SYSTEMS OF FORCES AND EQUILIBRIUM

In sport, more than one external force usually acts on the performer. In such a case, the effect produced by the combination of the forces, or the force system, will depend on their magnitudes and relative directions. Figure 3.3(a) shows the biomechanical system of interest, here the runner, isolated from the surrounding world.

The effects of those surroundings, which for the runner are weight and ground reaction force, are represented on the diagram as force vectors. Such a diagram is known as a free body diagram and should be used whenever carrying out a biomechanical analysis of force systems. The effects of such force systems can be considered in different ways.

Statics is a very useful and mathematically very simple and powerful branch of mechanics. It is used to study force systems in which the forces are in equilibrium, such that they have no resultant effect on the object on which they act (as in Figure 3.2(b)). Here B and G share the same line of action and are equal in magnitude but have opposite directions:

B=-G or

B+G=0 (3.2)

This approach may seem to be somewhat limited in sport, in which the net, or resultant, effect of the forces acting is usually to cause the object to accelerate, as in Figure 3.3(a). Here the resultant force can be obtained by moving F along its line of action (which passes through the centre of mass in this case), giving Figure 3.3(b).

Figure 3.3 Forces on a runner: (a) free body diagram of dynamic force system; (b) resultant force; (c) free body diagram with force not through centre of mass.

Basics of linear kinetics 87 As the resultant force in this case passes through the runner’s centre of mass,

the runner can be represented as a point (the centre of mass) because only changes in linear motion will occur for such a force system. The resultant of F and G will be the net force acting on the runner. By Newton’s second law of linear motion (see below), the net force equals mass times acceleration (m a) or:

F+G=ma (3.3)

More generally, as in Figure 3.3(c), the resultant force will not act through the centre of mass and a moment of force will then tend to cause the object (the runner in this case) to rotate. The magnitude of this moment of force about a point is the product of the force and its moment arm, which is the perpendicular distance of the line of action of the force from that point. Rotation will be considered in detail in Section 3.4.

It is possible to treat dynamic systems of forces, such as those represented in Figure 3.3, using the equations of static equilibrium. This involves the introduction of an imaginary, or inertia, force to the dynamic system, which is equal in magnitude to the resultant force but opposite in direction, to produce a quasi-static force system. This allows the use of the general equations of static equilibrium for forces (F) and moments of force (M):

F=0M=0 (3.4)

That is the sum () of all the forces, including the imaginary inertia forces, is zero and the sum of all the moments of force, including inertia ones, is also zero. The vector equations of static equilibrium (Equation 3.4) can be applied to all force systems that are static or have been made quasi-static through the use of inertia forces. The way in which the vector equations simplify to the scalar equations used to calculate the magnitudes of forces and moments of force will depend on the nature of the system of forces. Such force systems can be classified as follows.

Linear (collinear), planar (coplanar) and spatial force systems

Collinear systems consist of forces with the same line of action, such as the forces in a tug-of-war rope. No moment equilibrium equation (M =0) is relevant for such systems as all the forces act along the same line. Planar force systems have forces acting in one plane only and spatial force systems are three-dimensional.

Concurrent force systems

These are systems in which the lines of action of the forces pass through a common point. The collinear system is a special case. The runner in Figure 3.3(a) is an example of a planar concurrent force system and

spatial ones can also be found in sport and exercise movements. As all forces pass through the centre of mass, no moment equilibrium equation is relevant.

Parallel force systems

These have the lines of action of the forces all parallel and can be planar, as in Figure 3.2(a), or spatial. The tendency of the forces to rotate the object about some point means that the equation of moment equilibrium must be considered. The simple cases of first- and third-class levers in the human musculoskeletal system are examples of planar parallel force systems and are shown in Figure 3.4(a),(b).

The moment equilibrium equation in these examples reduces to the principle of levers. This states that the product of the magnitudes of the (muscle) force and its moment arm (sometimes called the force arm) equals the product of the resistance and its moment arm (the resistance arm), or F_mr_m=F_rr_r. The force equilibrium equation leads to F_j=F_m+ F_r and F_j=F_m-F_r for the joint force (F_j) in the first- and third-class levers of Figures 3.4(a) and 3.4(b) respectively. It is worth mentioning here that the example of a second-class lever often quoted in sports biomechanics textbooks—that of a person rising on to the toes treating the floor as the fulcrum—is contrived. Few, if any, such levers exist in the human musculoskeletal system. This is not surprising as they represent a class of mechanical lever intended to enable a Figure 3.4 Levers as examples of parallel force systems: (a) first-class lever; (b) third-class lever.

Basics of linear kinetics 89 large force to be moved by a small one, as in a wheelbarrow. The human

musculoskeletal system, by contrast, achieves speed and range of movement but requires relatively large muscle forces to accomplish this against resistance.

General force systems

These may be planar or spatial, have none of the above simplifications and are the ones normally found in sports biomechanics, such as when analysing the various soft tissue forces acting on a body segment. These force systems will not be covered further in this chapter.

The vector equations of statics (Equation 3.4) and the use of inertia forces can aid the analysis of the complex force systems that are commonplace in sports biomechanics. However, many biomechanists feel that the use of the equations of static equilibrium obscures the dynamic nature of force in sport, and that it is more revealing to deal with the dynamic equations of motion.

This latter approach will be preferred in this book. In sport, force systems almost always change with time, as in Figure 3.5.

This shows the vertical component of ground reaction force recorded from a force platform during a standing vertical jump. The effect of the force at any instant is reflected in an instantaneous acceleration of the performer’s centre of mass. The change of the force with time determines the way in which the velocity and displacement of the centre of mass change, and it is important to remember this!

Figure 3.5 Vertical component of ground reaction force during a standing vertical jump.