Summing Torques

The state of an object's rotation depends on the balance of torques created by the forces acting on the object. Remember that summing or adding torques acting on an object must take into account the vector nature of torques. All the muscles of a muscle group sum together to create a joint torque in a particular direction. These muscle group torques must also be summed with torques from antagonist muscles, ligaments, and external forces to determine the net torque at a joint. Figure 7.6 illustrates the forces of the anterior deltoid and long head of the biceps in flexing the shoulder in the sagittal plane. If ccw torques are positive, the torques created by these muscles would be positive. The net torque of these two muscles is the sum of their individual torques, or 6.3 N-m (60 • 0.06 + 90 • 0.03 = 6.3 N-m). If the weight of this person's arm multiplied by its moment arm created a gravitational torque of -16 N-m, what is the net torque acting at the shoulder? Assuming there are no other shoulder flexors or exten-

Figure 7.6. The shoulder flexion torques of anterior deltoid and long head of the biceps can be summed to obtain the resultant flexion torque acting to oppose the gravitational torque from the weight of the arm.

sors active to make forces, we can sum the gravitational torque (-16 N-m) and the net muscle torque (6.3 N-m) to find the resultant torque of -9.7 N-m. This means that there is a resultant turning effect acting at the shoulder that is an extension torque, where the shoulder flexors are acting eccentrically to lower the arm. Torques can be summed about any axis, but be sure to multiply the force by the moment arm and then assign the correct sign to represent the direction of rotation before they are summed.

Recall the isometric joint torques reported in Table 7.1. Peak joint torques during vigorous movement calculated from inverse dynamics are often larger than those measured on isokinetic dynamometers (Veloso & Abrantes, 2000). There are several reasons for this phenomenon, including transfer of energy from biarticular muscles, differences in muscle action, and coactiva-tion. Coactivation of antagonist muscles is a good example of summing opposing torques. EMG research has shown that iso-kinetic joint torques underestimate net agonist muscle torque because of coactivation of antagonist muscles (Aagaard, Simonsen, Andersen, Magnusson, Bojsen-Moller, & Dyhre-Poulsen, 2000: Kellis & Baltzopou-los, 1997, 1998).

ANGULAR INERTIA (MOMENT OF INERTIA)

A moment of force or torque is the mechanical effect that creates rotation, but what is the resistance to angular motion? In linear kinetics we learned that mass was the mechanical measure of inertia. In angular kinetics, inertia is measured by the moment of inertia, a term pretty easy to remember because it uses the terms inertia and moment from moment of force. Like the mass (linear inertia), moment of inertia is the resistance to angular acceleration. While an object's mass is constant, the object has an infinite number of moments of inertia! This is because the object can be rotated about an infinite number of axes. We will see that rotating the human body is even more interesting because the links allow the configuration of the body to change along with the axes of rotation.

The symbol for the moment of inertia is I. Subscripts are often used to denote the axis of rotation associated with a moment of inertia. The smallest moment of inertia of an object in a particular plane of motion is about its center of gravity (I0). Biomechani-cal studies also use moments of inertia about the proximal (IP) and distal (ID) ends of body segments. The formula for a rigid-body moment of inertia about an axis (A) is IA = 2mr2. To determine the moment of inertia of a ski in the transverse plane about an anatomically longitudinal axis (Figure 7.7), the ski is cut into eight small masses (m) of know radial distances (r) from the axis. The sum of the product of these masses and the squared radius is the moment of inertia of the ski about that axis. Note that the SI units of moment of inertia are kg-m2.

The formula for moment of inertia shows that an object's resistance to rotation depends more on distribution of mass (r2)

Axis

Activity: Moment of Inertia

Take a long object like a baseball bat, tennis racket, or golf club and hold it in one hand. Slowly swing the object back and forth in a horizontal plane to eliminate gravitational torque from the plane of motion.Try to sense how difficult it is to initiate or reverse the object's rotation. You are trying to subjectively evaluate the moment of inertia of the object. Grab the object in several locations and note how the moment of inertia changes.Add mass to the object (e.g., put a small book in the racket cover) at several locations. Does the moment of inertia of the object seem to be more related to mass or the location of the mass?

than mass (m). This large increase in moment of inertia from changes in location of mass relative to the axis of rotation (because r is squared) is very important in human movement. Modifications in the mo-

Axis

Figure 7.7. The moment of inertia of a ski about a specific axis can be calculated by summing the products of the masses of small elements (m) and the square of the distance from the axis (r).

ment of inertia of body segments can help or hinder movement, and the moment of inertia of implements or tools can dramatically affect their effectiveness.

Most all persons go through adolescence with some short-term clumsiness. Much of this phenomenon is related to motor control problems from large changes in limb moment of inertia. Imagine the balance and motor control problems from a major shift in leg moment of inertia if a young person grows two shoe sizes and 4 inches in a 3-month period. How much larger is the moment of inertia of this teenager's leg about the hip in the sagittal plane if this growth (dimension and mass) was about 8%? Would the increase in the moment of inertia of the leg be 8% or larger? Why?

When we want to rotate our bodies we can skillfully manipulate the moment of in ertia by changing the configuration of our body segments relative to the axis of rotation. Bending the joints of the upper and lower extremities brings segmental masses close to an axis of rotation, dramatically decreasing the limb's moment of inertia. This bending allows for easier angular acceleration and motion. For example, the faster a person runs the greater the knee flexion in the swing limb, which makes the leg easy to rotate and to get into position for another footstrike. Diving and skilled gymnastic tumbling both rely on decreasing the moment of inertia of the human body to allow for more rotations, or increasing the length of the body to slow rotation down. Figure 7.8 shows the dramatic differences in the moment of inertia for a human body in the sagittal plane for different body segment configurations relative to the axis of rotation.

Figure 7.8. The movement of body segments relative to the axis of rotation makes for large variations in the moment of inertia of the body. Typical sagittal plane moments of inertia and axes of rotation for a typical athlete are illustrated for long jump (a,b) and high bar (c) body positions.

Variations in the moment of inertia of external objects or tools are also very important to performance. Imagine you are designing a new unicycle wheel. You design two prototypes with the same mass, but with different distributions of mass. Which wheel design (see Figure 7.9) do you think would help a cyclist maintain balance: wheel A or wheel B? Think about the movement of the wheel when a person balances on a unicycle. Does agility (low inertia) or consistency of rotation (high inertia) benefit the cyclist? If, on the other hand, you are developing an exercise bike that would provide slow and smooth changes in resistance, which wheel would you use? A heavy ski boot and ski dramatically affect the moments of inertia of your legs about the hip joint. Which joint axis do you think is most affected?

The moment of inertia of many sport implements (golf clubs and tennis rackets) is commonly called the "swing weight." A longer implement can have a similar swing

weight to a shorter implement by keeping mass proximal and making sure the added length has low mass. It is important to realize that the three-dimensional nature of sports equipment means that there are moments of inertia about the three principal or dimensional axes of the equipment. Tennis players often add lead tape to their rackets so as to increase shot speed and racket stability. Tape is often added to the perimeter of the frame for stability (by increasing the polar moment of inertia) against off-center impacts in the lateral directions. Weight at the top of the frame would not affect this lateral stability, but would increase the moments of inertia for swinging the racket forward and upward. The large radius of this mass (from his grip to the tip of the racket), however, would make the racket more difficult to swing. Recent baseball/softball bat designs allow for variations in where bat mass is located, making for wide variation in the moment of inertia for a swing. It turns out that an individual

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