Abstract
This chapter is the culmination of the primer. To start, the linear momentum of a system of K particles and N rigid bodies is discussed. Similarly, the angular momenta and kinetic energy of such a system are developed. We then turn to the balance laws for a system and demonstrate how the balance laws can be used to determine the equations of motion, conservations of energy, angular momentum, and linear momentum in a range of examples. These examples include a double pendulum, a semi-circular cylinder rolling on a cart, a simple model for a spherical robot, and impact problems.
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Notes
- 1.
You should be able to consider special cases of these results: for example, cases where the system of interest contains either no rigid bodies or no particles. Additionally, we could establish a balance of angular momentum relative to the center of mass of the system (\(\mathbf{M} = \dot{\mathbf{H}}\)), but we leave this as an exercise. Such an exercise involves using \(\mathbf{F} = \dot{\mathbf{G}}\) and \(\mathbf{M}_O = \dot{\mathbf{H}}_O\). Its proof is similar to that used to establish the corresponding result for a single rigid body.
- 2.
See the time-lapse images of the falling cat in Crabtree [25] and Kane and Scher [55]. References to modern approaches to this problem can be found in Fecko [38] and Shapere and Wilczek [86].
- 3.
- 4.
For ease of notation, we drop the subscripts r and p used in the previous sections.
- 5.
- 6.
If we did, we would find that \(N_2\) and \(N_3\) are not constant and oscillate as the semi-circular cylinder moves back and forth along the upper surface of the cart. The possibility of one of these forces vanishing is also present. If this were to happen, then our assumption that
would be invalid and the tipping motion of the cart must also be considered. - 7.
It is useful to observe how the contributions from the mechanical powers of equal and opposite friction forces and normal forces cancel out.
- 8.
We leave it as an exercise to use the work-energy theorem to establish this conservation. A key intermediate step in this exercise is to establish that \(\mathbf{R}_{C_1}\cdot {\bar{\mathbf{v}}}_{1} - \mathbf{R}_{C_1} \cdot \mathbf{v}_2 = 0\).
- 9.
This is similar to the conservation of linear momentum in the cart–pendulum system discussed in Section 7.6.
- 10.
Paul Painlevé (1863–1933) was a French mathematician and politician. He is credited as being the first airplane passenger of Wilbur Wright in 1908.
for a single rigid body is discussed.
would be invalid and the tipping motion of the cart must also be considered.Author information
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O’Reilly, O.M. (2019). Systems of Particles and Rigid Bodies. In: Engineering Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-030-11745-0_10
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DOI: https://doi.org/10.1007/978-3-030-11745-0_10
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