Lecture 17 - Lecture 17 CURVILINEAR MOTION: GENERAL &...

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Dynamics: Lecture Notes for Sections 12.4-12.6 1 Lecture 17 CURVILINEAR MOTION: GENERAL & RECTANGULAR COMPONENTS MOTION OF A PROJECTILE Section 12.4-12.6 Ehab Zalok 2 CURVILINEAR MOTION: GENERAL & RECTANGULAR COMPONENTS Today’s Objectives : Students will be able to: 1. Describe the motion of a particle traveling along a curved path. 2. Relate kinematic quantities in terms of the rectangular components of the vectors.
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Dynamics: Lecture Notes for Sections 12.4-12.6 2 3 APPLICATIONS The path of motion of each plane in this formation can be tracked with radar and their 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 each plane at any instant? Should they be the same for each aircraft? 4 APPLICATIONS (continued) 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 ( F = ma ) of the car?
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Dynamics: Lecture Notes for Sections 12.4-12.6 3 5 GENERAL CURVILINEAR MOTION (Section 12.4) A particle moving along a curved path undergoes curvilinear motion . Since the motion is often three-dimensional, vectors are used to describe the motion. A particle moves along a curve defined by the path function, s. 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. If the particle moves a distance Δ s along the curve during time interval Δ t, the displacement is determined by vector subtraction : Δ r = r’ - r 6 VELOCITY Velocity represents the rate of change in the position of a particle. The average velocity of the particle during the time increment Δ t is v avg = Δ r / Δ 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 Δ s approaches the magnitude of Δ r as t 0, the speed can be obtained by differentiating the path function (v = ds/dt). Note that this is not a vector!
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Dynamics: Lecture Notes for Sections 12.4-12.6 4 7 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 Δ t, the average acceleration during that increment is: a avg = Δ v / Δ t = ( v - v’ )/ Δ t The instantaneous acceleration is the time- derivative of velocity: a = d v /dt = d 2 r /dt 2 A plot of the locus of points defined by the arrowhead of the velocity vector is called a hodograph .
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This note was uploaded on 02/28/2011 for the course MATH 101 taught by Professor Duke during the Spring '11 term at University of Ottawa.

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Lecture 17 - Lecture 17 CURVILINEAR MOTION: GENERAL &...

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