Problem 60 - Technical Mathematics with Calculus

Question

A shaft is known to have a diameter of 35.000 mm. You measure it and get a reading of \(34.725 \mathrm{mm} .\) What is the percent error of your reading?

Step-by-Step Solution

Answer

The percent error of your reading is 0.7857%.

Step 1: Calculate the Absolute Error

First, find the absolute error by subtracting the measured value from the known value. Absolute Error = Known Value - Measured Value = 35.000 mm - 34.725 mm.

Step 2: Convert Absolute Error to Absolute Percentage Error

To get the percentage error, divide the absolute error by the known value and then multiply by 100. Percentage Error = (Absolute Error / Known Value) x 100.

Step 3: Compute the Percent Error

Substitute the absolute error calculated in step 1 into the formula from step 2 to find the percent error. Percent Error = ((35.000 mm - 34.725 mm) / 35.000 mm) x 100.

Key Concepts

Absolute Error

When measuring physical quantities, there's almost always a discrepancy between the measured value and the true or accepted value. This discrepancy is known as the absolute error. Intuitively, it's the 'straight-up' difference between what you get and what you expect to get, with no regard for whether the error is over or under the true value.

In the context of our exercise where the diameter of a shaft is measured, the absolute error is calculated by the simple subtraction of the measured value (\(34.725 \text{ mm}\)) from the known value (\(35.000 \text{ mm}\)). This yields an absolute error of \( 35.000 \text{ mm} - 34.725 \text{ mm} = 0.275 \text{ mm}\), representing the exact amount by which the measurement deviates from the true dimension. Grasping this concept is essential, as it sets the stage for more complex calculations in accuracy and error analysis.

Measurement Accuracy

The concept of measurement accuracy refers to how close a measured value is to the true or accepted value. Accuracy is often confused with precision, which is about the repeatability or consistency of measurements. Simply put, accuracy is about hitting the target, whereas precision is about how consistently you can hit the same spot, whether it's on target or not.

In technical mathematics, especially when performing quality control in manufacturing or scientific experimentation, it's vital to aim for high accuracy to ensure that parts fit together properly or that experimental results are valid. The percent error is one of the ways to quantify accuracy. In our example with the shaft measurement, a lower percent error would mean a more accurate measurement. Students should note that accuracy is influenced by a variety of factors such as instrument calibration, user skill, and environmental conditions.

Technical Mathematics

The field of technical mathematics covers a wide range of mathematical techniques used in applied contexts, such as engineering, computer science, and the physical sciences. It includes concepts like percent error calculation, which are indispensable for evaluating the reliability of measured data.

Understanding technical mathematics is critical for anyone performing technical measurements or working with data, as it provides the tools to analyze and communicate the quality of findings. In the case of our shaft measurement, by calculating the percent error (\( ((35.000 \text{mm} - 34.725 \text{mm}) / 35.000 \text{mm}) \times 100 = 0.7857\text{%} \) we are applying these mathematical principles to gauge the reliability of the measurement. Learners should delve into this field with practical problems in mind, as real-world applications help cement abstract concepts into usable knowledge.