6.14: Summary
- Page ID
- 103473
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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 this chapter we have formulated a complete methodology to derive the kinetics of systems of one or more point masses under the influence of forces and constraints generated by constraining objects and massless mechanisms. Here we have made use of the kinematics from the previous chapter, introduced the important concepts force and mass, and discussed the procedure of sketching the problem and the FBDs to obtain and solve the EoMs. Let us summarise the main topics and equations in this chapter:
- Mass and force
- Mass and force are defined by Newton’s laws
\(-\sum \overrightarrow{\boldsymbol{F}}=m \overrightarrow{\boldsymbol{a}}\)
\(-\overrightarrow{\boldsymbol{F}}_{\text {action }}=-\overrightarrow{\boldsymbol{F}}_{\text {reaction }}\)
- Procedure for solving the EoMs
- Sketch, CS, FBD, projecting forces, determining the EoM
- Solving the EoM using kinematics and constraints
- Relative motion and IRF
- Forces, mechanisms and constraints
- Gravity: \(\overrightarrow{\boldsymbol{F}}_{g}=m \overrightarrow{\boldsymbol{g}}\)
- Friction: Eqs. \((6.28,6.29)\)
- Massless mechanisms: \(\sum \overrightarrow{\boldsymbol{F}}_{\text {ext }}=\overrightarrow{\mathbf{0}}\)
- Rods and ropes: \(\left|\overrightarrow{\boldsymbol{r}}_{A}-\overrightarrow{\boldsymbol{r}}_{B}\right|=L\)
- Spring: \(\overrightarrow{\boldsymbol{F}}_{k}=k \Delta L \hat{\boldsymbol{s}}\)
- Damper: \(\overrightarrow{\boldsymbol{F}}_{c}=c \frac{\mathrm{d} L}{\mathrm{~d} t} \hat{\boldsymbol{s}}\)
Although all kinetic problems can be solved using the general method outlined in this chapter, certain problems, in particular those where the forces only depend on position, are easier to solve using the concepts of work and energy, which will be discussed in the next chapter. If the forces only depend on time, or have a very short duration, the principle of impulse and momentum is useful, which will be discussed in chapter 8 .