Chapter 2 – kinematics of particles Tuesday, September 1, 2015: Lecture 4 Today’s Objective:...
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Transcript of Chapter 2 – kinematics of particles Tuesday, September 1, 2015: Lecture 4 Today’s Objective:...
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 = , ds/dt = /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
• 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 Fig 2
Fig 1
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
• 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 Or d = v/r
𝜃𝑎𝑛𝑑 𝛽𝑎𝑟𝑒𝑢𝑠𝑒𝑑 h𝑖𝑛𝑡𝑒𝑟𝑐 𝑎𝑛𝑔𝑒𝑎𝑏𝑙𝑦
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, andb. At point C