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Chapter12-Trangentail Acceleration

# Chapter12-Trangentail Acceleration - Kinematics of a...

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Kinematics of a Particle: Curvilinear Motion Lectures 19-20 (Chapter 12) Dr. Marina Milner-Bolotin [email protected] Check Blackboard for course web site!

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Part I: Lecture 19 – 2-D Kinematics (Curvilinear Motion) Goals: Describes the motion of a particle traveling along a curved path by relating the kinematic quantities in terms of the rectangular components of the vectors. Describes the motion of a particle traveling along a curved path by relating the kinematic quantities in terms of the tangential and normal components of the vectors. 2
How can we determine the velocity or acceleration of each plane at any instant? How can we determine its position or acceleration of the roller coaster at any instant? A Few Applications

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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 s along the curve during time interval t, the displacement is determined by vector subtraction : r = r’ - r CLM - Motion of a particle along a curved path which lies on a single plane. Vectors are used to describe the motion. Curvilinear Motion (CLM) POSITION
Velocity represents the rate of change in the position of a particle. The average velocity of the particle during the time increment t is: The instantaneous velocity is the time-derivative of position: The velocity vector , v , is always tangent to the path of motion. The magnitude of v is called the speed . Since the arc length s approaches the magnitude of r as t →0, the speed can be obtained by differentiating the path function ( v = d s /d t ). Note that this is not a vector! Velocity

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Acceleration Acceleration represents the rate of change in the velocity of a particle. The instantaneous acceleration is the time-derivative of velocity: If a particle’s velocity changes from v to v’ over a time increment t, the average acceleration during that increment is:
Rectangular Coordinates in 2D If we know x = f 1 ( t ) and y = f 2 ( t ), then at any given time we can combine coordinates x and y to obtain r . Similarly, we can calculate the first derivatives of x and y to obtain v , or their second derivatives to obtain a . Thus, curvilinear motion is the superposition of two simultaneous rectilinear motions in the x - and y -directions.

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Summary The average speed: The instantaneous velocity: IMPORTANT: when t 0 the direction of r approaches the direction of the tangent to the path; thus, the velocity v is always a vector tangent to the path. The average velocity:
Given: The motion of two particles (A and B) is described by the position vectors r A = [3 t i + 9 t (2 – t ) j ] m r B = [3( t 2 –2 t +2) i + 3( t – 2) j ] m Find: The point at which the particles collide and find their speeds just before the collision.

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