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Erschienen in:
Elementary Particle Dynamics Kapitel lesen Erstes Kapitel lesen

2019 | OriginalPaper | Buchkapitel

5. Power, Work, and Energy

verfasst von : Oliver M. O’Reilly

Erschienen in: Engineering Dynamics

Verlag: Springer International Publishing

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Abstract

The chapter starts with a discussion of the notions of power and work. Subsequently, we make these ideas more precise by defining the mechanical power of a force and, from this, the work done by the force during the motion of a particle. Next, the work-energy theorem is derived from the balance of linear momentum. It is then appropriate to discuss conservative forces, and we spend some added time discussing the potential energies of gravitational and spring forces. With these preliminaries aside, energy conservation is discussed and a second form of the work-energy theorem is established. Numerous examples are presented that show how to prove and exploit energy conservation if it occurs in a problem featuring the dynamics of a particle.

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Vorheriges Kapitel Friction Forces and Spring Forces
Nächstes Kapitel Momenta, Impulses, and Collisions
Fußnoten
1
Gaspard-Gustav de Coriolis (1792–1843) was a French engineer, mathematician, and scientist. His most famous contribution to science was the discovery of the Coriolis effect.
 
2
The Scotsman James Watt (1736–1819) played a seminal role in the development of the steam engine, and James P. Joule (1818–1889) was an English physicist who is famed for his discovery of the formula for the heat developed by passing a current through a conductor.
 
3
The easiest example that illuminates this point is to consider a particle moving on a horizontal line. The tangent vector is constant; say \(\mathbf{E}_x\). However, if the line is moving, say with a velocity \(v_c\mathbf{E}_y\), then the velocity vector of the particle is not \(v_x\mathbf{E}_x\), rather it is \(v_x\mathbf{E}_x + v_c\mathbf{E}_y = \left( \sqrt{v^2_x + v^2_c}\right) \mathbf{e}_t\), where \(\mathbf{e}_t\) is the unit tangent vector to the path of the particle.
 
4
This is a classical result that was known, although not in the form written here, to the Dutch scientist Christiaan Huygens (1629–1695) and the German scientist Gottfried Wilhelm Leibniz (1646–1716). These men were contemporaries of Isaac Newton (1643–1727).
 
5
A common error is to assume that \(\dot{E} = 0\) implies that \(E = 0\). It does not.
 
6
This solution can be found using the solution to (the unforced harmonic oscillator) \(\ddot{x} + \omega ^2_0 x = 0\) that is discussed in Section A.5.2 of Appendix A.
 
7
It can be shown for a loop with \(H = 35\) meters that \(v_A > \sqrt{2 g H} \approx 26.21\) meters per second in order for the particle to reach the top D of the loop. By way of background, 27 meters per second is approximately 60 mph, 35 meters is approximately 115 feet, and a value of \(H = 35\) corresponds to a value of the clothoid parameter \(a = 27.657\).
 
8
For a discussion on the relationships between brain trauma and g-forces in roller coaster rides, we refer the reader to [89] .
 
9
This identity was established in Section 5.5.2 where the potential energy of spring force was discussed.
 
Metadaten
Titel
Power, Work, and Energy
verfasst von
Oliver M. O’Reilly
Copyright-Jahr
2019
Buch
Engineering Dynamics

Print ISBN: 978-3-030-11744-3

Electronic ISBN: 978-3-030-11745-0

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-030-11745-0_5