Name: accuracy and approximation in calculations
Uploaded: 2023-09-22T11:49:19.000Z
Duration: 1 min 28 s
Channel: JoVE

K12 Standards

Texas Essential Knowledge and Skills

High School

Specialised Topic in Science

112.51.c.1.

112.51.c.2.C.

Accuracy, limits, and approximation

In engineering calculations, accuracy, limits, and approximation are used to represent a value.
Consider two square blocks of different dimensions. The accurate values of their areas can be obtained by rounding off their numbers to the nearest values based on significant figures.
Consider a right circular cone having a base radius, r, and height, h. If the cone is circularly sliced at a distance x from its vertex, such that the sliced element has a&#160; thickness, &Delta;x, then its volume can be determined.
By applying the limit from &Delta;v to dv and &Delta;x to dx, and neglecting higher-order differentials, the expression for the volume is obtained.
Consider an arc AB, subtending a small angle at O. The arc-length can be approximated to a base of a right-angle triangle.
If the length of the hypotenuse is unity, the arc-length equals sine theta, approximately equal to theta. In addition, cosine theta also approximately equals unity.
For theta of one degree, the approximate value for sin theta and tan theta is nearly the same.

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&#176; 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&#176; or 0.26 radians. However, for angles less than or equal to 15&#176; (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.

Accuracy, limits, and approximations are common in many fields, especially in engineering calculations. These concepts are imperative for ensuring that a ...

Education

JoVE Core

Mechanical Engineering

Unit 1

An Introduction to Statics

KEY TERMS AND CONCEPTS

accuracy and approximation in calculations

Accuracy and approximation are important in calculations, especially in engineering. Accuracy means how close a measured value is to the true or actual value. Repeated measurements are often used in engineering mechanics to help make results more precise and accurate.

The accuracy of a solution depends on empirical data and the results from different calculations or measurements. A cylindrical container filled with liquid is a simple example. Its volume can be found from the measured dimensions and a formula, but those dimensions cannot be measured with absolute precision in practice.

Approximations are useful when a function is too complex to solve exactly. The degree of approximation depends on the number of significant figures used in the measurements. More significant figures usually give a result that is closer to the true value.

Approximations can also introduce error, so the effect of the error must be understood. For small angles, usually less than or equal to 15° or 0.26 radians, sine, cosine, and tangent can be approximated with good accuracy. This accuracy becomes worse when the angle is greater than 15° or 0.26 radians.

Limits are another key idea in engineering calculations. A limit describes the behavior of a function as an independent variable approaches a specific value. For a function that grows toward infinity as x approaches a certain value, the limit helps define the maximum value that can be reached without passing that x value. Limits are useful for judging the stability and performance of systems in critical applications.

Accuracy and Approximation in Calculations

accuracy-and-approximation-in-calculations

an introduction to mechanics

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Accuracy and approximation are important in calculations, especially in engineering. Accuracy means how close a measured value is to the true or actual value. Repeated measurements are often used in engineering mechanics to help make results more precise and accurate. The accuracy of a solution depends on empirical data and the results from different calculations or measurements. A cylindrical container filled with liquid is a simple example. Its volume can be found from the measured...

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key terms and concepts

Learn how accuracy, significant figures, small-angle approximations, and limits help describe real measurements and function behavior in engineering calculations.

Accuracy and Approximation in Calculations | JoVE K12

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PHYS.2

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Chemistry and Physics

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IPC.2.A

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112.51.c.1.A.

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112.51.c.1.B.

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112.51.c.1.F.

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112.51.c.1.H.

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Accuracy and Approximation in Calculations

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