## Conservation of Angular Momentum

We have previously seen that, for a system of particles, conservation of moment of momentum dictated that in which Ho and Hc refer to the moment of momentum with respect to fixed point O and center of mass, respectively. The symbols Mo and Mc represent, respectively, the resultant moment of external forces about point O and the center of mass. When a rigid object undergoes planar motion, we have seen that its moment of momentum (angular momentum) can be expressed by the following simple...

## Moment Arms

Typical moment arms (d) of knee and ankle muscle-tendon systems At the knee Hamstring Muscle Group Semitendinosus 40 mm < d < 55 mm Semimembraneous 20 mm < d < 40 mm Biceps femoris d 20 mm Gastrocnemius lateralis 12 mm < d < 20 mm Gastrocnemius medialis 12 mm < d < 20 mm Rectus femoris 20 mm < d < 40 mm At the ankle Achilles tendon 40 mm < d < 52 mm Moment arms depend on the angles between the articulating bones, as discussed in Chapter 5. Consult Spoor et al. (1990) and...

## Internal Forces and the Human Body Complexity of the Musculoskeletal System

External forces that act on a body induce physical stresses within the body. These stresses have wide-ranging consequences. The gravity causes com-pressive stresses in our musculoskeletal system, slowly shortening us as we get older. An accidental fall could induce such high-level stresses in a bone that it could fracture. Even without the application of an external force, an object can be under stress. Consider, for example, a thick rubber tube that is free of loading, except the action of...

## Books

Biomechanics of Human Movement. New York Benchmark Press. Agur, A.M.R. 1991. Grant's Atlas of Anatomy. Baltimore Williams & Wilkins. Alexander, R.M. 1992. The Human Machine. New York Columbia University Press. Alexander, R.M. 1990. Animals. Cambridge Cambridge University Press. Alexander, R.M. 1988. Elastic Mechanisms in Animal Movement. Cambridge Cambridge University Press. Alter, M.J. 1996. Science of Flexibility. Champaign Human Kinetics. Borelli, G.A....

## Introduction

Humans possess a unique physical structure that enables them to stand up against the pull of gravity. Humans and animals utilize contact forces to create movement and motion. The biggest part of the human body is the trunk comprising on the average 43 of total body weight. Head and neck account for 7 and upper limbs 13 of the human body by weight. The thighs, lower legs, and feet constitute the remaining 37 of the total body weight. The frame of the human body is a tree of bones that are linked...

## Statics Tugof War Weight Lifting Trusses Cables Beams

An object that either moves with constant velocity or remains at rest is said to be in a state of static equilibrium. A building is in static equilibrium because its weight is balanced vertically by the upward ground force exerted on it. A ballerina keeps a delicate balance by positioning her center of mass on a vertical line that passes through the tip of her feet in contact with the floor. What are the conditions of static equilibrium How do we use the equations of static equilibrium to...

## Skeletal Muscle

Consider a living relaxed skeletal muscle fiber that is aligned vertically, with one end fixed the other free, as shown in Fig. 6.11. The length Lo of the fiber is measured in this force-free configuration. Then, a weight of W is attached to the free end. The fiber will elongate rapidly in response to the applied load and then will appear to reach a steady-state configuration. The length L of the fiber is measured at that instant. The experiment is continued in this fashion with the addition of...

## Newtons Laws of Motion and Their Applications

We have explored the physical content of the laws of motion in the introduction of this chapter. The mathematical operations presented in the preceding sections now allow us to write these laws in mathematical language. According to Newton's first law, the resultant force acting on a particle must be equal to zero when the particle is at rest or moving with constant velocity in an inertial reference frame in which 2F denotes the sum of all forces acting on the particle. Newton's second law...

## Rolling of an Abdominal Wheel on a Horizontal Plane

In this section, we consider the three-dimensional motion of an abdominal wheel on a flat surface. The solution is obtained by using the definition of angular momentum directly. Later we explain how the problem could be solved by expressing angular momentum in terms of the mass moment of inertia and angular velocity. Example 9.7. Abdominal Wheel. A simple mechanical device called an abdominal wheel was promoted in the 1980s as the miracle tool for strengthening abdominal muscles. It is composed...

## Problems

Consult an anatomy book to identify the primary functions of muscles and other anatomical structures of the shoulder, arm, forearm, thigh, and leg. Problem 1.2. Build a model of a spinal column using a nylon string, beads, rubber bands, and corks. How does the bead shape affect the curvature of the model spine Is it possible to construct a column using beads of spherical shape Note that the rubber bands, if they were to simulate muscles, would have to be prestretched before being...

## SF1 22 2mi ai33

In this equation, SF1 denotes the sum of external forces acting on B (external contact forces plus the gravitational body force). The term 22fij is Figure 3.1. A system of particles (B) moving with respect to reference frame E. Particles in B were marked with rectangles and particles outside of B with filled circles. The position vectors r o, r c, and rc o connect, respectively, the point O to particle i, the center of mass C to point i, and point O to the center of mass C. Also shown in the...

## A L d26dt2 et L d6dt2 er29c

Note that the speed of the particle v is given by the expression Example 2.4. Arm Movements in Aerobics. An aerobic instructor abducts her arm from downward vertical position to horizontal position at shoulder length in 0.6 seconds (s), at constant rate (Fig. 2.7a). Determine the velocity and acceleration of her elbow. Assume that the length of her upper arm is 0.38 m. Solution Since 0.6 s was required to traverse an angle of 2 radians at constant rate Thus, using Eqn. 2.9, velocity and...

## Principle of Impulse and Momentum

The impulse of a force F over a time interval At tf ti is the integral of the force F over the time interval The entity is a vector and has the units of N-s. If the force does not change direction during the time period At, the magnitude of the impulse is equal to the area under the F -time curve (Fig. 7.1). Figure 7.1. The plot of a magnitude of impulsive force F against time. The magnitude of the impulse generated by F is equal to the area under the force-time curve. The impulse acting on an...

## Addition and Subtraction of Vectors

Mechanics of human movement can best be explained with the use of vector notation and calculus. We present next a brief review of vector mathematics. A vector is a quantity that has both a magnitude and direction. Perhaps the vector most commonly known is body weight, which acts always in the direction pointing to the center of earth. Graphically, a vector is shown as a directed segment of a straight line. The length of the segment represents the magnitude of the vector. The direction of the...

## Throwing and Hitting and Falling

Andrews, J.R., Dillman, C.J., and Fleisig, G.S. 1993. Biomechanics of pitching with emphasis upon shoulder kinematics. J. Sports Phys. Ther. 18 402-408. Amis, A.A., Dowson, D., and Wright, V. 1980. Analysis of elbow forces due to high-speed forearm movements. J. Biomech. 13 825-831. Jackson, K.M., Joseph, J., and Wyard, S.J. 1978. A mathematical model of arm swing during human locomotion. J. Biomech. 11 277-289. Karlsson, D., and Peterson, B. 1992. Towards a model for force predictions in the...

## Moment of Momentum of a Tree of Rigid Bodies

In the analysis of some movements, the human body can be considered as a tree of rigid body segments. What is the angular momentum of a tree of rigid objects about the center of mass of the system We can address this question by computing the angular momentum H of each rigid body Bi in the system about the center of mass of the system by using Eqn. 9.13a where mi and Ci represent the mass and the center of mass of body Bi. Summing over all bodies in the multibody system, we obtain Hc mi rCi C X...

## Notation for Human Movement

Spatial positions of various parts of the human body can be described referring to a Cartesian coordinate system that originates at the center of gravity of the human body in the standing configuration (Fig. 1.2). The directions of the coordinate axis indicate the three primary planes of a standing person. The transverse plane is made up of the x1 and x3 axes. It passes through the hip bone and lies at a right angle to the long axis of the body, dividing it into superior and inferior sections....

## Limb Lengthening

Bones of a living person may behave quite differently from bone taken from a corpse and dried. Living bone is self-repairing. Alterations in the distribution of stress in a bone could yield in significant growth or remodeling. In the low-gravity situation of space flight, the compressive stresses acting on the bones are much less than that on earth, and bones lose thickness and strength. On the other hand, on earth, the bones of the leg, which carry the weight of the body, thicken with age....

## Elasticity of Collision Coefficient of Restitution

The examples of collision discussed in the previous sections of this chapter had two common features (a) the impulse of collision occurred in the direction of common normal to the contact area, and (b) after the collision the velocity at the point of contact was the same in both solids. There are many cases in real life where these assumptions do not hold. For example, in running, the impulse of collision has both a tangential and normal component at the surface of contact. Condition b is not...

## Physical Stress

To build a muscle, one must bring it to near exhaustion. What are the levels of force intensity (stress) in the muscle, when the muscle force peaks To answer this question, one must evaluate (a) the force generated by a muscle and (b) the cross-sectional area of the muscle normal to the direction of the muscle force. Average axial force intensity or average normal stress at a cross section is defined as where o-av denotes the average axial stress and F is the force acting on the cross-sectional...

## The Method of Inverse Dynamics

The method of inverse dynamics is increasingly being used to analyze the sequential ordering of body segment movements during an athletic event. The method is also useful to compute the joint moment that is resisted by muscle action in various modes of movement. Inverse dynamics is based on the experimental determination of velocity and acceleration terms that appear in the laws of motion. These laws are then used to evaluate the unknown forces and moments acting on parts of the body. As we...

## Initial Motion

If one of the supports of a body at rest suddenly gives way or is removed, the reactions at the other supports are instantaneously altered and the accelerations of the body are changed. To determine the altered reactions immediately after the removal of the support, we note that all the displacements and velocities resulting from the new accelerations are infinitely small and therefore can be neglected. The sudden removal of a support produces just the same effect as the sudden application of a...

## Moment of Momentum About a Stationary Point

Another example of vector product from the field of classical mechanics is a vector called the moment of momentum. Moment of momentum of a particle i about point O, H , is defined as where ri o is the position vector connecting point O to i, mi is the mass of particle i, and vi is its velocity, as measured with respect to a coordinate system E that is fixed on earth (Fig. 3.9). The moment of momentum of a particle of mass m tracing a circle of radius r with speed v is where O is the center of...

## Angular Impulse and Angular Momentum

The moment of momentum of a body B with respect to a point O that is fixed on earth was defined by the following integral over the mass of the body in which r and v are, respectively, the position and velocity of mass element dm in the inertial reference frame E (Fig. 7.7). We have shown (Chapter 3) in Eqn. 3.39b that in which the superscript c refers to the center of mass. According to the conservation of moment of momentum in which 2Mc and 2Mo denote the resultant external moment about the...

## B S f

The stable equilibrium of the humerus on top of the scapula. The arm, positioned vertically above the head, appears to defy the laws of gravity much like a ball standing on the nose of a seal (a). When a small lateral force is applied to a rod standing on one of its ends on a rough horizontal plane, the rod will begin to rotate and ultimately to lie flat on the plane (b). If, on the other hand, a translational acceleration is imposed on the supporting plane in the direction of...

## Laws of Motion A Historical Perspective

All living as well as nonliving objects that are large enough to be visible through a light microscope obey the laws of motion first formulated in mathematical terms by Sir Isaac Newton in 1687 in his book Philosophica Naturalis Principia Mathematica. The first of the three laws Newton formulated is about the resting state an object (body) remains at rest unless it is compelled to move by a force exerted on it. The first law stems from Galileo's assertion that a moving body thrown on a...

## Summary

The structural frame of the human body is a tree of bones that are linked together by ligaments in joints called articulations. There are 206 bones in the human body. Bone is a facilitator of movement and protector of the soft tissues of the body. Approximately 700 muscles pull on various parts of the skeleton, using the bones as levers to preserve a certain posture or to produce movement. These muscles are connected to the bones through cable-like structures called tendons or to other muscles...

## Examples from Weight Lifting

Weight lifting involves the rotation of a body segment against resistance. Because the exercise is done slowly (about one repetition per second), the inertial effects are neglected. We next present examples from the static analysis of weight lifting. Example 6.4. Deltoids. Deltoids are a shoulder muscle group that is located on the upper side of the arms. Deltoids originate at the bones of the shoulder (clavicle and scapula) and end at the outer midsection of Figure 6.5a,b. A man performing...

## Muscle Groups and Movement

There are layers of muscles in the muscular system. Muscles visible at the body surface are often called externus and superficialis, and they typically serve important functions to stabilize a joint or cause movement. With the naked eye it is often possible to identify the muscle group responsible for a certain action. Major muscle groups of the body are shown in Fig. 1.11. The axial musculature begins and ends on the axial skeleton. Belonging to the group of axial musculature are the muscles...

## Skeletal Tree

The human skeleton is divided into two parts the axial and the appendicular (Fig 1.4). The axial skeleton shapes the longitudinal axis of the human body. It is composed of 22 bones of the skull, 7 bones associated with the skull, 26 bones of the vertebral column, and 24 ribs and 1 sternum comprising the thoracic cage. It is acted on by approximately 420 different skeletal muscles. The axial skeleton transmits the weight of the head and the trunk and the upper limbs to the lower limbs at the hip...

## Physical Properties of Skeletal Muscle

Muscles are composed of bundles of long and thin cells that are called muscle fibers. Bundles of skeletal muscle fibers are encased by a dense fibrous connective tissue layer called the epimysium. Bundles are separated from each other by connective tissue fibers of the perimysium, and within each bundle the muscle fibers are surrounded by a delicate network of reticular fibers called the endomysium. Scattered satellite cells lie between the endomysium and the muscle fibers. These cells function...

## Moment of a Force

An important example of vector multiplication is the concept of moment of a force with respect to a point in space. The moment Mo is a measure of the capacity of force F acting on point P to cause rotation about point O. It is defined as follows where rp o is the position vector connecting point O to P (Fig. 3.6b). The magnitude of Mo is the product of the magnitude of the force and the perpendicular distance of the point O from the line of action of the force. Note that in the evaluation of...

## Human Body Dynamics Swinging Steel Ball While Standing On Turntable

A figure skater spins about her longitudinal axis b2 with constant angular speed of 5 rad s (Fig. P.9.1). She then begins to raise her arms over her head at a rate of 2 rad s. Determine the angular velocity of her arms (A) with respect to the inertial reference frame in E (EwA). Express this angular velocity in the auxiliary coordinate system B shown in the figure. Answer wA 5 (rad s) b2 + 2 (rad s) b3. Problem 9.2. The following equation relates the acceleration of two points in a...

## Applications to Human Body Dynamics Pole Vaulting

Pole vaulting is an exciting athletic event in which a vaulter clears a crossbar resting on two metal standards placed approximately 4 m apart (Fig. 8.10). On the ground in front of the crossbar is a small wedge-cut hole called the vaulting box that holds the end of the pole during the vault. Behind the standards is a landing pit that is at least 5 m wide. Back in 1877, the first championship was won with a vault of 2.92 m, but today vaulters reach the sky with much longer and flexible poles....

## Muscle Force in Motion

The first step in the analysis of forces acting on a body segment is to draw a free-body diagram of the segment. To this end, the part of the body is considered as distinct from the entire body, and all forces acting on the part of the body are identified. Then the equations of motion are used to gather information about the unknown muscle forces acting on the body part. The following example on the kicking of a soccer ball illustrates this technique. Example 6.1. Quadriceps Force Before...

## Joints of the Human Body

Human joints can be classified into three groups based on the range of motion permitted at the joint. An immovable joint is called synarthrosis in anatomy. These are the joints found between the bones of the skull and between teeth and the surrounding bone of the jaw. In the skull, the edges of the bones are interlocked and bound together by dense connective tissue. These joints are called sutures. The second group of joints, such as the distal articulation between tibia and fibula, allow for...

## Biarticular Muscles

Biarticular muscles act on two joints. These muscles include some of the major muscles of the upper and lower limbs. Hamstrings, a group of three muscles, constitute an important example for biarticular muscles. These muscles originate in the ischial tuberosity of the hipbone and insert into the bones of the lower leg. When the thigh and the hip are fixed, ham strings flex the knee. They also extend the hip through the movement of the thigh. The third function of the hamstrings is to raise the...

## The Lever Arm Of The Triceps With Respect To The Center Of Rotation Of Elbow

A man is performing dumbbell kickbacks to strengthen his triceps Fig. P.6.1 . He weighs 55 kg. His forearm and his hand together constitute 2.5 of his body weight. The length of his forearm is 26 cm. The lever arm of the triceps with respect to the center of rotation of elbow is 2.4 cm. The weight he carries is equal to 5 kg. Assuming that the man performs the dumbbell kickbacks slowly, determine the triceps force as a function of angle 6 his forearm makes with the vertical axis....

## Center of Mass and Its Motion

The center of mass of a body B, living or nonliving, is defined by the following equation 2m1 is the total mass in B, rc is the position vector for the center of mass of B, mi is the mass of the ith element in B, and r' is its position vector. These entities are shown in Fig. 3.1. Note that the center of mass is also commonly known as the center of gravity. Let us now perceive center of mass as if it were a particle in space. In reality, the center of mass may not correspond to any point of the...

## Moment Arm and Joint Angle

Moment lever arm of a muscle acting on a joint changes with the angle between the two bones articulating in that joint. Consider, for example, the moment arms of the forearm flexors, biceps, and the brachioradialis muscles shown in Fig. 6.7a. The moment arm of biceps is nearly equal to zero when the angle between the upper arm and the forearm is 180 . The moment arm increases as the joint angle 6 decreases from 180 toward 90 . What is the relation between the moment arm and the joint angle This...

## Internal Forces and the Human Body

Complexity of the Musculoskeletal System 6.1 Introduction 6.2 Muscle Force in Motion 6.3 Examples from Weight Lifting 6.4 Moment Arm and Joint Angle 6.5 Multiple Muscle Involvement in Flexion of the Elbow 164 6.6 Biarticular Muscles 6.7 Physical 6.8 Musculoskeletal Tissues Impulsive Forces and Crash Mechanics 7.1 Introduction 7.2 Principle of Impulse and Momentum 7.3 Angular Impulse and Angular Momentum 200 7.4 Elasticity of Collision Coefficient of Restitution 207 7.5 Initial Motion 7.6...

## Bone Cartilage and Ligaments

Bones are the parts of the human body that are most resistant to deformation. Unless they are broken or fractured, bones do not undergo significant shape changes during short periods. As such, they can be considered as rigid bodies in the analysis of movement and motion. In a rigid body neither the distance between any two points nor the angle between any three points changes during motion. The bone matrix is composed of collagen fibers and inorganic calcium salts decorating these fibers Fig....