0.0167\(^{\circ}\)

The influence of the Vernier scale equals 9 times the least count: \(9 \times 0.0167^{\circ} = 0.1503^{\circ}\). Hence, the total angle is the sum of these two contributions: \(58.5^{\circ} + 0.1503^{\circ} = 58.6503^{\circ}\). Rounded to two decimal places, this gives us the final angle of the prism as \(58.65^{\circ}\). This accurate measurement allows for precise optical experiments and analysis.

Main Scale Reading

The main scale reading is the initial measurement taken using the primary scale on a Vernier caliper or similar instrument. This reading forms the base value to which the more refined Vernier scale readings are added. In this exercise, the main scale already contributes a substantial measurement of \(58.5^{\circ}\). Consider this as the coarse measurement, which helps to gauge the range within which the precise value lies.
The main scale reading is crucial as it informs you about the larger division angles without any adjustments. Combining this with the Vernier dilation helps refine the result to a very precise measurement. This synthesis of the main scale and Vernier scale readings delivers an accurate final value, critical to drawing reliable conclusions in scientific and engineering applications.