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Unformatted text preview: S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L4 Curvilinear Motion. Cartesian Coordinates We will start by studying the motion of a particle . We think of a particle as a body which has mass, but has negligible dimensions. Treating bodies as particles is, of course, an idealization which involves an approximation. This approximation may be perfectly acceptable in some situations and not adequate in some other cases. For instance, if we want to study the motion of planets, it is common to consider each planet as a particle. This simplification is not adequate if we wish to study the precession of a gyroscope or a spinning top. Kinematics of curvilinear motion In dynamics we study the motion and the forces that cause, or are generated as a result of, the motion. Before we can explore these connections we will look first at the description of motion irrespective of the forces that produce them. This is the domain of kinematics . On the other hand, the connection between forces and motions is the domain of kinetics and will be the subject of the next lecture. Position vector and Path We consider the general situation of a particle moving in a three dimensional space. To locate the position of a particle in space we need to set up an origin point, O , whose location is known. The position of a particle A , at time t , can then be described in terms of the position vector , r , joining points O and A . In general, this particle will not be still, but its position will change in time. Thus, the position vector will be a function of time, i.e. r ( t ). The curve in space described by the particle is called the path , or trajectory . We introduce the path or arc length coordinate , s , which measures the distance traveled by the particle along the curved path. Note that for the particular case of rectilinear motion (considered in the review notes) the arc length coordinate and the coordinate, s , are the same. 1 Using the path coordinate we can obtain an alternative representation of the motion of the particle. Consider that we know r as a function of s , i.e. r ( s ), and that, in addition we know the value of the path coordinate as a function of time t , i.e. s ( t ). We can then calculate the speed at which the particle moves on the path simply as v = s ˙ ≡ ds/dt . We also compute the rate of change of speed as a t = ¨ s = d 2 s/dt 2 . We consider below some motion examples in which the position vector is referred to a fixed cartesian coordinate system....
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 Fall '09
 widnall
 Dynamics, Cartesian Coordinate System, Acceleration, Velocity, Polar coordinate system

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