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