Work and Force

FIELDS OF STUDY: Classical Mechanics

ABSTRACT: Work is the transformation of energy from one form into another. For instance, a water wheel turns the potential energy of water into the rotational energy of the spinning wheel and ultimately into the kinetic energy of a millstone grinding grain. Force is the way this energy is transferred between objects. The relationship between force, mass, and acceleration is the basis of Isaac Newton’s second law of motion and mechanical engineering from antiquity to the present.

PRINCIPAL TERMS

• centripetal force: for an object moving in a circular path, the force pulling the object toward the center of the circle.
• displacement: the absolute distance and direction between the starting and end points of an object’s motion, which ignores any twists or turn the object’s path may take; it is always equal to or less than the total distance traveled.
• joule: the International System of Units standard unit for energy; one joule (J) is equal to the energy transferred (i.e., work done) when applying a force of one newton across a distance of one meter.
• mechanical advantage: the amplification of force provided by a device or machine such as a lever or pulley.
• newton: the International System of Units standard unit of force; one newton (N) is the force required to accelerate a 1 kilogram object at a rate of 1 meter per second squared (1 kg∙m/s²).
• newton-meter: the International System of Units standard unit for torque; one newton-meter (N·m) is equal to the torque resulting at the axis from the force of 1 newton applied perpendicularly to an attached 1-meter-long moment arm (i.e., a lever).
• power: the rate at which work is done, or at which energy is consumed; the International System of Units standard unit for power is the watt (W), which is equivalent to one joule per second (J/s).
• torque: the measure of how much a force acting on an object causes it to rotate.

Energy, Force, and Work

Energy is a fundamental property of matter. In functional terms, it is a measure of how much work a system (such as a car engine or a human body) can perform. According to the first law of thermodynamics, the total energy of the universe is constant: it can be neither created nor destroyed. Instead, energy is transformed or transferred. When energy is transformed or transferred, work is being done. In physics, work does not need to be useful or desirable to be considered work. When the energy from a heavy wind blows over a tree, the tree is doing work.

Force is the vector by which energy is transferred. It is a push or a pull acting upon an object. When a cue ball hits another billiard ball, for example, it imparts a force to the ball it hits. This transfers some of the energy of the cue ball to the target, and the force causes the target to move. When the force (transferred energy) is sufficient to move an object some distance—to displace it—it is said to be doing work. Specifically, when dealing with forces, displacement, not distance, is the relevant value. Displacement measures the absolute distance relative to an object’s starting point and includes information about the direction it has traveled. Both displacement and force always have a direction attached. Furthermore, if there is no displacement after a force is applied to the object, the force fails to do work. An example of this would be a building, which experiences a constant downward force exerted on it by the earth’s gravity. The earth’s crust is exerting an equal but opposite force upward, so there is no net movement, and gravity is performing no work on the building.

Kinematics

Kinematics is the name for the subfield of classical mechanics concerned with energy, motion, force, and work. It has its origins in the thinking of classical Greek philosophers and mathematicians like Aristotle (ca. 384–ca. 322 BCE). Early modern astronomers such as Tycho Brahe (1546–1601), Galileo Galilei (1564–1642), and Johannes Kepler (1571–1630) contributed to the understanding of kinematics by studying the motion of the heavenly bodies. In the seventeenth century, Isaac Newton (1642–1727) wrote the Principia, which established the formal groundwork for what would be dubbed "classical mechanics" during the twentieth century. Classical mechanics are often referred to as "Newtonian mechanics." Over the course of the twentieth century, classical mechanics was superseded by the theory of general relativity and the new field of quantum mechanics as the most accurate methods of understanding kinematics, but classical mechanics still sees wide use in everyday situations.

Newton’s Laws

Newtown’s three laws of motion are as follows. First, an object in motion tends to stay in motion, and an object at rest tends to stay at rest unless acted upon by an external force. Second, the force (F) necessary to move an object is the product of the mass of the object (m) and the acceleration needed (a):

F = ma

Third, for every action there is an equal and opposite reaction.

At any given moment, a building on earth is experiencing the effects of all three laws: a building appears to be stationary, but as the earth spins on its axis, it imparts the energy of its motion to the building through the foundation that is sunk into the soil. Why does a building not simply fly off the surface after receiving the force from the earth’s spin, the same way a tennis ball on a string goes flying if spun and released? Gravity is a centripetal force pulling the building’s center of mass toward the center of the earth, bending its path of movement so that it does not move in a straight line but bends in a circle conforming to the rotation of the planet. This is the first law in action.

Technically, everything on the surface of the planet is constantly falling in an arc toward the center— accelerated by the force of gravity. According to the second law, this acceleration coupled with the mass of a very large object such as a skyscraper raises the question What keeps the object from falling through the earth’s crust toward its center?

Here the third law comes into play. For all the force generated by the falling building, the earth’s crust counterbalances the mass with an equal and opposite force.

Newton’s laws are useful not only in the abstract. They are excellent for making predictions about the world under everyday circumstances.

Sample Problem

In the Japanese sport of Sumo, two wrestlers compete to push each other out of a circular ring. Whoever pushes the other out first, wins. A standard sumo ring is 4.55 meters (14.9 feet) in diameter. The announcer declares the weight of each contestant: the bigger man weighs 160 kilograms (about 353 pounds), and the smaller man weighs 140 kilograms (about 309 pounds). Both men begin in the center of the ring.

The match begins. After thirty seconds of struggle, the bigger man steps out of the ring. A visual estimation puts his speed as he stumbles back across the line at a bit less than walking speed, about 1 meter per second (2.24 miles per hour). How much force, in newtons, did the smaller man use to push the bigger man from the ring?

Solve using Newton’s second law. Recall that acceleration is the change in velocity (Δv) over time (t):

a = Δv / t = (vend - vstart) / t

Answer:

The mass of the bigger man is provided, but his acceleration must be calculated. Logically, his starting velocity is zero. His ending velocity is an estimated 1 meter per second away from the center of the ring. Plug these values into the formula for acceleration and solve:

a = Δv / t

= (vend - vstart) / t

= (1 m/s – 0 m/s) / 30 s

= (1 m/s) / 30 s

≈ 0.033 m/s²

Next, substitute the acceleration back into Newton’s second law:

F = ma

= 160 kg × 0.033 m/s²

≈ 5.33 kg∙m/s² = 5.33 N

Note that in this instance, the mass of the smaller man and the distance traveled are irrelevant; what matters are the speed and direction in which the mass was moved, not what did the pushing or how far the mass was pushed.

Measuring Energy Consumption

Work and the energy used to perform it are valuable resources. Much of the work that takes place around us is fueled by electrical energy provided by utilities. Power is a measurement of work done or energy used over time, and power is what is most often measured when utilities keep track of energy consumption for billing purposes. Most homes, for instance, have an external device that tracks the home’s use of power in kilowatt-hours. A single kilowatt of power is equal to one thousand joules of energy consumed per second, and a kilowatt-hour measures that consumption in hour-long intervals. Power is also useful as a measure of the rate at which a system can perform work, so car engines and household machines often provide horsepower ratings. A horsepower is a nonstandard measurement originally used to compare the amount of power produced by an engine compared to the power a horse used in pulling.

Energy, Work, and Force in Everyday Life

All powered devices operate on the principles of force and work. Automobiles make use of the explosive force of gasoline combustion to move pistons in cylinders, each of which is fired in a cycle timed to turn a crankshaft that eventually is translated into the rotational work of the driveshaft. This rotational work is known as torque. Torque is important in almost every modern engine and is measured in newton-meters (N·m), which measure the amount distance of rotational movement relative to the force applied to a lever-like machine to produce rotation. The way the firing of the pistons becomes an amplified rotational force is functionally same as the way the steering wheel moves the steering rod, turning a relatively small force into a larger one through mechanical advantage. This advantage is the same advantage one gets by using a lever. Imagine a cylinder with a long rod sticking out at a right angle. Applying a force to the outer part of the attached rod would rotate the central cylinder much more than applying the same force to the rod but closer to the cylinder. Cars use the mechanical advantage offered by levers and wheels to amplify the force generated by exploding gasoline, creating a lot of torque with fairly little energy.

The Future of Kinematics

Twenty-first-century kinematics based on general relativity has transcended the classical mechanics used here in terms of precision and accuracy. Yet even astrophysicists and aeronautics engineers, who must fully map out and understand all of the forces acting on an object, use classical kinematics to estimate and validate their work. The simplicity of Newton’s equations and their accuracy in day-to-day situations means they will see continued use for years to come.

Bibliography

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Henderson, Tom. "Work, Energy, and Power." Physics Classroom. Physics Classroom, 2015. Web. 3 June 2015.

"Kinematics." Encyclopaedia Britannica. Encyclopaedia Britannica, 5 June 2013. Web. 3 June 2015.

Oxford Dictionary of Physics. 7th ed. Oxford: Oxford UP, 2015. Print.

Reuleaux, Franz. Kinematics of Machinery: Outlines of a Theory of Machines. North Chelmsford: Courier, 2012. Print.

Simanek, Donald. A Brief Course in Classical Mechanics. Lock-Haven U, Feb. 2005. Web. 28 Apr. 2015.