Introduction to Sports Biomechanics: Analysing Human Movement ...

INTRODUCTION TO SPORTS BIOMECHANICS
146
LINEAR VELOCITIES AND ACCELERATIONS CAUSED BY ROTATION
In this section, we show how the rotation of a body can cause linear motion. Consider
the quasi-rigid body of a gymnast shown in Figure 4.14, rotating about an axis fixed in
the bar, with angular velocity ω, and angular acceleration, α. The axis of rotation in the
bar is fixed and we will assume that the position of the gymnast’s centre of mass is
fixed relative to that axis. Therefore, the distance of the centre of mass from the axis of
rotation (the magnitude r of the position vector r) does not change.
The linear velocity of the centre of mass of the gymnast is tangential to the circle that
the centre of mass describes; its magnitude is given by v = ω r. Readers who are familiar
with basic vector algebra, or those of you who can follow Appendix 4.2, will appreciate
that the tangential velocity of the centre of mass, from the cross-product rule (Appendix
4.2), is mutually perpendicular to both the position (r) and angular velocity (ω) vectors.
The angular velocity vector has a direction, perpendicularly out of the plane of the page
towards you, determined by the right-hand rule of Figure 4.11.
The linear acceleration of the centre of mass of the gymnast has two components,
which are mutually perpendicular, as follows.
The first component, called the tangential acceleration, has a direction identical to
that of the tangential velocity and a magnitude α r.
The second component is called the centripetal acceleration. It has a magnitude ω 2 r
(= ω v = v 2 /r) and a direction given by the vector cross-product (Appendix 4.2) ω ×
v. This direction is obtained by letting the angular velocity vector (ω) rotate towards
the tangential velocity vector (v) through the right angle between them and using
the vector cross-product rule of Appendix 4.2. This gives a vector (ω × v) perpendicular
to both the angular velocity and tangential velocity vectors, as shown in
Figure 4.14. As the velocity vector is perpendicular to the longitudinal axis of
the gymnast, the direction of the centripetal acceleration vector (ω × v) is along the
longitudinal axis of the gymnast towards the axis of rotation. You should note that
a centripetal acceleration exists for all rotational motion. This is true even if
the angular acceleration is zero, in which case the angular velocity is constant; the
magnitude of the centripetal acceleration is also constant if the distance from the
axis of rotation to the centre of mass is constant.
ROTATION IN THREE-DIMENSIONAL SPACE
For two-dimensional rotations of human body segments, the joint angle may be defined
as the angle between two lines representing the proximal and distal segments. A similar
procedure can be used to specify the relative angle between line representations of
segments in three dimensions if the articulation is a simple hinge joint, but generally the
process is more complex. There are many different ways of defining the orientation
angles of two articulating rigid bodies and of specifying the orientation angles of the