Physics 160 Biomechanics
Angular Kinematics
Questions to think about
• Why do batters slide their hands up
the handle of the bat to lay down a
bunt but not to drive the ball?
• Why might an athletic trainer or
physical therapist want to measure
the range of motion of a joint?
Angular Kinematics
• Angular motion: all parts of a
body move through the same
angle
• Angular kinematics deals with
angular motion.
• Nearly all human movement
involves rotation of body
segments.
Qualitative Application of
Biomechanical Principle
– EX. Soccer kick
Hips (flexion,rotation) --> knee (extension)
Qualitative Application of
Biomechanical Principle
– EX. Soccer kick
Hips (flexion,rotation) --> knee (extension)
Qualitative Application of
Biomechanical Principle
– EX. Soccer kick
Hips (flexion,rotation) --> knee (extension)
Qualitative Application of
Biomechanical Principle
– EX. Soccer kick
Hips (flexion,rotation) --> knee (extension)
Qualitative Application of
Biomechanical Principle
– EX. Soccer kick
Hips (flexion,rotation) --> knee (extension)
Qualitative Application of
Biomechanical Principle
– EX. Soccer kick
Hips (flexion,rotation) --> knee (extension)
Qualitative Application of
Biomechanical Principle
– EX. Soccer kick
Hips (flexion,rotation) --> knee (extension)
Qualitative Application of
Biomechanical Principle
– EX. Soccer kick
Hips (flexion,rotation) --> knee (extension)
Qualitative Application of
Biomechanical Principle
– EX. Soccer kick
Hips (flexion,rotation) --> knee (extension)
Qualitative Application of
Biomechanical Principle
– EX. Soccer kick
Hips (flexion,rotation) --> knee (extension)
Qualitative Application of
Biomechanical Principle
– EX. Soccer kick
Hips (flexion,rotation) --> knee (extension)
Qualitative Application of
Biomechanical Principle
– EX. Soccer kick
Hips (flexion,rotation) --> knee (extension)
Qualitative Application of
Biomechanical Principle
– EX. Soccer kick
Hips (flexion,rotation) --> knee (extension)
Qualitative Application of
Biomechanical Principle
- EX. Soccer kick
Hips (flexion,rotation) --> knee (extension)
Qualitative Application of
Biomechanical Principle
– EX. Soccer kick
Hips (flexion,rotation) --> knee (extension)
Rotational Motion
General Motion Rotation and
Translation
Instantaneous Center of Rotation
The instantaneous
joint center changes
throughout any
motion of the knee.
Measuring Angles
• An angle is at the intersection of
two lines (and planes).
• Units of measurement
– Degrees (arbitrary units)
– Radians (fundamental ratio)
– Revolutions (one revolution =
360o)
• Circumference = 2πr; therefore,
there are 2π radians in 360o
Angular Kinematic Quantities
Units of angular measure
90 degrees 180 degrees 270 degrees 360 degrees
3π
π
π radians
2π radians
2 radians
2 radians
1
4
revolution
1
2
revolution
3
4
revolution 1revolution
Radians
• One radian is the angle at the center of a
circle described by an arc equal to the length
of the radius.
Converting Angles
• 1 radian = 57.3o
• 1 revolution = 360o = 2 π radians
=
θ (deg) (180 / π ) ×θ (rad )
• Examples:
– Convert 30o to radians.
– Convert 4 radians to degrees.
Relative versus Absolute Angles
Relative angle: the angle
formed between two
adjacent body
segments (on right)
Absolute angle: angular
orientation of a single
body segment with
respect to a fixed line
of reference (on left)
Absolute Angles
Definition of the sagittal
view absolute angles of
the trunk, thigh, leg,
and foot.
Rear Foot Angles
Definition of the absolute
angles of the leg and
calcaneus in the frontal
plane. These angles are
used to constitute the
rearfoot angle of the right
foot.
Standard Reference
Terminology
• Movements
– Angular - Circular movements around an axis of
rotation. Unit of reference --> angle (degrees or
radians).
• Rotation -turning of a bicycle wheel or limb
segments around joints
• Axis of rotation
>middle of circle
>center of joint
Def: imaginary line
about which rotation
occurs (pivot, hinge)
origin
Circular Reference Terms
• Circumference (C=2πr)
• Radius (r) – the distance from the axis of rotation to the
perimeter of the circle
• Diameter (D) – the distance between one side of a circle
to the opposite going through the axis of rotation
• Arc – A rotary distance between 2 angular positions
• Arclength – curvilinear distance covered on the perimeter
of an arc
• pi (π) = 3.14159…….
Angular Velocity
• Angular velocity is the rate of change of angular
displacement
change in angular position
ω =
change in time
=
ω
θ f − θi
=
t f − ti
∆θ
∆t
ω = angular velocity in [ rad / s ]
∆θ =
angular displacement in [ rad ]
∆t =
time in [ s ]
Example
A therapist examines the range of motion of an
athlete’s knee joint. At full extension, the angle
between the leg and thigh is 178o. At full
flexion, the angle between the leg and thigh is
82o. The leg is moved between these two
angles in 1.2 s.
What is the angular velocity of the leg?
Example
Figure skater Michelle Kwan
performs a triple twisting jump.
She rotates around her
longitudinal axis three times while
she is in the air. The time it takes
to complete the jump from takeoff
to landing is 0.8 s. What was
Michelle’s average angular
velocity in twisting for this jump?
Angular Acceleration
• Angular acceleration is the rate of change of
angular velocity.
change in angular velocity
α=
change in time
=
α
ω f − ωi
∆ω
=
∆t
t f − ti
α = angular accleration in [rad / s 2 ]
∆ω =
change in angular velocity in [rad / s ]
∆t =time in [ s ]
Example
When Josh begins his discus throwing motion, he
spins with an angular velocity of 5 rad/s. Just
before he releases the discus, Josh’s angular
velocity is 25 rad/s. If the time from the beginning of
the throw to just before release is 1 s, what is
Josh’s average angular acceleration?
Example
•
Randy Johnson is pitching a fastball at a speed of 103 mph. At
0.2 sec into his throw, the angular velocity of the left elbow is
260 °/sec. Two frames later, his elbow is extending at
1310°/sec. If the film speed is 30 frames/sec, what is the
angular acceleration of the elbow joint?
Example
Randy Johnson is pitching a fastball at a speed of
103 mph. At 0.2 sec into his throw, the angular
velocity of the left elbow is 260 °/sec. Two frames
later, his elbow is extending at 1310°/sec. If the film
speed is 30 frames/sec, what is the angular
acceleration of the elbow joint?
ω2
A
ω1
Example
A golf club is swung with an
average angular acceleration of
1.5 rad/s2. What is the angular
velocity of the club when it
strikes the ball at the end of a
0.8 s swing?
Linear and Angular Displacement
s
linear
=
distance radius of rotation × angular displacement
s = r × ∆θ
(∆θ must be in radians )
Example
If the arm segment has length 0.13 m and it
rotates about the elbow an angular
displacement of 0.23 radians, what is the
linear distance traveled by the wrist?
Linear and Angular Velocity
v
linear
velocity radius of rotation × angular velocity
=
v=
r ×ω
(m / s) =
(m) × (rad / s )
Example
Two baseballs are consecutively hit
by a bat. The first ball is hit 25 cm
from the bat’s axis of rotation, and
the second ball is hit 45 cm from the
bat’s axis of rotation. If the angular
velocity of the bat was 35 rad/s at
the instant that both balls were
contacted, what was the linear
velocity of the bat at the two contact
points?
Example
A tennis racket swung with an
angular velocity of 12 rad/s
strikes a motionless ball at a
distance of 0.5 m from the axis
of rotation. What is the linear
velocity of the racket at the point
of contact with the ball?
Linear and Angular Acceleration
a
linear
acceleration radius of rotation × angular acceleration
=
a=
r ×α
(m / s 2 ) =
(m) × (rad / s 2 )
Radial Acceleration
Since linear velocity is a vector its direction will change even if
the angular speed of an object is constant. The acceleration
associated with the changing direction of the velocity vector is
called radial acceleration or centripetal acceleration.
ar
radial
=
acceleration
ar
v2
=
(m / s 2 )
r
(linear velocity ) 2
radius of rotation
(m / s) 2
m
Tangential and Centripetal Acceleration
linear acceleration=tangential acceleration=aT
radial acceleration=centripetal acceleration=aC
Example
An individual is running
around a turn with an 11 m
radius at 3.75 m/s. What is
the runner’s centripetal
acceleration?
Example
A hammer thrower spins with an
angular velocity of 1700o/s. The
distance from her axis of rotation to the
hammer head is 1.2 m.
a) What is the linear velocity of the
hammer head?
b) What is the centripetal acceleration of
the hammer head?
c) If the distance to the hammer head
changes to 1.0 m does the centripetal
acceleration increase or decrease?