Newton’s Laws of Motion

These important laws of nature specify how the system of particles moves in res­ponse to external forces or moments. For the system (described in an inertial frame of reference), we write:

Подпись: (3.3) (3.4) dP = F

dt

Подпись: = M,dH

dt

where P and H are the linear and angular momenta of the system, respectively, and F and M are the externally applied sums of forces and moments. Because we are dealing with a system of individual particles, the force F must include the interaction forces between the particles that constitute the system. These occur in equal and opposite pairs, so that it is usually the case that only the forces acting on the boundary of the system must be considered. It may be helpful to review the treatment of systems of particles in textbooks on dynamics.

Remember that the coordinate system in which the momentum vectors, P and H, are expressed must be a Newtonian or inertial coordinate frame; that is, it cannot be accelerating or rotating. If it is necessary or convenient to use noninertial coordi­nates in describing the system motion, then it is necessary to introduce corrective terms (e. g., centripetal and Coriolis effects) into Eqs. 3.3-3.4. This normally is not required in the applications considered in this book because the emphasis is on steady flow without spin.

The linear and angular momentum for the system can be written as integrals over the mass or volume of the system as follows:

P= J V dm = JJJ pV dV (3.5)

system system

mass volume

1 /V

H= J r x V dm= JJJ p(r x V)dV, (3.6)

system system

mass volume

where V is the velocity vector at a point in the system glob and r locates the corre­sponding point in the inertial coordinate system.

Although both linear motion of the system as expressed by the linear momentum P and rotational motion expressed by the angular momentum H are defined here for completeness, only the former appears in most aerodynamics applications. Thus, we carry out further manipulation of Newton’s laws in detail only for the linear momentum changes of the system. The student should notice that completely analogous developments can be made for the angular momentum if needed in a particular application of interest, where angular motion or spin is an important feature.

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