Introduction to Sports Biomechanics: Analysing Human Movement ...

CAUSES OF MOVEMENT – FORCES AND TORQUES
Figure 5.17 Generation of rotation: (a) an eccentric force; (b) addition of two equal and opposite collinear forces acting
through the centre of mass, G; (c) equivalence to a ‘pure’ force and a torque; (d) checking of linear motion.
This, as shown in Figure 5.17(d), is merely a special case of an eccentric force. It is
best considered in that way to avoid misunderstandings that exist in the literature,
such as the misconception that O is the instantaneous centre of rotation. This last
sentence begs the question of where the instantaneous centre of rotation does lie in such
cases.
Consider a rigid body that is simultaneously rotating with angular velocity ω about
its centre of mass while moving linearly, as in Figure 5.18(a). The whole body has the
same linear velocity (v), as in Figure 5.18(b), but the tangential velocity owing to
rotation (v t) depends on the displacement (r) from the centre of mass, G, such that
v t = ω × r, as in Figure 5.18(c). Adding v and v t gives the net linear velocity
(Figure 5.18(d)) that, at some point P (which need not lie within the body) is zero; this
point is called the instantaneous centre of rotation and its position usually changes with
time.
Consider now a similar rigid body acted upon by an impact force, as in Figure
5.18(e), which is similar to Figure 5.17(d). There will, in general, be a point R that
will experience no net acceleration. By Newton’s second law of linear motion, the
magnitude of the acceleration of the centre of mass of the rigid body, mass m, is a g =
F/m. From Newton’s second law of rotation (see above), the magnitude (F c) of the
moment of the force F is equal to the product of the moment of inertia of the body
about its centre of mass (I g) and its angular acceleration (α). That is F c = I g α. This gives
a tangential acceleration (α r) that increases linearly with distance (r) from G. The
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