Lecture_2 - Lecture #2 Part #1 CURVILINEAR MOTION: GENERAL...

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Lecture #2 Part #1 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. In-Class Activities : Reading Quiz Applications General Curvilinear Motion Rectangular Components of Kinematic Vectors Concept Quiz Group Problem Solving Attention Quiz
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READING QUIZ 1. In curvilinear motion, the direction of the instantaneous velocity is always A) tangent to the hodograph. B) perpendicular to the hodograph. C) tangent to the path. D) perpendicular to the path. 2. In curvilinear motion, the direction of the instantaneous acceleration is always A) tangent to the hodograph. B) perpendicular to the hodograph. C) tangent to the path. D) perpendicular to the path.
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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?
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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 of the car?
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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. 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 Δ s along the curve during time interval Δ t, the displacement is determined by vector subtraction : Δ r = r ` - r
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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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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 . The acceleration vector is tangent to the hodograph, but not, in general, tangent to the path function.
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Lecture_2 - Lecture #2 Part #1 CURVILINEAR MOTION: GENERAL...

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