Rigid-body Dynamics

Type of physical science: Rigid-Body Dynamics, Solids, Inertia, Classical physics

Field of study: Mechanics

Rigid-body dynamics is the science that deals with the motion of a rigid body under the influence of forces seeking to cause motion. A rigid body is a body with a definite shape that does not change under the influence of a force, so that the particles composing the body stay in fixed positions relative to one another.

Overview

When dealing with an extended body, or one that has size, the assumption is made that, in the simplest cases, it is a point particle and undergoes only translational motion. Real extended bodies, however, may undergo rotational and other types of motion. For example, if a baseball bat is flipped into the air, its motion as it turns is clearly more complicated than that of a nonspinning tossed ball, which moves similarly to a particle. Every part of the bat moves differently, so it cannot be regarded as a particle that is tossed into the air; instead, it is regarded as a system of particles. A closer look reveals that there is one point in the bat that follows a simple parabolic path, just as a particle would if it were tossed into the air. In fact, that point moves as if the bat's total mass were concentrated there and the weight of the bat acted only there. This point is the center of mass of the bat. In general, the center of mass of a body (or a system of bodies) is the point that moves as though all the mass were concentrated there. The center of mass of a baseball bat lies along its central axis. It can be located by balancing the bat horizontally on an outstretched finger: The center of mass is on the bat's axis just above the finger on which the bat is balanced.

The concept of center of mass of a system of particles or a rigid body is important because it is directly related to the net force acting on the system as a whole. That is, the sum of all the forces acting on the system is equal to the total mass of the system times the acceleration of its center of mass. Thus the center of mass of a rigid body of mass M moves in the same way as a single particle of mass M would when acted on by the same net external force. Therefore, the translational motion of any body can be treated as the motion of a particle. This principle simplifies the analysis of the motion of a rigid body. Although the motion of various parts of the system may be complicated, often it is enough to know the motion of the center of mass. This principle can be extended to a rigid body's linear momentum (the product of its mass and velocity). The (linear) momentum of a rigid body equals the product of its total mass and the velocity of the body's center of mass. The rate of change of momentum is the net force acting on the body. In the absence of a force, the linear momentum remains conserved.

A concept similar to center of mass is center of gravity, which is the point upon which the force of gravity is considered to act. Of course, the force of gravity acts on all parts or particles of a body, but in order to determine the motion of a body as a whole, the entire weight of the body is assumed to act on the center of gravity. Strictly speaking, there is a conceptual difference between the center of gravity and the center of mass, but for practical purposes, they are generally the same point. A difference between the two would exist only if a body were so large that the acceleration due to gravity varied among its parts. The center of mass or center of gravity of an extended body can often be determined more easily through experimentation than by analysis, as was the case with the baseball bat, which was balanced on a finger to find its center of mass.

An ordinary object such as a baseball bat contains so many particles that it is best treated as a continuous distribution of matter. The particles then become differential mass elements of that rather solid body. The principal characteristic of a solid body is its rigidity. Normally, its size and shape vary only slightly under stress, compression, pull, push, twist, changes in temperature, and other forces. This naturally gives rise to an idealization of a perfectly rigid body, whose size and shape are permanently fixed. Such a body may be characterized by the requirement that the distance between any two points of the body remains fixed under forces such as those just mentioned. A rigid body is a set of particles, but the constraints between the particles are so numerous and of such a special character that the study of rigid bodies has evolved into a subject of great importance for its applications.

A particle is defined as the simplest mechanical system that can be represented in the mathematical scheme of mechanics by a point. A particle is described when its position in space is given and when the values of certain parameters such as mass and electric charge are given. These parameters must have constant values because they describe the internal constitution of the particle. If these parameters vary with the time, the object is not a simple particle. The position of a particle can, of course, vary with time. The position of a single particle may be specified in space by giving its distances from each of three mutually perpendicular planes. These three numbers are called the "Cartesian coordinates" of the particle. The particles that are considered the "elementary particles" of physics are not particles in the strict sense of the word because they must be described by internal parameters that are not constant over time. The spin of a particle is such a parameter. The spin, representing an intrinsic angular momentum (tendency of rotation), has some, though not all, of the properties of angular momentum of a rigid body about an axis through its center of mass.

A rigid body is a system of at least three particles, not lying in the same straight line, constrained to remain at fixed distances from one another. The number of coordinates needed to describe the configuration of a rigid body is six. The three noncollinear particles of a body have three coordinates each, but because of the constraint that they remain at fixed distances from one another, only six of the coordinates are free. Any other particle of the body also has three coordinates, but its distances from the first three particles are fixed, and so none of the additional coordinates is free. The six coordinates may be thought of as the three coordinates of any point in the body and the three coordinates needed to give the orientation of the body about that point.

The motion of a rigid body can be analyzed as the translational motion of its center of mass plus rotational motion about its center of mass. All points in the rigid body move in circles, and the centers of all these circles lie on a line called the axis of rotation. A rigid body rotating about an axis through its center of mass can be regarded as consisting of many particles located at various distances from the axis of rotation. If the mass of each particle is multiplied by the square of its respective distance from the axis of rotation and all these products are summed, the resulting quantity is designated by the moment of inertia or rotational inertia of the body, denoted by I . The moment of inertia plays the same role for rotational motion that mass does for translational motion. The rotational inertia of an object depends not only on its mass but also on how that mass is distributed with respect to the axis of rotation. When the mass is concentrated farther from the axis of rotation, the rotational inertia is greater. For rotational motion, the mass of a body cannot be considered as concentrated at its center of mass. If all the mass of an object is concentrated at a distance from the axis such that the product of the mass of the object and the square of this distance is the same as the moment of inertia of the object about this axis, this distance is called "radius of gyration," usually denoted by k.

Torque, responsible for a rotational motion of a body about an axis, is equal to the product of the moment of inertia and the angular acceleration. Angular acceleration is the rotational force analogous to linear acceleration. A body that rotates while its center of mass undergoes translational motion will have both translational and rotational kinetic energies. They are respectively half of the product of the mass of the body and the square of the velocity of the center of mass and half of the product of the moment of inertia and the square of the angular velocity. An example of this situation is a wheel rolling down a hill. The angular momentum is the product of the moment of inertia of the body about an axis and the angular velocity of the body about the same axis. The rate of change of the angular momentum is the torque acting on the body and causing the rotation. In the absence of any torque, the angular momentum remains conserved.

The law of conservation of angular momentum is a very important tool in understanding the dynamics of rigid bodies, including incredibly small systems in which the angular momentum is quantized. A physical quantity is said to be quantized if it can have only certain discrete values and all intermediate values are prohibited. There is an important difference between the classical (unquantized) and quantum properties of systems with respect to angular momentum. If a direction in space is defined (for example, by putting the bodies in a magnetic field that points in a fixed direction), there is not necessarily a connection between the angular momentum of a classical object and this direction. The classically spinning objects can have their angular momentum pointing in their original directions. The quantum mechanically spinning objects have their angular momentum oriented only in specific directions determined by the magnetic field and their original spinning directions. The dynamics of a rigid rotator in quantum mechanics is handled in an idealized situation (which does not exactly occur in nature). Some of the simple quantum rigid rotators are systems of two atoms of equal mass separated by a fixed distance (like a dumbbell) rotating as a whole about their midpoint fixed in space. There are many more quantum mechanical rigid systems besides those with a dumbbell-like shape. They can be treated with the coordinate system discussed earlier.

Applications

The principle of rigid-body dynamics is the basis of many familiar and interesting devices in everyday life that involve both translational and rotational motions. For example, a simple rigid body such as a wrench translates and rotates in a constrained manner along a horizontal surface. Its center of mass will always be observed moving in a straight line, which is the direction of the force applied to the wrench. If its motion is considered in two dimensions as the action of two forces directed perpendicular to each other, its center of mass will move in a parabolic curve.

The path of a football projected at an angle into the air or that of a diver from the springboard into the water are also examples of both transitional and rotational motion. Knowing the center of mass of a body when it is in various positions could be of great use. If high jumpers clearing the bar can get into a position in which their center of mass lies outside their body, they can clear a bar that is higher than their actual, normal center of mass. They can achieve such positions by laying their body flat above the bar and parallel to it and first letting their arms and legs hang down, then lifting their arms and legs during the last stage in clearing the bar.

Another example can be found in rocket technology. A rocket is fired into the air to fall on a certain target. When the rocket reaches its highest point at a horizontal distance D from its starting point, it separates into two parts of equal mass. Part one falls vertically to the ground, but part two lands at a distance of 3D from the starting point. Had the rocket not separated, the entire rocket would have fallen at a distance of 2D from the starting point. The path of the center of mass of the system continues to follow the parabolic path that is midway between the two distances. Therefore, while the first part lands at a distance of D, the second part will fall at a distance of 3D in order to land the center of mass at a distance of 2D.

Many interesting phenomena involving rigid-body dynamics can be understood based on how they conserve angular momentum. Figure skaters executing a spin rotate slowly when their arms are outstretched. When the skaters pull their arms in close to them, they suddenly spin much faster. This can be understood from the variation of the moment of inertia in the two positions. Recall that when the skaters pull their arms in, their moment of inertia is reduced. Because the angular momentum remains the same, if the moment of inertia decreases, the angular velocity increases because the angular momentum is the product of moment of inertia and the angular velocity. Similarly, as divers leave the springboard, the first push gives them an initial angular momentum about their center of mass. When they curl themselves into the tuck position, they rotate one or more times faster. They then stretch out again, increasing their moment of inertia and reducing their angular velocity to a smaller value before entering the water. The moment of inertia can reduce by as much as a factor of 3.5 from the upright to tuck position.

The law of conservation of angular momentum of a rigid body is really appreciated when applied to a gyroscope, an instrument that forms the basis of guidance systems for mariners and aircraft pilots. The rapidly spinning wheel is mounted on a set of bearings so that when the mount moves, no net torque acts to change the direction of the angular momentum. Thus the axis of the wheel remains pointed in the same direction in space. The spacecraft Voyager 2, on a 1986 flyby mission to Uranus, was set into unwanted rotation by that wheel effect every time its tape recorder was turned on at high speed. The ground staff had to program the onboard computer to turn on counteracting thrust jets every time the tape recorder was turned on or off.

Context

The origin of the field of rigid-body dynamics is lost in antiquity. The first long-term written records of observations of celestial objects as rigid bodies were made by Mesopotamians around 4000 b.c.e., and Muslim scientists had begun work in this area during the medieval period. However, the modern, detailed understanding of dynamic phenomena began with Sir Isaac Newton in the eighteenth century. His calculus-based approach to solving the equations arising from laws of motion forms the foundation of modern mechanics, which made it possible to understand rigid-body dynamics at the most basic level and to exploit it more comprehensively.

The Newtonian mechanics for rigid bodies were historically developed for nonquantum and nonrelativistic systems, that is, for objects of large size and in ordinary motion (moving significantly slower than the speed of light). Conservation laws for rigid-body motion (and for all systems) hold beyond the limitations of Newtonian mechanics. They hold for bodies whose speed approaches that of light, where the theory of relativity reigns, and they remain true in the world of subatomic physics, where quantum mechanics reigns. No exception to the laws of conservation developed for "classical" rigid-body dynamics has ever been found when applied to those domains. The classical theory of rigid-body dynamics assumes even greater significance when applied to quantum mechanics. The quantum field theory of solids, for example, enables scientists to predict the thermal and mechanical properties of materials.

Stars produce nuclear energy in their cores at a very high rate. When this energy decreases, exhausting the nuclear fuel, the star may eventually begin to collapse, building up pressure in its interior. When this happens to the Sun, the collapse may reduce its radius from 700,000 kilometers during its full bloom to an incredibly small value of a few kilometers. At that stage, the Sun will become a neutron star--its material would be compressed to an incredibly dense gas of neutrons. During this shrinking, its angular momentum would not change. Because its moment of inertia would be greatly reduced, its angular speed would experience a corresponding increase from one revolution per month to as many as 600 to 800 revolutions per second.

The dynamics of a rigid body, in general, are determined by the linear momentum and angular momentum. The motion of the center of mass is determined by the momentum rate change, or the total force acting on the body. To discuss the rotational motion, it is convenient to refer to a set of principal axes. The rotational motion about a fixed axis is determined by the angular momentum rate change about the axis, or the total torque acting about the axis. The relation between the momentum and velocity of the center of mass is described by the mass of the body. The relation between the angular momentum and angular velocity is described by the moment of inertia.

Principal terms

ANGULAR MOMENTUM: The product of a body's moment of inertia and its angular velocity; the rate of change of angular momentum is the torque acting on the body responsible for the rotational motion of the body about an axis

CENTER OF MASS: A point in the body that moves (behaves) as though all the mass of the body were concentrated there and all external forces were applied there

DYNAMICS: The science that deals with the motion of systems of bodies under the influence of forces; dynamics deals with the causes of motion

MOMENT OF INERTIA OR ROTATIONAL INERTIA: The capacity of a body to resist rotational acceleration; plays the same role for rotational motion that mass does for translational motion

MOMENTUM: The product of the mass of a body and its velocity; the rate of change of momentum of a body is equal to the force acting on it

RIGID BODY: An aggregate of material particles in which the interaction of particles is such that the distance between any two particles remains constant with time in an ideal situation; in practice, slight variations in size and shape are accepted

Bibliography

Corben, H. C., and Philip Stelle. Classical Mechanics. New York: John Wiley & Sons, 1950. This book develops the theory of real linear vector spaces to treat the motion of a rigid body, which paves the way for later study of quantum mechanics.

French, A. P., and E. F. Taylor. An Introduction to Quantum Physics. Massachusetts Institute of Technology Introductory Physics Series. New York: W. W. Norton, 1978. This book on quantum physics provides clear explanations, appropriate diagrams, and examples from various disciplines, including atomic physics and astrophysics. Includes an exhaustive list of references.

Giancoli, D. C. Physics: Principle with Application. Englewood Cliffs, N.J.: Prentice-Hall, 1991. A noncalculus, algebra-based introductory textbook. Offers an understanding of the basic concepts of physics, with many applications.

Halliday, David, R. Resnick, and J. Walker. Fundamentals of Physics. New York: John Wiley & Sons, 1997. A standard calculus-based introductory physics textbook that offers a good discussion of basic concepts and their applications.

Kibble, T. W. B. Classical Mechanics. New York: McGraw-Hill, 1966. The second half of this book is devoted to the mechanics of systems of particles and rigid-body dynamics. The crucial role of the conservation laws is particularly stressed. Also discusses the relationship between the symmetry and conservation laws.