Pendulums

Type of physical science: Pendulums, Inertia, Momentum, Oscillatory systems, Classical physics

Field of study: Mechanics

A pendulum is a classic example of an oscillatory system with a well-defined periodic motion caused by application of a restoring torque under conditions of mechanical energy conservation.

Overview

A simple pendulum can be formed by taking an inelastic string of a given length, attaching one end of the string to a small point mass, securing the other end to a rigid support, and allowing the string and mass to hang freely beneath the support. This pendulum's motion under displacement from equilibrium is a classic example of what is often referred to as a periodic, or oscillatory, motion in that the motion is repetitive in both its temporal and spatial characteristics. It retraces the same path as it swings and requires the same amount of time, called the period of the motion, to complete each cycle of oscillation. The motion of such a mechanical system is similar to that of a typical linear harmonic oscillator, namely a mass attached to a spring. Although the linear harmonic oscillator repeats a translational motion under conditions of conservation of energy, the simple pendulum repeats a rotational motion about its suspension point, also under conditions of conservation of energy.

To set up the initial conditions for pendulum release--an angular displacement from the vertical or equilibrium position--work must be done against gravity by an external agent. That work provides the pendulum with potential energy. By the work-energy principle, that potential energy is converted into kinetic energy as the pendulum gains angular velocity under influence of an angular acceleration provided by the pendulum's weight component, which is perpendicular to the tension acting along the string. That weight component is not counterbalanced by any other force and acts through a perpendicular distance equal to the pendulum's length, thereby creating a torque that attempts to restore the pendulum to equilibrium. Starting from rest, as the pendulum swings, it decreases its potential energy, providing an equivalent increase in kinetic energy. (Because only potential energy changes are pertinent to determination of work, potential energy can be measured relative to any arbitrarily chosen reference value, in this case the system's equilibrium position, at which the reference value of potential energy is zero.) As the pendulum swings through its equilibrium, its potential energy has vanished, having been conservatively transformed into kinetic energy. Therefore the pendulum has maximum kinetic energy and maximum angular velocity as it passes through that point. As the pendulum continues past equilibrium and climbs uphill, it gains potential energy at the expense of kinetic energy, thereby increasing its angular displacement from the vertical and diminishing its angular velocity until it reaches an angle equal to that of the original angle of release. At this point, all the kinetic energy the pendulum possessed at equilibrium has been conservatively transformed back into potential energy. This constitutes half a cycle of oscillation. The pendulum achieves a maximum angle on the other side of vertical from that of original release with a momentarily vanishing angular velocity, but the restoring torque forces the pendulum to attempt to achieve equilibrium again. The pendulum once again passes through equilibrium with maximum angular velocity directed opposite of what it was when the pendulum first passed through equilibrium, having again conservatively transformed all potential energy to kinetic energy. The swing continues until the pendulum returns to its original configuration, total potential energy at an angular displacement equal to the original angle of release, thereby completing one full oscillation. Because this configuration is identical to the initial condition of the pendulum, the oscillation begins again. As the system is conservative, oscillations continue ad infinitum, each oscillation taking place within the same amount of time--the period of the motion.

The period is observed to be independent of pendulum mass. Also, provided that the initial angle of release is relatively small (less than about fifteen degrees), that period is also observed to be independent of angle of release. What the period does depend on is the value of the acceleration due to gravity and the pendulum's length. If the angle of release is greater than fifteen degrees, the period has a measurable dependence on that initial angle.

This behavior is described by solving the equation of motion, a differential equation that indicates how the dynamics or energetics of pendulum motion varies with time or angular displacement. This behavior can be mathematically determined by generating the differential equation of the motion through two approaches, examining the nature of the restoring torque that seeks to return the pendulum toward equilibrium or looking at the nature of mechanical energy conservation, the conservative transformation from potential to kinetic to potential energy as the pendulum oscillates between maximum angular displacements on either side of equilibrium.

If the pendulum is displaced by an initial angle from the vertical, tension is directed along the line of the string that supports the pendulum weight and secures it to the point about which oscillations occur. The pendulum's weight is directed vertically down rather than totally along the line of the string. Therefore, the weight makes the same angle with the direction of the string as does the string with respect to the vertical. The weight force can be resolved into two parts, one along the line of the string and the other perpendicular to it. The component parallel to the string is totally balanced by tension in the string. Therefore, no radial motion of the pendulum exists along the line of the string. The attached mass always remains at a distance equal to the string's length from the suspension point. However, the weight force component that is perpendicular to the line of the string points back toward the equilibrium position following the arc along which the pendulum swings. That force is the pendulum's weight multiplied by the sine of the angle the string makes with respect to the vertical. That angle will vary as the pendulum swings, ranging from plus to minus the original angle of release. That unbalanced force component acts through a perpendicular distance equal to the length of the string back to the point of suspension. The differential equation of the motion can be obtained by equating this restoring torque, which is the product of the pendulum length and the unbalanced component of its weight, to the product of the pendulum's moment of inertia and its resulting angular acceleration. This is a differential equation in that the torque depends upon the angle the pendulum makes with the vertical, which is a function of time, and the pendulum's acceleration is the time rate of change of the time rate of change of that angle, hence the second derivative of angular position taken with respect to time.

The alternative approach is to determine the pendulum's total mechanical energy, which can be done by measuring the state of the motion at any instant. The initial condition, at which point all energy is potential, is principally convenient, and the value is determined by the pendulum's weight, length, and maximum angular displacement. At any other point than at the maximum angular displacement (where all the total mechanical energy is in the form of potential energy) or equilibrium position (where all the total mechanical energy is in the form of kinetic energy), the energy is partially potential and partially kinetic, with the sum of the kinetic and potential equal to that of the potential energy at initial release. This approach generates a differential equation of motion that can be solved for the time rate of change of the pendulum's angle measured with respect to the vertical; that is, the pendulum's angular velocity, which is the first derivative of the angle taken with respect to time. Once angular velocity is determined, the desired solution--the angular displacement as a function of time--can be found by integration.

In either case, the solution is identical. In both cases, the mathematics is somewhat complicated by the fact that neither differential equation of motion is an ordinary linear differential equation with constant coefficients. The complete solution requires elliptic integrals or a numerical approach to approximate the solution. Nonlinearity in the dynamical equation arises because of the presence of the sine of the angle the pendulum makes with respect to the vertical. If the pendulum is restricted to small oscillations (under fifteen degrees), the exact equation can be approximated by replacing the angle's sine by the angle itself expressed in radians. In this case the equation of motion's form is that of an ordinary linear differential equation with constant coefficients and follows the same nature as the simple harmonic motion displayed by a linear oscillator.

Applications

Before the advent of more sophisticated mechanical and electrical means, oscillatory systems such as the linear harmonic oscillator and simple pendulum were used to accurately measure time. In the real world, internal friction is always present, creating a damping force that produces a torque and results in nonconservative work. The effect of this damping influence is to remove mechanical energy from the system. With each successive swing, although the frequency is not altered, the pendulum's maximum angular displacement diminishes. Over time, the pendulum eventually comes to rest at its equilibrium position when the system's total mechanical energy has vanished. That energy can be replaced on each oscillatory cycle by a mechanical or electrical driving mechanism specifically tuned to supply energy to the oscillating system at the rate at which nonconservative work is being done, thereby maintaining the system's oscillatory motion.

Precise measurements of the oscillations of a pendulum permit accurate determination of the local value of the earth's gravitational acceleration. The acceleration due to gravity is directly proportional to the pendulum's length and inversely proportional to the square of the period of oscillation. Further, because the period of the motion is determined by both the gravitational acceleration and the pendulum's length, it is possible to design a pendulum that requires one second to complete a full oscillation. This device is a convenient means of keeping track of time. By watching this pendulum and counting the number of oscillations, it is possible to determine how much time has elapsed. On Earth, where the acceleration due to gravity has a value of 9.81 meters per second squared at sea level, a one-second pendulum would have a length of 24.85 centimeters. However, on the Moon, where the acceleration due to gravity is only about one-sixth as strong as on Earth, a one-second pendulum would be only 4.14 centimeters long.

Suppose, instead of a pointlike mass, an extended object (one with significant physical extent, possibly even an asymmetrical shape) is suspended from the pendulum string. What has been created is often referred to as a physical, or compound, pendulum. With this device, the distribution of matter within the extended object must be taken into consideration, but conservation of mechanical energy is preserved. A detailed analysis of this motion yields the observation that there are two points relative to the physical pendulum's center of mass for which suspension of the object at those points yields precisely the same period of oscillation. The points are called centers of oscillation.

Another type of pendulum, a torsional pendulum, involves a twisting or torsional oscillation. Its behavior is similar to that of the simple pendulum in that (apart from internal frictional losses) the system oscillates between equal amplitudes of twist measured relative to the equilibrium, or relaxed, position. If a disk-shaped mass is suspended from a wire of arbitrary length and rotated away from its equilibrium condition in a horizontal plane perpendicular to the supporting wire, the twisted wire provides a restoring torque directly proportional to the amount of twist present when the torsional pendulum is released. The differential equation of the motion is identical to that for the simple pendulum under the small angle approximation, and the motion is strikingly similar to the simple linear harmonic motion of a spring-mass system set into oscillation. The constant of proportionality between the twist angle and restoring torque is referred to as the torsion constant. The disk's moment of inertia is determined by its mass and radius. The period and natural frequency of oscillation are determined by the square root of the ratio of the torsion constant to that moment of inertia.

A conical pendulum can be produced using a simple pendulum. Instead of oscillating in a plane containing the equilibrium position, the position of suspension, and the initial position from which the pendulum is released, the pendulum mass executes a circular motion in a plane perpendicular to the axis and the string traces out a cone shape. The pendulum is given an initial push perpendicular to the plane in which the simple pendulum would oscillate if released from rest. Although it is not intuitively obvious, the dynamics of the conical pendulum are exactly the same as the description of vehicles safely negotiating banked roadways at constant speed. In each case the speed around the circle is determined by the radius of the circular motion and the tangent of an angle; the angle the string makes with the vertical in the case of a conical pendulum and the angle of inclination in the case of a banked roadway.

Pendulums can be used to generate chaotic systems. Instead of using a string, attach a spring between the support and the pendulum mass. Because the spring is capable of linear oscillations, the length of the pendulum no longer remains constant. Regardless of how the chaotic pendulum is set into motion, it displays a complex mixture of oscillatory behavior attributable to a simple pendulum and linear expansion and contraction characteristic of the spring.

Another type of chaotic pendulum system can be made by constructing a pendulum that can undergo conical motion. Use a metallic ball for the pendulum mass, and place at least three strong flat magnets on a base just below the circular plane in which the conical pendulum mass will move. Set the pendulum in motion. Because of the magnetic influence, the pendulum will chaotically orbit between the magnets, displaying a much more complicated motion than an ordinary conical pendulum.

Context

In physics, a number of central themes pervade areas that at first thought would appear to have little in common. A variety of problems can be described by strikingly similar differential equations. In addition, many problems display similar driving mechanisms, such as an influence that attempts to restore the system toward equilibrium. The linear harmonic oscillator and pendulum are two such systems. Understanding the behavior of either system enables the investigation of many other phenomena that are governed by equations of similar form or display oscillatory behaviors. Many other phenomena can be described in terms of pendulum-like characteristics.

Characteristics of pendulum motion were known for centuries before the development of Newtonian mechanics. That an ideal pendulum system repeats a motion characterized by energy conservation was understood before development of the calculus that is necessary to solve equations describing a pendulum's angular motion as a function of time. However, the pendulum's unique temporal period, which depends on its length but not its mass, was a property that could be used without being able to fully describe the motion.

Oscillatory motions like those of a pendulum are an important example of energy transfer mechanisms. In the simple pendulum, initial potential energy is transformed to kinetic energy, which reaches its maximum value as the pendulum swings through its equilibrium position, and then that kinetic energy is restored to potential energy as the pendulum achieves maximum angular displacement on the other side of equilibrium. The pendulum repeats the process in the reverse order and returns to its starting position. In the absence of nonconservative influences, this energy transfer mechanism completely conserves the available mechanical energy; it just transforms it from potential to kinetic to potential energy as the oscillations continue. Energy transfer is a central theme in physics and a useful technique in engineering applications.

Complex mechanical systems can be designed that make use of the simple pendulum's capability to oscillate and transfer energy. Two identical length simple pendulums suspended close to each other from a common support can be coupled by attaching a spring to where the pendulum strings attach to the pendulum masses. Each simple pendulum, if left unattached to the other, would oscillate with the same frequency. Connecting the two pendulums provides a more complicated means for energy transfer. If one pendulum is displaced from its equilibrium position and released from rest, initially only the displaced pendulum displays angular motion. However, energy is transferred through the connecting spring to the second pendulum. Both pendulums develop an oscillatory motion. At one point the first pendulum appears to stop, and only the second one undergoes angular motion. The process then repeats itself and returns to the originally observed motion. An analysis of this complicated system's behavior reveals that any motion that develops from the initial release is the superposition of two more simplified motions called normal modes. One normal mode is described by both pendulums moving in the same direction with equal amplitude. The other normal mode is described by both pendulums moving in opposite directions--toward each other--with equal amplitudes. The first normal mode is said to be in phase, the second normal mode is out of phase. Each normal mode is characterized by its own frequency. This type of approach is common for describing complicated wave phenomena and oscillatory systems.

Principal terms

AMPLITUDE: The maximum angular displacement from the vertical or equilibrium position

ANGULAR DISPLACEMENT: The measure of deviation from the vertical or equilibrium position

DAMPING FORCE: A nonconservative influence that removes mechanical energy and diminishes oscillation amplitude, eventually bringing the pendulum to equilibrium

KINETIC ENERGY: An energy of motion that can be conservatively transformed to energy of position or be consumed as work done by nonconservative forces

MOMENT OF INERTIA: The property of a system that resists rotation under application of an external torque

NATURAL FREQUENCY: The number of complete oscillations undergone per unit time; determined by pendulum length and mass

OSCILLATION: A repetitive motion with a well-defined period and amplitude

PERIOD: The time to complete one oscillation; inverse of natural frequency

POTENTIAL ENERGY: The energy of position in a field of force that can be conservatively transformed to energy of motion

RESTORING TORQUE: The application of a force at a distance from an axis about which a system can rotate, directed in such a way as to attempt to restore equilibrium

Bibliography

Feynmann, Richard P., R. B. Leighton, and M. Sands. The Feynmann Lectures on Physics. Vol. 1. Reading, Mass.: Addison-Wesley, 1963. A landmark series of lectures on classical mechanics given by Richard P. Feynmann, the Nobel Prize-winning physicist from the California Institute of Technology. Presents pendulum motion as a system displaying oscillatory motion under conservation of mechanical energy.

Halliday, David, Robert Resnick, and Jearl Walker. Fundamentals of Physics: Extended Version. New York: John Wiley & Sons, 1993. Venerable text for college students. Calculus-based study of general physics. Chapter 14 provides detailed derivations, sample problems, and illustrations of pendulum motion as a classic example of an angular oscillatory system.

Ostdiek, Vern J., and Donald J. Bord. Inquiry Into Physics, 2d ed. St. Paul, Minn.: West, 1991. A thorough survey of physics, not based on calculus. Particularly strong in its description of the pendulum and illustration of how its repetitive motion can be used to control the speed of a clock mechanism.

Symon, Keith R. Mechanics. Reading, Mass.: Addison-Wesley, 1971. This excellent textbook provides a full solution to the pendulum problem, both in the limit of small oscillations and in general. Places the pendulum problem in its proper perspective in the development of Newtonian mechanics, illustrating its importance as a system undergoing oscillatory rotational motion.