Introduction to Sports Biomechanics: Analysing Human Movement ...

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The ways of obtaining body segment data, including masses and locations of centres of mass, were briefly outlined in Chapter 4. The positions of the segmental end points are obtained from some visual record, usually a video recording of the movement. For the centre of mass to represent the system of segmental masses, the moment of mass (similar to the moment of force) of the centre of mass must be identical to the sum of the moments of body segment masses about any given axis. The calculation can be expressed symbolically as: m r = Σm i r i, or r = Σ(m i /m)r i, where m i is the mass of segment number i, m is the mass of the whole body (the sum of all of the individual segment masses Σm i); (m i/m is the fractional mass ratio of segment number i; and r and r i are the position vectors, respectively, of the centre of mass of the whole body and the centre of mass of segment number i. In practice, the position vectors are specified by their two-dimensional (x, y) or three-dimensional (x, y, z) coordinates. Table 5.1 (see page 221) shows the calculation process for a two-dimensional case with given segmental mass fractions and position of centre of mass data (see also Study task 3). FUNDAMENTALS OF ANGULAR KINETICS Almost all human motion in sport and exercise involves rotation (angular motion), for example the movement of a body segment about its proximal joint. In Chapter 3, we considered the kinematics of angular motion. In this section, our focus will be on the kinetics of such motions. Moments of inertia CAUSES OF MOVEMENT – FORCES AND TORQUES In linear motion, the reluctance of an object to move (its inertia) is expressed by its mass. In angular motion, the reluctance of the object to rotate depends also on the distribution of that mass about the axis of rotation, and is expressed by the moment of inertia. The SI unit for moment of inertia is kilogram-metres 2 (kg m 2 ); moment of inertia is a scalar quantity. An extended (straight) gymnast has a greater moment of inertia than a piked or tucked gymnast and is, therefore, more ‘reluctant’ to rotate. This makes it more difficult to somersault in an extended, or layout, position than in a piked or tucked position. Formally stated, the moment of inertia is the measure of an object’s resistance to accelerated angular motion about an axis. It is equal to the sum of the products of the masses of the object’s elements and the squares of the distances of those elements from the axis of rotation. Moments of inertia can be expressed in terms of the radius of gyration, k, such that the moment of inertia (I) is the product of the mass (m) and the square of the radius of gyration: I = m k 2 . The moments of inertia for rotation about any point on a three-dimensional rigid body are normally expressed about three mutually perpendicular axes of symmetry; the moments of inertia about these axes are called the principal moments of inertia. For the sports performer in the anatomical reference position, the three principal axes of 191
INTRODUCTION TO SPORTS BIOMECHANICS 192 inertia through the centre of mass correspond with the three cardinal axes. The moment of inertia about the vertical axis is much smaller – about one-tenth – of the moment of inertia about the other two axes and that about the frontal axis is slightly less than that about the sagittal axis. It is worth noting that rotations about the intermediate principal axis, here the frontal axis, are unstable, whereas those about the principal axes with the greatest and smallest moments of inertia are stable. This is another factor making layout somersaults difficult, unless the body is realigned to make the moment of inertia about the frontal, or somersault, axis greater than that about the sagittal axis. For the sports performer, movements of the limbs other than in symmetry away from the anatomical reference position result in a misalignment between the body’s cardinal axes and the principal axes of inertia. This has important consequences for the generation of aerial twist in a twisting somersault. BOX 5.3 LAWS OF ANGULAR MOTION The laws of angular motion are analogous to Newton’s three laws of linear motion. Principle of conservation of angular momentum (law of inertia) A rotating body will continue to turn about its axis of rotation with constant angular momentum unless an external torque (moment of force) acts on it. The magnitude of the torque about an axis of rotation is the product of the force and its moment arm, which is the perpendicular distance from the axis of rotation to the line of action of the force. Law of momentum The rate of change with time (d/dt) of angular momentum (L) of a body is proportional to the torque (M) causing it and has the same direction as the torque. This is expressed symbolically by the following equation, which holds true for rotation about any axis fixed in space: M = dL/dt. If we now rearrange this equation by multiplying by dt, and integrate it, we obtain: ∫M dt = ∫dL = ∆L; that is, the impulse of the torque equals the change of angular momentum (∆L). If the torque impulse is zero, this equation reduces to: ∆L = 0 or L = constant, which is a mathematical statement of the first law of angular motion. The last equation in the previous paragraph can be modified by writing L = I ω where I is the moment of inertia and ω is the angular velocity of the body if, and only if, the axis of rotation is fixed in space or is a principal axis of inertia of a rigid or quasi-rigid body. Then, and only then, M = d(I ω)/dt. Furthermore if, and only if, I is constant (for example for an individual rigid or quasi-rigid body segment): M = I α, where α is the angular acceleration. The restrictions on the use of the equations in this paragraph compared with the universality of the equations in the previous paragraph should be carefully noted. The angular motion equations for three-dimensional rotation are far more complex than the ones above and will not be considered in this book.
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Introduction to Sports Biomechanics
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First edition published 1997 This e
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Contents List of figures x List of
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Study tasks 216 Glossary of importa
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FIGURES 1.26 Underarm throw - femal
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FIGURES 5.9 Typical path of a swimm
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Tables 2.1 Examples of slowest sati
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Preface Why have I changed the cove
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Introduction MISSING TEXT The first
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INTRODUCTION principal movements in
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INTRODUCTION TO SPORTS BIOMECHANICS
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Figure 2.2 ‘Principles’ approac
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Figure 2.13 Identifying critical fe
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Figure 6.8 Main skeletal muscles: (
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INDEX 282 balls air flow past 173 b
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INDEX 284 energy 219 kinetic 219 mi
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INDEX 286 isokinetic contraction 24
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INDEX 288 muscle length-tension rel
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INDEX 290 three-dimensional 146-7,
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INDEX 292 validity 220 vantage poin
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