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# lecture+notes+2 - 14:440:222 Dynamics(Lecture Note#2...

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14:440:222 Dynamics (Lecture Note #2) Instructor: Prof. Peng Song Rutgers University Today’s Lecture Curvilinear motion Rectangular coordinates

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Applications The path of motion of a plane can be tracked with radar and its x, y, and z coordinates (relative to a point on earth) recorded as a function of time. How can we determine the velocity or acceleration of the plane at any instant? Applications (cont’d) A roller coaster car travels down a fixed, helical path at a constant speed. How can we determine its position or acceleration at any instant? If you are designing the track, why is it important to be able to predict the acceleration of the car?
General Curvilinear Motion A particle moving along a curved path undergoes curvilinear motion. Since the motion is often three-dimensional, vectors are used to describe the motion. The position of the particle at any instant is designated by the vector r = r (t). Both the magnitude and direction of r may vary with time. A particle moves along a curve defined by the path function, s. If the particle moves a distance Ds along the curve during time interval t, the displacement is determined by vector subtraction: r = r’ - r Velocity Velocity represents the rate of change in the position of a particle. The average velocity of the particle during the time increment t is v avg = r / t . The instantaneous velocity is the time-derivative of position v = d r /dt . The velocity vector, v , is always tangent to the path of motion. The magnitude of v is called the speed. Since the arc length Ds approaches the magnitude of r as t0, the speed can be obtained by differentiating the path function (v = ds/dt). Note that this is not a vector!

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Acceleration Acceleration represents the rate of change in the velocity of a particle. If a particle’s velocity changes from v to v’ over a time increment t, the average acceleration during that increment is: a avg = v / t = ( v - v’ )/ t The instantaneous acceleration is the time-derivative
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