1. Questions to think about • • Why might an athletic trainer or
physical therapist want to measure the range of motion of a joint? •

2. 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. • •

3. Biomechanical Principle
Qualitative
Application of
Biomechanical Principle
– EX.
Soccer kick
Hips (flexion,rotation) --> knee (extension)

4. General Motion Rotation and
Translation

5. Instantaneous Center of Rotation The instantaneous
joint center changes throughout any motion of the knee.

6. Measuring Angles 360o) • 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

7. Angular Kinematic Quantities Units of angular measure π 2π radians
90 degrees
180 degrees
270 degrees
360
degrees π 2 3π 2
radians π
radians
radians
2π radians
1revolution 1 4 1 2 3 4
revolution
revolution
revolution

8. Radians
One radian is the angle at the center of a circle described by an arc equal to the length of the radius. •

9. Converting Angles • θ (deg) = (180 / π ) ×θ (rad ) 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.

10. 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)

11. Absolute Angles Definition of the sagittal
view absolute angles of the trunk, thigh, leg,
and foot.

12. Rear Foot
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.
Angles

13. Standard Reference Terminology >middle of circle•
Movements
– Angular - Circular movements around an axis rotation. Unit of reference --> angle (degrees radians). of or
• Rotation -turning of a bicycle
segments around joints
• Axis of rotation
>middle of circle
>center of joint
wheel or
limb
origin
Def:
imaginary line
about which rotation
occurs (pivot, hinge)

14. Circular Reference Terms Circumference (C=2πr)
Radius (r) – the distance from the axis of perimeter of the circle
Terms •
rotation to the •
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) = ……. • •

15. Angular Velocity ω = change in angular position ω = θ f − θi = ∆θ ti •
displacement
is the rate of change of
angular
ω = change in angular position
ω = θ f − θi = ∆θ
ω = angular velocity in [rad / s]
∆θ = angular displacement in [rad ]
∆t = time in [s]
change in time t − ti ∆t f

16. 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?

17. Angular Acceleration α = change in angular velocity α = ω f − ωi = ∆ω•
Angular acceleration
is the
rate of
change of
angular
velocity.
α = change in angular velocity
α = ω f − ωi = ∆ω
α = angular accleration in [rad / s 2 ]
∆ω = change in angular velocity in [rad / s]
∆t = time in [s]
change in
time
t f
− ti ∆t

18. 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?

19. 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?

20. Linear and Angular Displacement
s = linear distance = radius of rotation × angular displacement
s = r × ∆θ
(∆θ
must be in radians)

21. 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?

22. Linear and Angular Velocity v = linear velocity =
radius of rotation × angular velocity
v = r ×ω
(m / s) = (m) × (rad / s)

23. 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?

24. 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?

25. Linear and Angular Acceleration
a = linear acceleration = radius of
rotation × angular acceleration
a = r ×α
(m / s2 ) = (m) × (rad / s2 )

26. Radial Acceleration ar = radial (m / s 2 ) = (linear velocity)2
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
called radial acceleration or centripetal acceleration. is
(linear velocity)2
r r m
ar = radial
acceleration =
radius of
(m / s)
rotation v
a =
(m / s 2 ) =

27. Tangential and Centripetal Acceleration
linear acceleration=tangential acceleration=aT
radial acceleration=centripetal acceleration=aC

28. 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?

29. Example
A hammer thrower spins with an angular velocity of 1700o/s. The
distance from her axis of rotation to
hammer head is 1.2 m.
What is the linear velocity of the hammer head?
What is the centripetal acceleration the hammer head?
If the distance to the hammer head
the a) b) of c)
changes to 1.0 m does the centripetal
acceleration increase or decrease?