1.8: Accuracy, limits, and approximation
Accuracy, limits, and approximations are common in many fields, especially in engineering calculations. These concepts are imperative for ensuring that a given value is as close as possible to its true value.
Accuracy is defined as the closeness of the measured value to the true or actual value. In engineering mechanics, repeated measurements are taken during theoretical or experimental analyses to ensure that the result is precise and accurate.
The accuracy of any solution is based on the empirical data and the results obtained from different calculations or measurements. Consider a cylindrical container filled with a liquid. The volume of the liquid can be determined by measuring the dimensions of the container and calculating the volume using an appropriate formula. However, in practice, it is impossible to measure the dimensions of the container with absolute precision.
Approximations are particularly useful when dealing with complex functions that are challenging to solve analytically. The degree of approximation refers to the number of significant figures used in the measurements. When more significant figures are used, the result is more accurate. While approximations are useful, one must be careful when using them, as they can introduce errors that affect the accuracy of the calculation. This means that it is essential to understand the error introduced by the approximation and how it may affect the result.
For example, for a relatively small angle, usually less than or equal to 15° or 0.26 radians, approximations can be used to simplify the sine, cosine, and tangent functions. Notably, the accuracy of these approximations decreases when the angle is greater than 15° or 0.26 radians. However, for angles less than or equal to 15° (0.26 radians), these formulas have good accuracy and produce values that are almost equal to the exact value.
The concept of limits is equally important in engineering calculations. Limits are used to ascertain the behavior of a function, particularly as an independent variable approaches a specific value. Consider a function that approaches infinity as x approaches a certain value. The limit defines the maximum value that can be reached without surpassing the specified value of x. Limits are particularly useful in determining the stability and performance of systems in critical applications.
