Curvilinear Motion: Normal and Tangential Components Explained
Dive into the intricacies of curvilinear motion with a focus on the normal and tangential components. Learn about the complexities of velocity, acceleration, and coordinate systems in dynamic particle motion.
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Presentation Transcript
Chapter 2 kinematics of particles Tuesday, September 1, 2015: Lecture 4 Today s Objective: Curvilinear motion Normal and Tangential Components
Plane Curvilinear Motion: Velocity Note the direction of acceleration. It s not predictable!
Acceleration Note: The curve isn t the path of the particle, it s a plot of the velocity!
Normal and Tangential Coordinates Used when Motion is along a curve n-t coordinates are most effective ds = d , ds/dt = d /dt = v
Normal and Tangential Coordinates From Fig 2, as dv approaches 0, the direction of dv becomes perpendicular to the tangent a = v det/dt + dv/dt et Fig 1 The second term on the r.h.s. represents acceleration component in the tangential direction In the 1st term on RHS, et has a magnitude of 1, but the direction is changing with the motion, so this is not a constant vector and det/dt 0 Fig 2
Normal and Tangential Coordinates Let us find what is det/dt det = d (et = 1) en det/dt= d /dt en The direction of et is along the tangent to the curve, where as, det points toward the center of the curve. det/dt = angular velocity (d /dt) x en = (w?)en =?/?en d /dt = angular velocity of the particle = w = v/
Normal and Tangential Coordinates Circular Motion a = v det/dt + dv/dt et = (v) (v/ ) en + dv/dt et = v2/ en + atet a = anen +atet For a circular motion, v = r ?? Or d?/?? = v/r ?? ? ??? ? ??? ???? ?????? ????????
Problem 2. 101 Given: N = 45 rpm. Find v and a of point A.
Problem 2.122 Given: The particle P starts from rest at point A at time t = 0 and changes its speed thereafter at a constant rate of 2g as it follows the horizontal path as shown. Determine the magnitude and direction of its total acceleration: a. Just after point B, and b. At point C
