Energy & Work

The idea in plain language

Energy comes in many forms: kinetic, gravitational, thermal, elastic, electrical, chemical, electromagnetic, nuclear and more.^[12] It keeps changing from one form into another, but the total amount stays the same.^{[12, 17]}

Work is how a force passes energy on. In physics, work is done on an object when energy is transferred to it, which happens when a force acts on something that moves from one position to another.^[1]

The US Energy Information Administration (EIA) gives a simple definition of energy itself: scientists define energy as the ability to do work.^[16] MIT’s physics course notes instead use the conservation of energy as their working definition of energy.^[12] The OpenStax textbook calls the law of conservation of energy a very good bookkeeping device.^[8]

Work: energy moved by a force

The work done by a force can be positive, negative or zero: positive when the force points generally along the motion, negative when it points generally against it, and zero when it acts at right angles to it.^[1] Work is a scalar: it has a size and a sign, but no direction.^{[12, 1]}

The unit of work is a unit of force times a unit of length, newtons times metres, and this combination is called the joule (J).^[1] The US National Institute of Standards and Technology (NIST), in its guide to SI units, lists the joule as the unit of energy, work and amount of heat.^[15]

The work done by a constant force of gravity on an object depends only on the object’s weight and how far up or down it moves, and work done lifting something is returned when it comes back down.^[1]

One subtle rule: the displacement that counts is that of the point where the force is applied, not that of the body.^[12]

Kinetic energy: the energy of motion

The kinetic energy of a moving object is one-half of its mass times the square of its speed.^[2] Because the speed is squared, kinetic energy rises steeply as things go faster.^[12] In an MIT example, car A speeds up from 10 to 20 mph and car B, of the same mass, from 50 to 60 mph; car B gains much more kinetic energy.^[12] By our own calculation, B’s gain is 11/3 times A’s, about 3.7 times, because 60² − 50² = 1,100 while 20² − 10² = 300.^[12]

Kinetic energy depends only on speed, not on the direction of motion, and it can never be negative.^{[12, 2]} Its value does depend on the frame of reference it is measured from.^[2]

OpenStax works out that an 80 kg athlete running at 10 m/s has 4.0 kJ of kinetic energy.^[2]

The formula ½mv² is the classical one; at speeds comparable to the speed of light, the special theory of relativity requires a different expression.^[2]

The work-energy theorem

The work-energy theorem ties work and motion together: the net work done on an object equals the change in its kinetic energy.^{[3, 12]} Positive net work speeds an object up, negative net work slows it down, and zero work leaves its speed unchanged.^{[3, 12]} Every force acting on the object must be included, and no force that does not act on it, or the answer comes out wrong.^[3] One advantage of the theorem is that it gives the final speed for a frictionless surface of any shape.^[3]

Potential energy: stored energy

Throw a football upward and the kinetic energy it loses becomes gravitational potential energy of the football-Earth system; on the way down, that potential energy is converted back into kinetic energy.^[5]

The change in potential energy between two points is defined as the negative of the work done by the force.^[5] Only these differences have physical meaning; where to put the zero is a choice.^[5] A 75 kg hiker who climbs 147 m from the base of a hill to its summit has about 108 kJ more gravitational potential energy at the top.^[5]

Springs store elastic potential energy, and the energy that launches an arrow is elastic potential energy stored in the bowstring.^{[5, 10]}

Conservative and dissipative forces. A force is conservative if the work it does between two points is the same for any path, or, equivalently, zero around any closed path.^[6] Only conservative forces have a potential energy.^[6] Friction and air resistance are dissipative, non-conservative forces: they take energy away from a system, energy that cannot be got back.^[6]

Conservation of energy

Kinetic energy plus potential energy is called mechanical energy.^{[7, 13]} It stays constant unless forces from outside the system, or non-conservative forces, do work, and then it changes by the work those forces do.^{[7, 13]} In an OpenStax example, a 15 kg panel breaks off a helicopter hovering at 1 km and hits the ground at 45 m/s; most of its starting 147 kJ of potential energy was lost to air resistance.^[7]

The EIA puts the wider law simply: energy changes form, but the total amount of energy in the universe stays the same.^[17] OpenStax states that the total energy of an isolated system always remains constant.^[8]

The forms energy takes. Thermal energy is the internal kinetic energy of the random motion of atoms and molecules, and it is related to temperature.^[8] Fuels such as gasoline and food have chemical energy, a potential energy that comes from their molecular structure.^[8] When surfaces rub, their atoms vibrate, and the energy of those vibrations is converted into heat.^[11]

No conversion is perfect. A car engine converts the chemical energy in gasoline into mechanical energy, but much of the engine’s power goes into waste heat, which is why cars need radiators.^{[17, 4]} Every conversion produces some unusable energy, and the final result of most energy transformations is waste heat.^{[17, 8]}

How sure is it? So far no experiment has contradicted the conservation of energy; whenever measurements seemed to, new forms of energy were discovered or recognised.^[8] OpenStax adds that the law cannot be proven from basic principles, although no exceptions have ever been found.^[8] Others link conservation laws to symmetry: a result of Emmy Noether’s, sometimes called Noether’s theorem, proves a relationship between symmetries in physics and conservation principles.^[24] In a preprint on teaching symmetry, C. T. Hill and L. M. Lederman write that the conservation law corresponding to time-translation symmetry is the law of conservation of energy.^[26]

Power: how fast energy moves

Power is the rate of doing work, or equally the rate at which energy is transferred.^{[4, 12]} It is measured in watts: one watt is one joule per second.^{[4, 15]} The power of everyday devices is also given in horsepower, where 1 hp = 746 W.^[4]

Watts measure power at a given moment, while watt-hours measure how much electricity is used over a period of time.^[18]

The math (optional)

• Work. For a constant force F at an angle θ to a straight displacement d, the work is W = F·d·cos θ; in general, work is the integral of the force along the path.^[1] Its lawn mower example, a 75.0 N push at 35° below the horizontal over 25.0 m of level ground, gives W = (75.0 N)(25.0 m)cos 35.0° = 1.54 × 10³ J.^[1]
• Kinetic energy. K = ½mv², and the work-energy theorem reads W(net) = K(final) − K(initial).^{[2, 3]}
• Potential energy. Near Earth’s surface U = mgy + constant; for a spring of stiffness k stretched or squeezed by x, U = ½kx² + constant.^[5]
• Mechanical energy. E = K + U, and its change equals the work done by non-conservative forces.^[7] By our own calculation, the falling panel lands with ½ × 15 × 45² ≈ 15.2 kJ of kinetic energy, so about 147 − 15.2 ≈ 132 kJ (OpenStax rounds this to 130 kJ) was dissipated by air resistance.^[7]
• Units. 1 J = 1 N·m and 1 W = 1 J/s.^[15] One kilowatt-hour is 3.6 × 10⁶ J, the thermochemical calorie is exactly 4.184 J, and the kilocalorie used in nutrition is 4.184 × 10³ J.^[15] The electronvolt is the kinetic energy an electron gains passing through a potential difference of 1 V in vacuum.^[15]

Common misconceptions

“Energy conservation means saving energy.” In everyday use it means using less energy, a different idea from the physical law, which says energy is neither created nor destroyed.^{[8, 17]}

“Kinetic energy and momentum are the same idea.” Unlike kinetic energy, momentum depends equally on an object’s mass and velocity.^[9] A closed system always conserves momentum; it might also conserve kinetic energy, but very often it does not.^{[10, 14]}

“A newton metre is always a joule.” NIST gives the unit of torque (moment of force) as the newton metre rather than the joule.^[15]

History

In his Principles of Philosophy, Descartes argued that the quantity of motion in the world stays constant, measuring it by a body’s speed times its size.^[21] Leibniz publicly attacked this principle and argued that vis viva (mv²), or living force, was an adequate measure of force; for him it measured a body’s ability to bring about effects through its motion.^[21] Real collisions are never perfectly elastic, which seemed to count against conserving vis viva; Leibniz answered that the apparently lost vis viva had passed to the smaller parts of the bodies.^[21]

MacTutor quotes a summary of Christiaan Huygens’ achievements saying that the ideas of mass, weight, momentum, force and work were finally clarified in his treatment of impact, centripetal force and the compound pendulum.^[22] According to OpenStax, the name kinetic energy was given to energy of motion in the eighteenth century.^[2]

Before James Prescott Joule, most scientists thought heat was a fluid called caloric, according to the Science and Industry Museum.^[25] In his family’s brewery cellar, Joule found that water stirred by a paddlewheel got slightly warmer, and concluded that the work done by a falling weight had been converted into heat through the paddlewheel’s friction.^[25] In 1850 he published On the Mechanical Equivalent of Heat for the Royal Society.^[25] The museum calls Joule, born in Salford in 1818, the first person to prove that heat is a form of energy, and the SI unit of energy is named after him.^{[25, 12]}

In 1847 Hermann von Helmholtz published Über die Erhaltung der Kraft, a paper on the mathematical principles behind the conservation of energy.^[23] MacTutor says he built on earlier work by Sadi Carnot, Clapeyron, Joule and others, and showed that where energy seems to be lost, as in collisions or muscle contraction, it is in fact converted into heat.^[23]

What energy is, and open questions

MIT’s notes add that if all the known energy changes in a process do not add up to zero, either the formulas for energy are wrong or a new kind of energy change has been found.^[12] One open problem is the apparent acceleration of the universe’s expansion, which has been attributed to dark energy, an energy type without a clearly known source.^[12]

Going further

Next on the map are Momentum and Friction; heat engines belong to Heat & Thermodynamics. The free OpenStax and MIT courses listed below have many more worked examples.