Applied Mathematics B154
TW B154 (Group 3 & 4)
Mechanics: Dynamics
Curvilinear motion:
Polar and Cylindrical components
Quotation
The more important fundamental laws and
facts of physical science have all been
discovered, and these are now so firmly
established that the possibility of their ever
being supplanted in consequence of new
discoveries is exceedingly remote [...] our
future discoveries must be looked for in the
sixth place of decimals.
Light Waves and their Uses (1903)
Albert Michelson (1852-1931)
Review questions
Changes in particle speed and changes in
particle direction are neatly separated in the
acceleration components in n-t coordinates.
Explain.
Which vector component in n-t coordinates
mimics one-dimensional motion, but on a
curvilinear path?
How do we get the radius of curvature from
angular displacement on a circular arc?
Which component results from projecting
acceleration in the direction of velocity?
New questions
How are the polar coordinates and the
cylindrical coordinates defined?
What are the radial and the transverse
components of a vector?
What are the expressions for velocity and
acceleration in polar coordinates?
What is meant by the angular velocity of a
particle? Angular acceleration?
Polar coordinates
Any planar position can be specified using
a radial coordinate r, which extends outward
from the fixed origin O (or pole) to the particle,
and a transverse coordinate θ, which is the
counterclockwise direction between a fixed
reference line and the r-axis.
Position in polar coordinates
The unit vector in the
radial direction is ur.
In the transversal
direction it is uθ.
The polar position of a
particle in the plane is
defined by the vector:
r=r u r
Velocity in polar coordinates
(Planar) velocity may be
written in its two polar
components as:
v=v r ur v  u
where
and
v r = ṙ
v θ=r θ̇
Polar acceleration
(Plane) acceleration may
be written in its two polar
components, as
a=a r ur a u
where
and
2
˙
a r = r̈−r 
a =r ̈2 ṙ ̇
Angular velocity and acceleration
Angular velocity θ̇ is the time rate of
change in angle θ, measured in radians per
second. (It is different from the transverse
or θ-component of velocity r θ̇ )
Angular acceleration θ̈ is the second time
derivative of θ, measured in radians per
second squared. (It is different from the
transverse or θ-component of acceleration
r θ̈+2 ṙ θ̇ )
Problem 12-157
A particle moves along a circular path of
radius 300 mm. If its angular velocity is
2
θ̇=(3t ) rad/s, where t is in seconds,
determine the magnitudes of the
particle's velocity
and acceleration
when θ = 45°.
The particle starts from rest when θ = 0°.
Problem 12-158
A particle moves along a circular path of
0.5t
radius 5 m. If its position is θ = ( e ) rad,
where t is in seconds,
determine the magnitude of the
particle's acceleration
when θ = 90°.
Problem 12-160
The position of a particle is described by
r = ( 300e-0.5t ) mm and θ = ( 0.3t2 ) rad,
where t is in seconds.
Determine the magnitudes of the
particle's velocity
and acceleration
at the instant t = 1.5 s.
Assignment
Completion: before class on 16 August
Study Hibbeler §12.8 and the four examples
Work through the class problems again
Look at the six Fundamental Problems
Do problems 156 & 159 on your own
P12-156 result: Velocity 2.40 m/s, acceleration
19.3 m/s2