Problem 3 - Master Resource Book in JEE Main Physics

Question

For measuring depth of a beaker using vernier callipers, observed readings are given as $$ \begin{array}{c|c|c} \hline \text { S.No. } & \text { MSR (cm) } & \text { VSD } \\ \hline \text { 1. } & 0.5 & 8 \\ \text { 2. } & 0.5 & 4 \\ \text { 3. } & 0.5 & 6 \\ \hline \end{array} $$ If zero error is \(-0.03 \mathrm{~cm}\), then mean corrected depth is (a) \(0.56 \mathrm{~cm}\) (b) \(0.59 \mathrm{~cm}\) (c) \(0.53 \mathrm{~cm}\) (d) \(0.52 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The mean corrected depth is 0.59 cm.

Step 1: Calculate Least Count of Vernier Calipers

The least count (LC) of a vernier caliper is calculated by the formula: \[ LC = 1 ext{ main scale division} - 1 ext{ vernier scale division} \] For example, if 1 main scale division (MSD) is 1 cm and there are 10 vernier scale divisions (VSD) in 1 MSD, then: \[ LC = rac{1}{10} ext{ cm} = 0.1 ext{ cm} \] Since specific details are not given, we'll assume LC is commonly 0.01 cm for vernier calipers.

Step 2: Calculate Individual Depth Readings

For each observation, calculate the depth measurement using the formula: \[ ext{Measurement} = ext{MSR} + ( ext{VSD} imes ext{LC}) \] For observation 1: \[ = 0.5 + (8 imes 0.01) = 0.5 + 0.08 = 0.58 \text{ cm} \] For observation 2: \[ = 0.5 + (4 imes 0.01) = 0.5 + 0.04 = 0.54 \text{ cm} \] For observation 3: \[ = 0.5 + (6 imes 0.01) = 0.5 + 0.06 = 0.56 \text{ cm} \]

Step 3: Calculate Mean Depth

Calculate the mean depth by averaging the calculated measurements: \[ ext{Mean Depth} = \frac{0.58 + 0.54 + 0.56}{3} = \frac{1.68}{3} = 0.56 \text{ cm} \]

Step 4: Apply Zero Error Correction

Account for the zero error correction by subtracting the negative zero error from the mean depth: \[ ext{Corrected Depth} = 0.56 + (-(-0.03)) = 0.56 + 0.03 = 0.59 \text{ cm} \]

Step 5: Determine the Correct Value

The corrected mean depth of the beaker is calculated as 0.59 cm. Hence, the correct choice among the options is (b) 0.59 cm.

Key Concepts

Least Count

When working with vernier calipers, understanding the concept of "Least Count" is crucial. The least count is the smallest measurement that can be accurately read using the instrument. In simple terms, it is the resolution of the device. It determines the precision of the vernier calipers and tells you how finely you can measure.
The formula for finding the least count of vernier calipers is: \[ \text{LC} = 1 \text{ main scale division} - 1 \text{ vernier scale division} \] Typically, the main scale division (MSD) is marked in millimeters or centimeters, while the vernier scale division (VSD) is subdivided further. To better understand this, think of each MSD as one whole unit, and VSDs offer more precision by subdividing the whole unit. Commonly, the least count for many vernier calipers is found to be 0.01 cm. This level of precision allows for detailed and accurate readings, essential for various scientific and engineering applications. Understanding and utilizing the least count correctly can make a significant difference in the quality of your measurements.

Zero Error

Even the most precisely manufactured instruments are not immune to small systematic errors. One such prevalent error in vernier calipers is the "Zero Error". When the jaws of the caliper are closed, ideally, the zero on the main scale should align perfectly with the zero on the vernier scale. However, due to wear and manufacturing imperfections, they may not line up, resulting in what is referred to as zero error. Zero error can be positive or negative. A positive zero error occurs when the vernier zero is to the right of the main scale zero, and a negative zero error happens when it is to the left. For reliable measurements, this error must be identified and corrected. The given exercise specifies a zero error of \(-0.03 \text{ cm}\), meaning the reading is consistently 0.03 cm less than it should be. Therefore, you add this value when calculating the corrected measurement.
Identifying and adjusting for zero error ensures that your final readings are accurate and reflect the true dimensions you are trying to measure.

Mean Depth Calculation

Calculating the depth or thickness of an object using vernier calipers involves several readings, as seen in the exercise provided. Initially, for each observation, you will determine the depth by using the formula: \[ \text{Measurement} = \text{MSR} + (\text{VSD} \times \text{LC}) \] Where MSR stands for Main Scale Reading and VSD is the Vernier Scale Division read. Once individual readings are obtained, the next step is to find the mean depth, which represents the average of all measurements taken.
To find the mean depth, you sum up all the calculated values and then divide by the number of observations. In this example, after computing all three readings, the mean was found to be 0.56 cm. However, don't forget to apply the zero error correction. Since there is a negative zero error of -0.03 cm, you'll need to add this discrepancy back, resulting in a corrected mean depth of 0.59 cm. Calculating the mean depth accurately allows for a reliable estimate of the object's true dimension, which is significant in precise scientific work.