Manual Problem solving without help

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If you analyze where mistakes are most often made and the problem is not solved properly, you can see that:

- the variables are not converted into a unit compatible with those of the other variables

- incorrect equations are identified or they are solved incorrectly

- the given variables are incorrectly linked to those of the equation or calculated in the meantime

In order to prevent this, it is advisable to carry out the solution in the manual case in a systematic and orderly manner and to write it down in a form that identifies each step.

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Variable identification

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From the statement of the problem, the given variables must be identified, with their values and units:

Variables, their values, and units can be entered on the form to neatly document the solution process.

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Convert variable value to MKS

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As it is necessary that all the variables are in the same system of units, the corresponding transformation must be carried out.

To do this, the value is converted and the result of the calculation is entered in the corresponding column of the form. In case the value is already in the selected unit system, its value is simply written again.

This can be done either with the corresponding conversions or if they are not available using applications such as https://www.convertunits.com/.

In this example the MKS unit system (meters-kilograms-seconds) is being used, but others can also be used.

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Identify equation and variable

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Based on the network of equations and the given variables, the equation that can be used to calculate the intermediate variables must be identified.

It is to remember that equations can only be used in which there is only one unknown variable (blank node) and all the rest are given (green node) or have already been calculated (orange node).

On the other hand, the equations that are used come from books, material or forms and are written with variables that do not necessarily coincide with those of the model. This problem is solved later under what we call translate.

For the moment, the equation and the variable to be calculated are entered in our form with the nomenclature that it has from where it is obtained.

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Solve equation

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If necessary, you must solve the equation for the variable you want to calculate.

This can also be done with an algebraic calculation program like wxMaxima that can be downloaded for free. In this case, the solve command is used, entering the equation to the variable to be solved.

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Enter the solution to the form

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To document the solution, the solved equation must be entered into the form:

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Translate the equation

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It must be indicated for each variable of the equation to which variable of the problem it corresponds. To do this, you must understand the physical meaning of each variable in the equation and understand which variable of the problem it corresponds to.

It is key to base it on the meaning and not on the similarity of the symbols used since this similarity can be coincidental.

Seeking to define the variables of the problem based on those of the equations is many times impossible both because of the fact that different equations can use the same symbols for variables that physically mean different things. On the other hand, the same equation can be used on multiple occasions for different variables of the problem.

For all these reasons it is necessary for each application of an equation to establish an equivalence with variables of the problem and thus translate the equation.

In the case of the form, it is assumed that the different variables given are assigned the letters a, b, c, etc., while the intermediate variables are assigned the letters u, v, w and for the final result the letter z. With this, the equation must be rewritten as a function of these variables.

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Calculate the intermediate variable

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After the translation, it was defined exactly which variable should assume which value and the calculation of the intermediate variable can be performed directly.

An alternative to carry out the calculation is the use of wxmaxima using the ev function that allows entering the formula and in a list the variables and their values.

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Equation for the final result

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As the last calculation may require a conversion to the units in which the solution is required, a somewhat different final calculation has been included at the end of the form.

Analogously to the previous calculations, the equation and the variable that correspond to the sought result must be identified. Both must be recorded on the far left of the bottom edge of the form.

Additionally, the equation must be solved, which in turn is entered in the corresponding box.

The solution of the equation can also be done with the help of wxmaxima as explained in the first equation used.

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Translation of the final equation

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As in the previous cases, it must be indicated for each variable of the equation to which variable of the problem it corresponds.

Remember that this is necessary because the symbols of the variables of the problem will not necessarily correspond to those of the equation that is used.

In the case of the form, it is assumed that the different variables given are assigned the letters a, b, c, etc., while the intermediate variables are assigned the letters u, v, w and for the final result the letter z. With this, the equation must be rewritten as a function of these variables.

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Calculation of the final result in MKS

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As in the previous cases, the value of the result in MKS must be calculated with the translated equation.

An alternative to carry out the calculation is the use of wxmaxima using the ev function that allows entering the formula and in a list the variables and their values.

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Convert result to required unit

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The result calculated up to this point is in the units of the system that has been chosen to work (example MKS).

If the result is requested in different units, it is necessary to convert the result of the calculation to the required units.

This can be done either with the corresponding conversions or if they are not available using applications such as https://www.convertunits.com/.

In this example the MKS unit system (meters-kilograms-seconds) is being used, but others can also be used.

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