5.11: Summary
Let us summarise the kinematic methods that have been discussed in this chapter.
- Constraint equations
- Using geometry for determining constraint equations.
- Taking time derivatives of constraint equations for the position, velocity and acceleration of point masses and objects. See e.g. Equation 5.5.
- Equations of motion
- Solving equations of motion using initial conditions and constraint equations.
- Kinematics along the path curve
- Integration and differentiation with respect to time to obtain the path coordinate \(s(t)\), path velocity \(v_{s}(t)\) and path acceleration \(a_{s}(t)\).
- Integration and differentiation with respect to path coordinate \(s\) to obtain the functions \(v_{s}(s)\) and \(a_{s}(s)\).
- Analysing segmented kinematics along the path curve using (5.98), (5.99) and (5.100).
- Integration and differentiation with respect to the path velocity \(a_{s}(v)\)
- Kinematics in 3D coordinate systems
- Describing a path curve and the motion with the position vector \(\overrightarrow{\boldsymbol{r}}(t)\) of a point mass in 3D using Cartesian and cylindrical coordinates and work with natural \(t, n, b\) coordinates in different parametrisations.
- Properly draw coordinate systems, path coordinates and unit vectors. Draw position, velocity and acceleration vectors and project them on the coordinate axes, obtaining their components, magnitudes and angles.
- Determine the velocity vector \(\overrightarrow{\boldsymbol{v}}(t)\) and acceleration vector \(\overrightarrow{\boldsymbol{a}}(t)\) for a given motion \(\overrightarrow{\boldsymbol{r}}(t)\), irrespective of its parametrisation (see Sec. 5.9.6). In Cartesian coordinates use (5.54-5.56), in cylindrical coordinates use (5.60-5.68), and in natural \((t, n, b)\) coordinates use (5.83-5.86).
- Determine all components of the position and velocity vectors by integrating twice for a given \(\overrightarrow{\boldsymbol{a}}(t)\) in Cartesian coordinates by using Equation 5.57 and Equation 5.58.
- Know that the direction of the velocity vector \(\overrightarrow{\boldsymbol{v}}\) is always tangential to the path curve, while the acceleration vector \(\overrightarrow{\boldsymbol{a}}\) is not.
- Solve kinematic problems in 3D using numerical integration and differentiation.
In the next chapter we will discuss the kinetics of point masses, deriving their dynamics from the effect of forces via Newton’s second law.