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Explain Dimensional consistency in mechanical engendering, with equation.
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Solution:

Dimensional consistency means that the units associated with the numerical values on each side of an equality sign match.

In paper-and-pencil calculations, keep the units adjacent to each numerical quantity in an equation so that they can be combined or cancelled at each step in the solution.

You can manipulate dimensions just as you would any other algebraic quantity.

By using the principle of dimensional consistency, you can double check your calculation and develop greater confidence in its accuracy.

Of course, the result could be incorrect for a reason other than dimensions. Nevertheless, performing a double-check on the units in an equation is always a good idea.

Dimensional consistency can be illustrated by as simple a calculation as finding the weights of two objects, the first having a mass of 1 slug and the second having a mass of 1 LBM. In the first case, the weight of a 1-slug object is,

$ \begin{aligned}\\ w &=(1 \mathrm{slug})\left(32.174 \frac{\mathrm{ft}}{\mathrm{s}^2}\right) \\\\ &=32.174 \frac{\text { slug } \cdot \mathrm{ft}}{\mathrm{s}^2} \\\\ &=32.174 \mathrm{lb}\\ \end{aligned}\\ $

This object, having a mass of one slug, weighs 32.174 lb. For the calculation to be dimensionally consistent, an intermediate step is necessary to convert m to the units of slug using Equation,

$ \begin{aligned}\\ m &=(1 \mathrm{lbm})\left(3.1081 \times 10^{-2} \frac{\text { slugs }}{\mathrm{lbm}}\right) \\\\ &=3.1081 \times 10^{-2} \text { slugs }\\ \end{aligned}\\ $

In the second case, the weight of the 1-lbm object becomes,

$ \begin{aligned}\\ w &=\left(3.1081 \times 10^{-2} \text { slugs }\right)\left(32.174 \frac{\mathrm{ft}}{\mathrm{s}^2}\right) \\\\ &=1 \frac{\text { slug } \cdot \mathrm{ft}}{\mathrm{s}^2} \\\\ &=1 \mathrm{lb}\\ \end{aligned}\\ $

The principle of dimensional consistency will help you to make the proper choice of mass units in the USCS.

Generally speaking, the slug is the preferred unit for calculations involving Newton’s second law ( f = ma), kinetic energy ( 0.5 mv2 ), momentum (mv), gravitational potential energy (mgh), and other mechanical quantities.

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