Classical or applied mechanics

Definition:An ancient branch of the physical sciences with roots at the foundations of Western scientific thought, mechanics has applications in a variety of modern scientific fields. Classical mechanics studies the physical laws and forces that govern the movements and interactions of objects; it covers the motion and behavior of all bodies, from celestial to biological and engineered. Applied mechanics is the branch of physics and engineering that studies the practical and engineering applications of principles of classical mechanics.

Basic Principles

English physicist and mathematician Isaac Newton is known as the father of mechanics for his formulation of the laws of motion, first published in his book Philosophiae Mathematica Principia Naturalis (1687). Newton’s laws laid the foundation for the modern scientific understanding of motion using mathematical analysis. Newton also formulated one of the first scientific theories of gravitation, which became the foundation for the scientific understanding of planetary movement and the earth’s orbit around the sun.

Between the seventeenth century and the twentieth century, the principles of mechanics were reformulated by leading mathematicians, creating new approaches to mechanical analysis that are still used. French mathematician Joseph-Louis Lagrange presented the first important reformulation of Newtonian mechanics in the 1780s; fifty years later, Irish mathematician William Hamilton presented a reformulation of Lagrange’s work that helped to connect classical mechanics with the movement of atoms and molecules. In the twentieth century, physicists developed quantum mechanics to study the motion and behavior of fundamental particles; this soon became the other major branch of mechanics, alongside classical mechanics.

In the twenty-first century, classical mechanics continues to provide the basis for the physical understanding of motion. It has been divided into numerous subfields, including celestial mechanics, which studies the motion of celestial bodies; continuum mechanics, which studies the behaviors of continuous materials; and statistical mechanics, which studies the thermodynamic and mechanical principles of individual molecules and atoms and how those principles translate into the observable behaviors of physical bodies.

Core Concepts

Newtonian Mechanics. Newton’s three laws of motion describe the fundamental forces acting upon bodies in motion and define the mathematical relationships between these fundamental forces and observed patterns of movement.

The first law states that an object does not change in its state of either rest or motion unless acted upon by an outside force. Therefore, if an object is at rest, it will remain at rest unless some external force compels it to move; similarly, a moving object will continue moving in the same direction and at the same speed unless its motion is changed by another force. The first law also defines the concept of inertia, which is defined as an object’s tendency to resist changes in momentum or motion.

The second law defines the concept of force, describing it as equal to the change in an object’s momentum of movement. Momentum can be understood as the product of an object’s mass multiplied by its velocity, or speed of movement through space. In essence, the second law says that the acceleration of an object through space will be proportional to the force exerted on the object. A greater force will result in greater acceleration.

The third law is often summarized with the phrase “for every action, there is an equal and opposite reaction.” In terms of mechanical movement, the law states that when two bodies exert force on one another, the reactions of both objects are equal in terms of intensity but opposite in terms of direction. Reaction engines, such as those that power the flight of jet aircraft and rockets, are based on this principle, using the opposing force created by the firing of jet thrusters to propel an object through space.

Measurement Techniques.Research in classical mechanics relies on methods and tools used to measure the variables that contribute to patterns of movement. Physicists and engineers work together to develop measurement methods and equipment used to derive values for these parameters.

Velocity measurements are generally completed using laser or radar measurements, which gauge velocity by measuring the distance traveled by an object over time. Alternatively, high-speed cameras can be used to measure velocity.

Mass measurements can be conducted by using scales to gauge an object’s weight, which can then be compared to the object’s size and volume. The mass of gasses and liquids can be obtained using specialized equipment such as a thermal conductivity detector, which estimates the mass of a sample of gas by measuring thermal conductivity compared to a sample of gas with a known mass.

Force measurements are generally conducted using force gauges, which are machines that can translate the force placed on a target sensor into electric signals that provide a digital readout of the force applied.

Acceleration is measured using an accelerometer, an electrochemical device that works by translating the “squeezing” force applied to the device during acceleration into electric signals that can be used to measure the intensity of the movement. Accelerometers can be used to measure both the vibration of an object and its dynamic or directional acceleration.

Differential Equations.Differential equations are mathematical formulations that relate to the differential of a quantity, that is, how quickly the quantity changes with respect to another quantity. Differential equations begin with a function, which is a mathematical process that can be used to yield a certain value. Solving a differential equation yields a different function that describes the relationship between the original function and one or more derivatives. A partial differential equation is a differential equation in which there is more than one independent variable used in the formulation of the equation.

The theory and mathematical processes behind differential equations are central to the study of physics and mechanics. Newton’s second law of motion, for example, is based on mathematical formulations using differential equations. Differential equations also form the basis for the scientific understanding of fluid dynamics, which is an important topic in mechanics. A variety of naturally occurring processes, including atmospheric airflow, heat transfer between the earth and the atmosphere, and the movement of ocean currents, can be described and analyzed using differential equations.

Gravity and Gravitational Theory.Newton also contributed to mechanics the formulation of the laws of gravitation, which explain how gravity affects bodies in motion. Newton’s law of universal gravitation describes the attraction between two bodies with mass as being both proportional to the product of the masses of the two bodies and inversely proportional to the square of the distance between them. Newton’s theory of universal gravitation was later replaced by Albert Einstein’s theory of general relativity, which says that gravity is a result of a curvature in space-time that results from the mass of an object distorting space and time in the object’s vicinity.

While the theoretical basis for the gravitational effect was altered by the introduction of Einstein’s theory of general relativity, the equations that Newton derived to explain the gravitational attraction of bodies are still used in calculations of physical relationships. Accurate calculations of gravitational relationships are important to a number of scientific fields, including astrophysics and engineering.

Mechanical Work and Thermodynamics. In physics, work is defined as the product of a given force and the distance over which the force takes effect. The concept of work in physics is closely linked to the concept of energy and the equations of thermodynamics, which explain the relationship between matter and energy. Equations governing work in physics also must take into account any counterforces that act against the direction of the initial force. For example, to model the distance a bullet will fly when fired from a gun, physicists need to know the initial force applied to the bullet, the gravitational drag on the bullet as the result of the mass of the earth, and the viscosity of the atmosphere through which the bullet is traveling.

Physicists and engineers use the basic equations governing the physical principle of work to calculate the movement of a variety of objects. Aerospace engineers, for example, calculate variables including force, gravity, torque, and displacement to map the trajectory of aircraft, satellites, and other objects propelled through space. Equations related to work are also used in the biological sciences to calculate and model physical movements, producing equations that relate the use of energy to the amount of physical activity that can be completed in a certain situation.

Continuum Mechanics.Continuum mechanics is a subfield of classical and applied mechanics that deals with the behavior of bodies that are modeled as continuous materials, rather than discrete objects. Among the most important subfields of continuum mechanics are the fields of fluid mechanics and fluid dynamics, which study the behavior of liquids and gases, for example. A variety of specific measurement and analysis techniques have been developed for research into fluid and continuum mechanics, including wind tunnels, which allow researchers to model the behavior of airflow and wind currents, and laser Doppler velocimetry (LDV), which uses the reflection of laser beams to measure the speed of a moving liquid.

Applications Past and Present

Celestial Mapping and Exploration.The modern science of astronomy was built on the understanding of gravity and planetary motion developed through classical mechanics research. Newton used his laws of motion and his concept of universal gravitation to complete some of the earliest mathematical models of planetary motion. Lagrange, one of the other major figures in the development of classical mechanics, also applied mechanical principles to orbital movement.

The orbital path of a planet can be represented by a mathematical formula relating the planet’s mass to the mass of other astronomical bodies, such as stars, moons, and other planets, to calculate the gravitational pull acting on each body. Further mechanics formulations can be used to represent elliptical orbits and other variations in orbital path based on the gravitational interaction of multiple objects. The behavior of stars and planets as a result of gravitational fluctuations can also be used to study objects in distant solar systems. For example, scientists use variations in the movement of distant planets to detect which planets are surrounded by orbiting moons. Similarly, distant stars exhibit variations in movement when they are orbited by planets, and astronomers can use these minor variations in movement to map other solar systems.

In addition, classical mechanics is used extensively in engineering satellites and spacecraft for space exploration. The design of propulsion systems for rockets and shuttles is based directly on classical mechanics models of force, inertia, acceleration, and other factors that influence bodies in motion through the fluid of the atmosphere. Mechanics specialists also help to plot the trajectory of spacecraft to achieve orbit around the earth or other planets. For a spacecraft to achieve orbit around a planet or satellite, it is necessary to calculate precisely the angle of entry and approach that will be needed to compensate for gravitational pull and will allow the spacecraft to maintain orbital distance from a planet.

Sports Engineering. The design of baseball bats, golf balls, hockey pucks, and footballs is based on engineering that utilizes principles of classical mechanics. For example, the path of a golf ball is determined by energy imparted from the club as it impacts the surface of the ball. As the ball makes its way through the air, its trajectory is determined by the angle of impact, the spin of the ball, and a variety of other factors, including wind speed and direction. The depressions on the surface of a golf ball are designed to reduce drag on the ball by altering the reducing turbulence on the leading edge and therefore increasing initial lift. Engineer William Taylor utilized equations from aerodynamics research to invent the dimpled surface design for modern golf balls. Classical dynamics equations can be used to model each aspect of a golf stroke, from the position of the player’s body and angle of swing to the ultimate speed, path, and movement of the ball as it travels through the air.

Another example of applied mechanics in sports engineering is the design of shoes for various types of sporting activities. In recreational and professional running, for example, the soles of the shoes must be designed in such a way that they produce friction in contact with the ground surface, so that runners can push away from the ground to gain momentum. However, if the soles produce excess friction, runners will not be able to alternate steps quickly. Equations derived from mechanics research can be used to model the forces and counterforces involved in the impact of shoes against different types of surfaces. Using this data, engineers can attempt to mold the soles of shoes in such a way as to maximize initial friction while still allowing for rapid alternation of steps.

Hydraulic Engineering.Hydraulics is a field of engineering that utilizes the principles of continuum or fluid dynamics for a variety of applications. Hydraulic engineers design methods of extracting energy from fluids by utilizing pressure and heat to produce potential energy, which can then be harvested to produce electric or mechanical energy. One example is the hydraulic pump, a machine that pressurizes and transports liquid through a system of tubes and reservoirs. The pressurized liquid can be used to complete mechanical work, such as turning a rotor or powering the movement of another object. In some applications, hydraulic pumps can be used to power the generation of electricity by transitioning the mechanical energy in the pressurized liquid into electric energy through a generator.

Another application of hydraulic engineering is the design of hydroelectric dams or wave-power generators. Hydroelectric dams use hydraulic pumps to fill a reservoir with water such that the water has potential energy based on the gravitational forces pulling it to the surface. The gravitational and potential energy of the water in the filled reservoir can then be used to rotate turbines connected to an electric generator. In this way, the rotation of the earth, represented by gravitational forces, can be used to generate electric power. Similarly, wave-power generators use hydraulic pumps and rotary generators to translate the kinetic energy in the movement of waves into electric currents. Wave power has the potential to become a major avenue of future energy development.

Materials Science. Materials science and engineering is the branch of classical mechanics research that studies the mechanical properties of various materials, especially with regard to the effects of motion and stress on material components. Thousands of practical applications have emerged from materials science, including the design and composition of automobile frames and bodies and refinements in the shape and structure of airplane wings. Materials-science specialists may study the way that different types of materials deform when subjected to pressure from airflow, for example. This data can be used to enhance the aerodynamic qualities of various vehicles or projectiles. Alternatively, physicists specializing in materials research may investigate the effects of increasing pressure on different materials, which can be used to refine designs used for submersible objects or pipes used in hydraulic engineering.

The design of “smart” materials provides a modern example of materials-science applications. Smart materials are materials that respond to certain external stimuli by undergoing a change in properties. For example, engineers and physicists have created a variety of “self-healing” materials that automatically correct for minor damage caused by wear and mechanical stress. The design of self-healing materials generally involves utilizing polymers with chemical agents that can re-form chemical and physical bonds broken as a result of stress. While self-healing materials is a relatively new branch of research, there are thousands of potential applications in engineering, biomedical research, and commercial technology. Another example of smart-materials research is the development of thermoelectric technology, or materials that can convert changes in temperature into electric signals. The development of thermoelectric materials has the potential to majorly impact the energy industry, allowing engineers to create more efficient methods of harvesting energy from heat sources, such as solar or geothermal radiation.

Social Context and Future Prospects

Classical and applied mechanics are fundamental to the modern scientific understanding of the world. In modern science, mechanics specialists rarely concentrate solely on classical mechanics, tending instead to specialize in one or more subfields. Because mechanics is applicable to a wide variety of scientific fields and engineering projects, research into basic mechanics still plays an important role in technological development and research.

Modern research in physics often falls into the category referred to as multiphysics, which is the application of methods or models from multiple disciplines to create models of complex phenomena. The theoretical and practical innovations of classical mechanics research are being combined with other forms of scientific inquiry and other models of physics to create innovative research techniques. The future of applied physics lies partially in the modern blending of theoretical models.

A relatively recent development in physics research involves the investigation of nonlinear dynamic systems, sometimes called chaotic systems. Such systems appear random in behavior but can be analyzed using complex models and equations that account for subtle variables in initial conditions and track how changes in these variables lead to overall system-wide behaviors. The principles of classical mechanics are essential to the modern understanding of chaotic-systems behavior and help scientists to devise methods used to model complex systems for enhanced predictability. Examples of chaotic systems found in nature include atmospheric gas flow, population dynamics of organisms, and the activity of the brain.

Bibliography

Finn, John Michael. Classical Mechanics. Sudbury: Jones, 2009. Comprehensive introduction to the study of classical and applied mechanics, suitable for advanced students of physics and mathematics. Contains sections on statistical dynamics and fluid mechanics.

Kibble, Tom W. B., and Frank H. Berkshire. Classical Mechanics. 5thed.River Edge: World Scientific, 2004. Detailed overview of classical mechanics that introduces Newton’s laws of motion, relativity theory, and basic types of motion, with information regarding applications to biology and engineering.

McCall, Martin. Classical Mechanics: From Newton to Einstein; A Modern Introduction. New York: Wiley, 2011. Written for undergraduate students of the physical sciences. Presents the history and basic principles of classical mechanics, including applied mechanics research, plus the basics of relativity theory as it applies to celestial mechanics.

Morin, David. Introduction to Classical Mechanics: With Problems and Solutions. New York: Cambridge UP, 2008. Introductory text to mechanics written for undergraduate students that provides a guide to solving and interpreting differential equations for classical mechanics problems.

"Newton's Laws of Motion." Glenn Research Center, National Aeronautics and Space Administration, www1.grc.nasa.gov/beginners-guide-to-aeronautics/newtons-laws-of-motion/. Accessed 29 Sept. 2023.

"Newtonian Mechanics." Harvard Natural Sciences Lecture Demonstrations, Harvard University, sciencedemonstrations.fas.harvard.edu/newtonian-mechanics. Accessed 29 Sept. 2023.

Phillips, Anthony C. Introduction to Quantum Mechanics New York: Wiley, 2003. Basic introduction to quantum mechanics that discusses the types of problems researched by quantum-mechanics specialists and discusses the theoretical implications of quantum theory.

About the Author

Micah L. Issitt, BS, is a professional freelance writer and journalist specializing in writing on the life sciences and sociology. He has written numerous articles covering environmental sciences and the history of environmental science in the United States.