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

2. Coordinate Systems

3. Plane Curvilinear Motion: Velocity
Note the direction of acceleration. It’s not predictable!

4. Acceleration
Note: The curve isn’t the path of the particle, it’s a plot of the velocity!

5. Rectangular Vector Coordinates

6. 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

7. 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

8. 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

9. 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

10. Problem
Given: N = 45 rpm. Find v and a of point A.

11. 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