Abstract
As a prelude to the discussion of a system of particles, the linear and angular momenta of a single particle are introduced in this chapter. In particular, conditions for the conservation of these kinematical quantities are established. This is followed by a discussion of impact problems where particles are used as models for the impacting bodies.
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Notes
- 1.
Johannes Kepler (1571–1630) was a German astronomer and physicist who, based on observations of the orbits of certain planets, proposed three laws to explain planetary motion. His three laws are the most famous of his many scientific contributions.
- 2.
For further treatments of these equations, see, for example, Arnol’d [2] and Moulton [65] . Discussions of this central force problem can also be found in every undergraduate dynamics text, for example, in Meriam and Kraige [63, Section 13, Chapter 3] and Riley and Sturges [79, Section 15, Chapter 5]. You should notice that these texts assume that the motion of the particle is planar.
- 3.
We leave it as an exercise to show that the total energy and angular momentum \(\mathbf{H}_O\cdot \mathbf{E}_z\) are also conserved when the spring is replaced by an inextensible string.
- 4.
An example illustrating these three vectors is shown in Fig. 6.8. We also note that for many problems these vectors will coincide with the Cartesian basis vectors.
- 5.
It should be clear that, given the preimpact velocity vectors, the balance laws only provide six equations from which one needs to determine the six postimpact velocities and the linear impulses of \(\mathbf{F}_{1_d}\) and \(\mathbf{F}_{1_r}\). The introduction of the coefficient of restitution e and the assumption of a common normal velocity \(v_{II}\) at time \(t = t_1\) gives two more equations that render the system of equations solvable. That is, these two additional equations close the system of equations.
- 6.
- 7.
The model we discuss is a prototypical example in control theory for illustrating Zeno behavior (see, e.g., Liberzon [60] ). The term Zeno behavior is in reference to the Greek philosopher Zeno of Elea and his famous dichotomy paradox. He is perhaps more famous for his paradox about the tortoise and the hare.
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O’Reilly, O.M. (2019). Momenta, Impulses, and Collisions. In: Engineering Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-030-11745-0_6
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DOI: https://doi.org/10.1007/978-3-030-11745-0_6
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