2.6: How to solve problems
- Page ID
- 103431
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In general you can use the following flowchart [9] to solve problems in a structured way. The steps will be outlined in more detail in Ch. 14.
Follow these steps to solve a problem:
- Make sure you read the theory, at least up to the point needed to solve the problem.
- Read the entire problem description carefully, from beginning to end.
- Write down a list of variables given and a second list of variables to be calculated.
- Determine the type of problem.
- Determine for yourself whether this is a problem of kinematics, kinetics, or other.
- Break the problem down into sub-problems as appropriate and search for the required theory subsection in the book. Read this part again if needed.
- Extract the general methods and formulae that need to be applied. Write down the equation number(s) from the book. Make sure you use the original, generic form of the formulae (not a similar-looking version from an example or a problem solution).
- Make drawings, such as FBDs, or kinematic diagrams of your system. Use different views where needed. Clearly define at least one coordinate system including unit vector directions and a coordinate origin. Specify it if the coordinate system is moving or rotating.
- Optionally create a table of variables where you clearly associate the variables in the problem and in your drawings with the variables given in the theoretic formulae.
- Check all assumptions that need to hold for the formulae to be applicable
- Write down all assumptions you made for your specific problem, and highlight those that can only be checked later.
- In many problems you have to solve for one or more scalar variables. In general, you have \(E\) equations, and need to solve a system with \(U\) unknowns. To check if you have sufficient information \((E)\) to solve the problem apply the following steps:
a) Count the number \(U\) of scalar unknowns. Be careful to include all unique components of a vector or a matrix individually.
b) Count the number \(E\) of scalar equations given by the equations you found.
c) Compare \(E\) and \(U\).
- If \(E=U\), continue.
- If \(E<U\), do not start solving yet. Instead, find more equations, either in the theory or in the problem itself. Are there kinematic relationships between variables that can reduce \(U\) or increase E?
- If \(E>U\), continue, but make a note to come back to check if some of the equations do not contradict each other and if all of them are correct.
d) Solve the system of equations. Do this symbolically as much as possible, do not insert any numbers.
e) Check the original assumptions. Do they indeed hold? If not, go back to step 11, make other assumptions and solve the problem again.
f) Once you have found a result, conduct multiple plausibility checks. Also re-insert your solution into your original equations, especially if \(E>U\). If in doubt about your result, start again at step 1.
g) If requested compute the numerical values of the \(U\) unknowns using the \(E\) obtained equations and the given numerical quantities.