ANGULAR KINEMATICS
Angular Kinematics
‹
Chapter 6
Description of angular motion
without regard to the causes
(forces) of the motion
Angular motion- motion about an
axis of rotation
‹ Observation = angular kinematics
‹ Most human motion is angular
‹
ANGLE
GENERAL MOTION
composed of two lines that
intersect at a point called the
vertex.
‹ knee angle is between thigh
and leg segments.
‹ Goniometers
‹
Combination of linear and
angular (rotational) motion.
Not described by one single plane/axis
Motion is “non-planar”
ABSOLUTE ANGLE
RELATIVE ANGLE
The included angle between
‹
angle of orientation
of a body segment
with respect to a
fixed line of
reference
‹
describes the
orientation of a
segment in space
longitudinal axes of the two segments.
Describe the amount of motion at a
joint without regard to position in
space.
From ARP, all joint angles = 0
1
ANGULAR DISTANCE
vs DISPLACMENT
UNITS OF MEASURE
Ang. Distance:The total of all angular
changes measured following its exact path
Angular Displacement (θ)- is the difference
between the initial and final positions of the
rotating object
θ = θfinal - θinitial
CCW = +
Degree (o)
3600
‹ Revolution
1 Rev = 3600
‹ Radian - measure of an angle at
the center of a circle described
by an arc equal to the length of
the radius of the circle (57.3
deg)
‹
CW = -
∏/2
ANGULAR SPEED
Radians
Quantified in
multiples of pi
(π)
‹ 1 circle = 1 rev
= 360 degrees =
2π
‹ Radians are
used as a
conversion
factor between
linear and
angular units
‹
0
180
∏
360
2∏
Speed = angular distance
time
σ
=
φ
3∏
2
∆t
Scalar quantity, not used much
ANGULAR VELOCITY
ω = change in angular displacement
change in time
ω=
θ final - θ initial
t final - t initial
θ
=
- rate of change of angular velocity
with respect to time
α
rad/s
= ω final - ω initial
timefinal - timeinitial
t
Vector – must show +/- to indicate direction
o/s
ANGULAR ACCELERATION
α
=
ω
t
- o/s2
- rad/s2
Vector – must show +/- to indicate direction
2
Units of Measure
Linear
Displacement Velocity
Acceleration
Meters
Meters/
second
Meters/
second2
Radians/
second
Radians/
second2
Angular Radians
RIGHT HAND RULE
→Linear vectors represented with straight
lines…but, this method is not practical for
angular vectors…so we use the RIGHT HAND
RULE
→Direction of the angular motion vector
placing the curled fingers of the right hand in
the direction of the rotation.
→Vector now representing the motion would be
perpendicular to the plane of rotation
RELATIONSHIP BETWEEN
ANGULAR & LINEAR MOTION
DISPLACEMENT
s = rφ or d=rθ
Greater r between a pt on a rotating body & the
axis of rotation, the greater the linear distance
undergone by that point during angular motion
lin displcmt = radius of rot * ang displcmt
Validity of Relationship
1. Linear distance & radius of
rotation must be quantified in the
units of length
‹ 2. Angular distance must be
expressed in radians
‹
‹
Radians disappear because they only
serve as a conversion factor between
linear and angular measurement
Symbols
Angular speed
σ
Angular displacement
Angular distance
θ
φ
Change in…
∆
Angular velocity
ω
Angular acceleration α
Curvilinear distance s
radius
r
VELOCITY
VT = rω
V = linear (tangential) velocity of a
point
ω Must be expressed in rad/s
V must be expressed in radius units
over appropriate units of time
3
The average speed of a point on a rotating object =
average angular velocity of the object * the radius
Performance Goals
‹
RA
DI
IS
AX
Va
US
‹
‹
Direct an object
accurately while
imparting a large
velocity
Greater the r =
greater linear v
imparted to the
ball (Golf)
Trade-off
Vb
Linear & Angular Acceleration
‹
‹
Resolve into components
Tangential: directed along path of
motion
• Represents change in linear speed for a
body traveling on a curved path
• Speed of projection @ release greatly
affects range of projectile – maximize prior
to release if throwing for distance or speed
• Once released, tangential acceleration = 0
ACCELERATION
Tangential & Angular acceleration
aT = α r
Centripetal (radial) acceleration:
-Rate of change in direction of a body in
angular motion
-Always directed toward the center
v2
ar =
r
Summary
Most movements involve rotation
(angular)
‹ Angular kinematic quantities are
interrelated to their linear
counterparts
‹
‹
Homework
4