Question

The diameter of a cylinder is measured using a Vernier callipers with no zero error. It is found that the zero of the Vernier scale lies between \(5.10 \mathrm{~cm}\) and \(5.15 \mathrm{~cm}\) of the main scale. The Vernier scale has 50 divisions equivalent to \(2.45 \mathrm{~cm}\). The \(24^{\text {th }}\) division of the Vernier scale exactly coincides with one of the main scale divisions. The diameter of the cylinder is (A) \(5.112 \mathrm{~cm}\) (B) \(5.124 \mathrm{~cm}\) (C) \(5.136 \mathrm{~cm}\) (D) \(5.148 \mathrm{~cm}\)

Short answer

5.124 cm

Step-by-step solution

Understanding Vernier Callipers Reading

To find the diameter of the cylinder using a Vernier callipers, add the main scale reading (MSR) to the Vernier scale reading (VSR). The MSR is the last main scale division before the zero of the Vernier scale, in this case, between 5.10 cm and 5.15 cm. VSR is found using the formula: VSR = (coinciding Vernier division) * (value of one main scale division) / (number of divisions on Vernier scale).

Calculate the Value of One Main Scale Division (MSD)

Since 50 divisions of the Vernier scale equal 2.45 cm on the main scale, one main scale division (MSD) is 2.45 cm divided by 50, which equals 0.049 cm: MSD = 2.45 cm / 50 divisions.

Calculating the Vernier Scale Reading (VSR)

The 24th division of the Vernier scale coincides with the main scale, so multiply the 24 by the value of one main scale division and divide by 50: VSR = 24 * 0.049 cm / 50.

Calculate the Main Scale Reading (MSR)

The MSR is the last full cm reading before the zero of the Vernier scale. From the given information, it's 5.10 cm since the zero is between 5.10 cm and 5.15 cm.

Determine the Diameter of the Cylinder

Add the MSR and VSR to get the total reading: Diameter = MSR + VSR. This yields the diameter of the cylinder in cm.

Key concepts

Measurement Uncertainty

Understanding measurement uncertainty is essential for precision and accuracy in scientific experiments. It is the expression of the lack of exactness in a measurement, implying that there is a range within which the true value lies. Measurements can never be exact due to limitations in the measuring instruments, human perception, and other factors affecting the reading.

For example, when using a Vernier caliper to measure the diameter of a cylinder, the uncertainty comes from two main sources. The first is instrument precision, which refers to how finely a tool can measure — a Vernier caliper with more divisions will have lesser uncertainty. The second source is user interpretation, which involves reading the scale and determining where the Vernier scale lines up with the main scale. To reduce the measurement uncertainty, it's crucial to follow a systematic approach and understand how the reading is taken and calculated, as shown in the step-by-step solution provided for measuring the cylinder's diameter.

To improve upon this, ensure the caliper is calibrated appropriately, avoiding any zero error which affects the final reading. Careful observation and taking multiple readings can also help in reducing uncertainty and increase the reliability of the measurement.

Precision Instruments

In the realm of measurements, precision instruments like the Vernier caliper offer an advanced level of detail in quantifying dimensions. These instruments are designed to measure more precisely than standard rulers due to their unique scale, which provides readings to fractions of a millimeter.

The Vernier caliper consists of two scales: the main scale and the Vernier scale. The main scale offers a coarse measure, while the Vernier scale allows for a refined measurement by indicating where between two main scale marks the measurement falls. The innovation of the Vernier scale lies in its slight misalignment with the main scale, offering a subtle gradation that helps pinpoint measurements more precisely.

For better comprehension of the instrument's use, visualize the technique of obtaining a measurement, where the main scale provides the initial number and the Vernier scale denotes how many additional increments to count. Using these instruments effectively in experiments and problem solving takes practice, but mastering their use is vital for obtaining accurate and precise measurements in various scientific fields.

Physics Problem Solving

Effective physics problem solving requires a methodical approach, particularly when it comes to experiments and measurements. The critical steps usually include the identification of relevant principles, analysis of the problem, careful execution of the measurement, and precise calculations.

Let's break down the key stages in the context of measuring an object's dimension using a Vernier caliper:

• Comprehension: First, it is crucial to understand the mechanism of the instrument, such as a Vernier caliper in this case.
• Setting: Proper placement of the object and zeroing of the instrument ensures that systematic errors are minimized.
• Measurement: Observing the alignment of scales and noting readings carefully helps in acquiring the correct value.
• Calculation: This involves applying formulas or principles to process those observations into the final measurement, as seen with the Vernier scale reading (VSR) and main scale reading (MSR) calculations.
• Verification: Cross-checking the results and considering measurement uncertainty helps validate the reliability of the measurement.

This structured approach, when coupled with an understanding of the tools and their limitations, allows students to tackle physics problems more confidently and achieve consistent results in their experiments.