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3.5: Newton's Laws of Motion

These laws can be summarized in two statements one, for every action by body \(A\) on Body \(B\) there is opposite reaction by body on body \(A\).  Two, which can expressed in mathematical form as \[\sum F = \frac{D\left(mU\right)}{Dt}\tag{57}\] It can be noted that \(D\) replaces the traditional \(d\) since the additional meaning which be added. Yet, it can be treated as the regular derivative. This law apply to any body and any body can "broken'' into many small bodies which connected to each other. These small ``bodies'' when became small enough equation (57) can be transformed to a continuous form as \[\sum F = \int_{V} \frac{D\left(\rho U\right)}{Dt}dV\tag{58}\] The external forces are equal to internal forces the forces between the ``small'' bodies are cancel each other. Yet this examination provides a tool to study what happened in the fluid during operation of the forces. Since the derivative with respect to time is independent of the volume, the derivative can be taken out of the integral and the alternative form can be written as \[\sum F = \frac{D}{Dt}\int_V \rho U dV \tag{59}\] The velocity, \(U\) is a derivative of the location with respect to time, thus, \[\sum F = \frac{D^{2}}{Dt^{2}}\int_V \rho r dV \tag{60}\] where \(r\) is the location of the particles from the origin. The external forces are typically divided into two categories: body forces and surface forces. The body forces are forces that act from a distance like magnetic field or gravity. The surface forces are forces that act on the surface of the body (pressure, stresses). The same as in the dynamic class, the system acceleration called the internal forces. The acceleration is divided into three categories: Centrifugal, \(\omega \times \left(r \times \omega\right)\), Angular, \(r \times \dot{\omega}\), Coriolis, \(2\left(U_{r} \times \omega\right)\). The radial velocity is denoted as \(U_{r}\).

Contributors

  • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.