1. Basic Biomechanics, (5th edition) by Susan J. Hall, Ph.D.
Chapter 11
Angular Kinematics of Human Movement

2. Measuring Angles What is a relative angle?
angle at a joint formed between the longitudinal axes of adjacent body segments
AKA joint angle
the straight, fully extended position at a joint is regarded as zero degrees

3. Measuring Angles
The relative angle at the lead knee tends to be smaller during sprinting than during distance running.

4. Measuring Angles What is an absolute angle?
angular orientation of a body segment with respect to a fixed line of reference
reference lines are usually vertical or horizontal

5. Measuring Angles
The absolute angle of the trunk with respect to vertical is often a quantity of interest in studies of lifting as related to low back pain.

6. Angular Kinematic Quantities
What is angular displacement?
change in angular position
the directed angular distance from initial to final angular position
the vector equivalent of angular distance
measured in units of degrees, radians, or rotations

7. Angular Kinematic Quantities
What is a radian?
radius
1 radian
The size of the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.

8. Angular Kinematic Quantities
Units of angular measure
90 degrees
radians
revolution 2 14
180 degrees
radians
revolution
270 degrees
radians
revolution
360 degrees
radians
1revolution
3 2  2 12 34

9. Angular Kinematic Quantities
What is angular velocity?
the rate of change in angular position
angular displacement angular velocity = time 
 = t
degrees radians
measured in units of s or s

10. Angular Kinematic Quantities
What is angular acceleration?
the rate of change in angular velocity
 ang.velocity
angular acceleration = time
2 - 1
 = t
deg rad
measured in units of s or s

11. Example:
Identify the angular displacement, the angular velocity, and the angular acceleration of the second hand on a clock over the time interval in which it moves from the number 12 to the number 6. Provide answers in both degree- and radian-based units

12. Solution: The movement is clockwise
so θ = -180° or -π rad or rad
 = θ/t
 = -180°/30s = -6 deg/s or
–π/30 rad/s or rad/s
 = /t = 0 (constant velocity)

13. Relationships Between Linear and Angular Motion
What is the relationship between linear and angular displacement?
The greater the radius between a given point on a rotating body and the axis of rotation, the greater the linear distance traveled by that point during an angular motion.

14. Relationships Between Linear and Angular Motion
The larger the radius of rotation (r), the greater the linear distance (s) traveled by a point on a rotating body.
s = r r1 r2 s2 s1 

15. Relationships Between Linear and Angular Motion
What is the relationship between linear and angular velocity?
Since velocity is displacement over time, linear and angular velocity are related by the same factor that relates displacement: the radius of rotation (r).
v = r

16. Example:
A 1.2 m golf club is swung in a planar motion by a right‑handed golfer with an arm length of 0.76 m. If the initial velocity of the golf ball is 35 m/s, what was the angular velocity of the left shoulder at the point of ball contact? (Assume that the left arm and the club form a straight line and that the initial velocity of the ball is the same as the linear
velocity of the club head at impact.)

17. Solution:
v = r or  = v/r
 = 35 m/s /(1.2 m m)
 = rad/s

18. Relationships Between Linear and Angular Motion
What is the relationship between linear and angular acceleration?
The acceleration of a body in angular motion can be resolved into two perpendicular linear acceleration components.

19. Relationships Between Linear and Angular Motion
What is tangential acceleration?
component of acceleration of angular motion directed along a tangent to the path of motion
represents change in linear speed
v2 - v1
at = t at

20. Relationships Between Linear and Angular Motion
What is radial acceleration?
component of acceleration of angular motion directed toward the center of curvature
represents change in direction v2
ar = r ar

21. Example:
An ice skater increases her speed from 10 m/s to 12.5 m/s over a period of 3 s while coming out of a curve of 20 m radius. What are the magnitudes of her radial, tangential, and total accelerations as she leaves the curve. (Remember that ar and at are the vector components of total acceleration.)

22. Solution: ar = v2/r ar = (12.5 m/s)2 / 20 m ar = 7.81 m/s2
at = (v2 - v1) / t
at = (12.5 m/s - 10 m/s) / 3 s
at = 0.83 m/s2
a2 = ar2 + at2
a = 7.85 m/s2

23. Angular Kinematics of Human Movement
Chapter 11
Angular Kinematics of Human Movement