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:

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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:

58.65 degree

58.77 degree

58.59 degree

59 degree

Least count $(LC) = \frac{0 .5 degree}{30}$

Reading = Main scale reading + vernier scale reading

$= 58.5 + 9 \times \frac{0.5}{30}$

= 58.65 degree

This question involves reading a spectrometer to measure the angle of a prism. A spectrometer uses a vernier scale for precise measurements. Let's break down the given data and the calculation step-by-step.

Given Data:

Main Scale Reading (MSR) = 58.5°
Vernier Scale Reading (VSR) = 09 divisions
1 Main Scale Division (MSD) = 0.5°
Total Vernier Scale Divisions = 30
30 Vernier Scale Divisions = 29 Main Scale Divisions

Step 1: Find the Least Count (LC) of the Vernier Scale

The least count is the smallest measurement that can be taken with the instrument. It is calculated using the formula:

L C = \frac{1}{N} \times 1 M S D

Where N is the number of divisions on the vernier scale.

However, the standard formula is:

L C = \frac{1 M S D}{N}

But we are given a special case: 30 VSD = 29 MSD. This means the vernier scale is constructed so that N divisions on the vernier scale are equal to (N-1) divisions on the main scale. This is a common design to make the least count even smaller. The formula for least count in this case is:

L C = \frac{1 M S D}{N}

Let's calculate the value of 1 MSD and N:

1 M S D = 0.5 °

N = 30

Now, plug these values into the formula:

L C = \frac{0.5 °}{30} = \frac{1}{60} ° = 0.01667 °

Step 2: Calculate the Vernier Scale Contribution

The reading from the vernier scale is found by multiplying the Vernier Scale Reading (VSR) by the Least Count (LC).

V e r n i e r C o n t r i b u t i o n = V S R \times L C

V e r n i e r C o n t r i b u t i o n = 9 \times \frac{0.5 °}{30} = 9 \times \frac{1}{60} ° = \frac{9}{60} ° = 0.15 °

Step 3: Find the Total Reading

The total angle of the prism is the sum of the Main Scale Reading and the Vernier Scale Contribution.

T o t a l R e a d i n g = M S R + (V S R \times L C)

T o t a l R e a d i n g = 58.5 ° + 0.15 ° = 58.65 °

Final Answer: The angle of the prism is $58.65 °$ .

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