Chapter 2 – kinematics of particles Tuesday, September 1, 2015: Lecture 4 Today’s Objective: Curvilinear motion Normal and Tangential Components

Coordinate Systems

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!

Rectangular Vector Coordinates

Normal and Tangential Coordinates Used when Motion is along a curve n-t coordinates are most effective ds = ρ dβ, ds/dt = ρ dβ/dt = ρ 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 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 1 Fig 2

Normal and Tangential Coordinates Let us find what is det/dt det = dß (et = 1) en det/dt= dß/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 (dß/dt) x en = (w 𝜌)en =𝑣/𝜌 en dß/dt = angular velocity of the particle = w = v/ρ

Normal and Tangential Coordinates Circular Motion a = v det/dt + dv/dt et = (v) (v/ρ) en + dv/dt et = v2/ρ en + at et a = an en +at et For a circular motion, v = r 𝑑𝜃 𝑑𝑡 Or d𝜃/𝑑𝑡 = v/r 𝜃 𝑎𝑛𝑑 𝛽 𝑎𝑟𝑒 𝑢𝑠𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑐ℎ𝑎𝑛𝑔𝑒𝑎𝑏𝑙𝑦

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: Just after point B, and At point C