Lecture 21 - Lecture 21 EQUATIONS OF MOTION NORMAL AND...

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Dynamics: Lecture Notes for Sections 13.5-13.6 1 Lecture 21 EQUATIONS OF MOTION: NORMAL AND TANGENTIAL COORDINATES CYLINDRICAL COORDINATES Section 13.5-13.6 (pp 127-163) Problems 13.54,69,71,77,83,93,105 Ehab Zalok 2 EQUATIONS OF MOTION: NORMAL AND TANGENTIAL COORDINATES Objectives : Students will be able to: 1. Apply the equation of motion using normal and tangential coordinates.
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Dynamics: Lecture Notes for Sections 13.5-13.6 2 3 APPLICATIONS Race tracks are often banked in the turns to reduce the frictional forces required to keep the cars from sliding at high speeds. If the car’s maximum velocity and a minimum coefficient of friction between the tires and track are specified, how can we determine the minimum banking angle ( θ ) required to prevent the car from sliding? 4 NORMAL & TANGENTIAL COORDINATES (Section 13.5) When a particle moves along a curved path , it is convenient to write the equation of motion in terms of normal and tangential coordinates . The normal direction (n) always points toward the path’s center of curvature . In a circle, the center of curvature is the center of the circle. The tangential direction (t) is tangent to the path, usually set as positive in the direction of motion of the particle.
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Dynamics: Lecture Notes for Sections 13.5-13.6 3 5 EQUATIONS OF MOTION This vector equation will be satisfied provided the individual components on each side of the equation are equal, resulting in the two scalar equations: F t = ma t and F n = ma n . Here F t & F n are the sums of the force components acting in the t & n directions, respectively. Since the equation of motion is a vector equation , F = m a , it may be written in terms of the n & t coordinates as F t u t + F n u n = m a t + m a n Since there is no motion in the binormal (b) direction , we can also write F b = 0. 6 NORMAL AND TANGENTIAL ACCERLERATIONS The tangential acceleration , a t = dv/dt, represents the time rate of change in the magnitude of the velocity . Depending on the direction of F t , the particle’s speed will either be increasing or decreasing. The normal acceleration , a n = v 2 / ρ , represents the time rate of change in the direction of the velocity vector. Remember, a n always acts toward the path’s center of curvature. Thus,
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Lecture 21 - Lecture 21 EQUATIONS OF MOTION NORMAL AND...

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