## Introduction

During movement, the velocity in a human body is not uniform. For example, during running, the arms and legs rotate relative to the trunk, and therefore the velocity varies from point to point in each of the limbs. What are nature's laws of motion for objects in which the velocity is not uniform? In this chapter, we address this question. Our starting point is that any object in space can be thought of as a collection of small mass elements. Therefore, any object can be idealized as a system of particles. For each particle in the system, Newton's laws of motion hold. Summing over these equations, we arrive at an important principle in mechanics that is called the conservation of linear momentum. The linear momentum was characterized as "the quantity of motion of a body" in Newton's Principia. The linear momentum of a system of particles is the product of the mass of the system and the velocity of its center of mass. The center of mass is defined as the point of support (pivot, fulcrum) at which an object would be in a delicate balance under the action of gravity. In 1589, at the age of 25, Galileo published a treatise on the center of mass of solids. Almost a century later, Borelli devised an experiment to determine the position of center of mass of humans. His book On the Movement of Animals (1680) includes a sketch of a man lying on a seesaw in perfect balance. Borelli correctly predicted that the center of mass of the man to be placed right on top of the fulcrum to achieve static equilibrium.

The concept of center of mass (or center of gravity) has inspired the use of similar concepts in fields other than mechanics. For example, the U.S. Census Bureau assigned Butte County, South Dakota, as the geographic center of the United States. The geographic center is defined as the point at which the surface of a geographic entity would balance if the surface were a plane of uniform weight per unit area. According to the Census Bureau, the population center of the United States is Crawford County, Missouri. This is the point at which a flat, weightless map of the United States would balance if weights of equal value were placed on it with each weight representing the location of one person at a specified date. Unlike the geographic center of the United States, the position of the center of mass of a human body is not stationary but varies with body movement. Aware of this fact, dancers learn to manipulate the movement of the body around the center of mass. Once airborne during a jump, dancers reposition their arms and legs to give the impression they can defy gravity and remain suspended in air.

The conservation of linear momentum yields no information about the rate of rotation of a solid body or the average rate of rotation of a system of particles. How do we obtain an equation that relates the rate of rotation to the external forces acting on a body? To obtain such an equation, we multiply the equations of motion for each particle in the body with an appropriate weight and sum over all particles. Similar procedures are employed in statistics to evaluate the extent of scattering of data. The concepts presented in this chapter are those most fundamental to mechanics. We illustrate the use of the conservation principles with examples involving human movement and motion.

## Getting Started With Dumbbells

The use of dumbbells gives you a much more comprehensive strengthening effect because the workout engages your stabilizer muscles, in addition to the muscle you may be pin-pointing. Without all of the belts and artificial stabilizers of a machine, you also engage your core muscles, which are your body's natural stabilizers.

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