Daniel Funch Wohns

Article / Updated 09-27-2022

Many people find Physics to be a difficult subject to approach. Well now, you have some tools to help you along the way. This handy list of physics equations organized by topic contains the most common equations you’ll run across:

Article / Updated 03-26-2016

In physics, the potential energy of an object depends on its position. A boulder has more potential energy when it’s at the top of a hill than when it’s rolling down. In your physics class, you may be asked to interpret or draw an energy diagram. An energy diagram shows how the potential energy of an object depends on position and tells you all kinds of things about the motion of the object. Here’s an example energy diagram for the boulder: The potential energy curve shows how much potential energy the boulder has at each position. The boulder has more gravitational potential energy higher up the hill, so the curve also shows the shape of the hill. But what is that horizontal line going straight across the diagram? That’s the total energy of the boulder. If the boulder is sitting precariously at the top of the hill at xtop, its potential energy is equal to its total energy. A slight breeze gives the boulder a nudge, and it starts rolling down the hill. When it reaches the bottom of the hill at xbottom, it has less potential energy. The rest of its energy is kinetic energy, and you can read exactly how much kinetic energy the boulder has from the diagram — the kinetic energy is just the distance between the potential energy curve and total energy line. As you watch the boulder roll up the other hill toward you, you wonder how high the boulder will roll. You know the boulder will stop when its kinetic energy is zero, or when the total energy is equal to the potential energy. You go to the place where the potential energy curve and total energy line cross, and take one more step up the hill. As you turn around, you see the boulder slow down, momentarily stop just in front of you, and roll away back down the hill. Phew! Here’s what you should keep in mind about energy diagrams: The total energy doesn’t change. The height of the potential energy curve is the potential energy of the object, and the distance between the potential energy curve and the total energy line is the kinetic energy of the object. The object will turn around where the total energy line and potential energy curve cross. If you start the object at a different location or with a different initial kinetic energy, the total energy line can shift up or down. The potential energy curve is a property of the object and whatever it’s interacting with. The object can never be at a location where the potential energy curve is above the total energy line. The object feels a force pulling it down the slope toward the location with lower potential energy.

Article / Updated 03-26-2016

The Carnot Cycle is an important concept in physics. No heat engine is 100 percent efficient. The amount of work you get out of a heat engine is always less than the heat that goes into the engine. The second law of thermodynamics says that the most efficient possible heat engine can’t convert all the input heat into work. Real heat engines always have some losses due to friction or interaction with their environment. A theoretical heat engine that has no such losses is known as a Carnot engine — and even a perfect Carnot engine can’t convert all the input heat into work. A Carnot engine consists of a gas, a hot reservoir, and a cold reservoir. The gas repeatedly heats and cools and expands and contracts in a cycle known as the Carnot cycle. The Carnot cycle has four steps: The gas expands isothermally (at constant temperature). The gas absorbs heat from the hot reservoir. As the gas expands, it does work on its surroundings equal to the heat it absorbs. The temperature of the gas is equal to the temperature of the hot reservoir in this stage. The gas cools adiabatically (with no heat flow). The gas continues to expand, doing work on its surroundings, cooling as it transfers internal energy to those surroundings. Adiabatic cooling and heating can be very nearly achieved if the walls of the container holding the gas are well insulated. Once it reaches the temperature of the cold reservoir, the gas begins to expand isothermally. The surroundings do work on the gas, and heat is transferred into the gas. The gas compresses and warms up adiabatically. The surroundings continue to do work on the gas, but no heat is transferred into the gas during this stage. The stage ends when the temperature of the gas reaches the temperature of the hot reservoir again. You can see how the pressure and volume change during the Carnot cycle. The work done in one cycle is the shaded area of the graph. Pressure and volume in the Carnot cycle. The gas is back at the temperature, pressure, and volume it started at, ready for the cycle to start all over again. If you want a heat pump rather than a heat engine, no problem — the ideal Carnot cycle is completely reversible, so you just need to run your heat engine in reverse.

Article / Updated 03-26-2016

When you’re solving a physics problem with a bunch of forces pointing every which way, the easiest way to keep everything straight is to draw a free-body diagram. A free-body diagram is a diagram that depicts the directions and types of forces acting on an object. A Mona Lisa isn’t required — if you want to find the forces acting on a cow standing on the side of a hill, feel free to represent the cow as a square (you can add some spots to your square cow if you’re feeling particularly artistic). The important part of the diagram is to draw all the forces acting on the object. A rather useful convention is to draw these vectors from the center of mass of the object, pointing away from that center. It’s also helpful (but not always possible when starting a problem) to draw vectors with lengths proportional to their magnitudes. A cow standing on a hill has three forces acting on it: the force of gravity, the normal force, and a frictional force. The gravitational force points down, the normal force is perpendicular to the hill, and friction points up the hill. A square cow. A free-body diagram makes it easy to use Newton’s second law, The free-body diagram has all the forces and their directions, which is all the information you need to find the net force. The easiest way to add vectors is to add their components, so you’ll usually want to find the components of one or more of the force vectors in your free-body diagram. Great! You’ve already got a diagram showing which way each of your vectors points. You can draw the components of your forces on your free-body diagram by using dashed lines (or next to it if your diagram is getting crowded). You can also use free-body diagrams to solve torque problems. When you’re solving a torque problem, you also need to keep track of what part of the object the forces acts upon. Torque is force times the lever arm, so if you’re dealing with angular motion, you can use your free-body diagram to figure out all the distances and angles you need to find the torque.

Article / Updated 03-26-2016

Physicists love to go out and measure properties of the world around us. If you are hoping to get a better grasp on physics, these properties are useful information. Here are some of the most useful physical constants: Gravitational constant: Avogadro’s number: Boltzmann’s constant: Gas constant: Stefan-Boltzmann constant:

Article / Updated 03-26-2016

Through the centuries, physics has had thousands of heroes — people who furthered the field in some way or another. In this chapter, you take a look at ten physics heroes who’ve done their bits to make physics what it is today. And just because age has its privileges, these are arranged in chronological order by birth date. Galileo Galilei Galileo Galilei (1564–1642) was an Italian physicist, mathematician, astronomer, and philosopher. He was an important person in the Scientific Revolution — at various times, people called him the father of modern observational astronomy, the father of modern physics, and even the father of science. He was the first to employ a pendulum as a timekeeping device, and he laid the groundwork for both Newton’s three laws of motion and Einstein’s theories of relativity. He also studied the motion of objects undergoing constant acceleration. Galilei is perhaps best known for his improvements to the telescopes and the consequent observations he was able to make. Among his other achievements were the confirmation of the phases of Venus, the discovery of the four largest satellites of Jupiter (now named the Galilean moons), and the observation and analysis of sunspots. Famously, Galilei supported the heliocentric view of the solar system, which says the planets orbit around the sun, not Earth. That was a tough stance to take in 1610, and he got into trouble for it with the Catholic Church, which in 1616 declared it “false and contrary to Scripture.” In 1632, he was tried by the Roman Inquisition, found guilty of heresy, and forced to recant. He spent the rest of his life under house arrest. Modern physicists can be glad that kind of thing doesn’t go on much anymore. Robert Hooke Like many early physicists, Robert Hooke (1635–1703) had his finger in many pies — he was a scientist, architect, investor, and so on. He’s best known for his law of elasticity, Hooke’s law, which says that the restoring force on an object undergoing an elastic pull is proportional to the displacement of the object and a constant, often called the spring constant. Hooke experimented in many different fields, however — in fact, he was the first person to use the term cell to refer to the basic unit of life. Originally very poor, he grew quite wealthy through his investments. He was very active after the Great Fire of London, surveying the ruins in organized maps. He was also a well-known architect, and buildings he designed still survive in England. Sir Isaac Newton Sir Isaac Newton (1643–1726) was an exceptional genius. He was an English physicist, mathematician, astronomer, natural philosopher, and theologian. His accomplishments include the following: Laying the groundwork for most of classical mechanics Formulating the law of universal gravitation Formulating the three laws of motion Building the first practical reflecting telescope Developing a theory of color based on prisms Discovering an empirical law of cooling Studying the speed of sound Sharing the credit with Gottfried Leibniz for the development of differential and integral calculus Demonstrating the generalized binomial theorem, an ancient mathematical problem of the expansion of the sum of two terms into a series Developing Newton’s method for approximating the roots of a function Adding to the study of power series Newton greatly influenced three centuries of physicists. In 2005, the members of Britain’s Royal Society were asked who had the bigger effect on the history of science and made the greater contribution to humankind — Sir Isaac Newton or Albert Einstein. The Royal Society chose Newton. Benjamin Franklin Benjamin Franklin (1706–1790) is familiar to most people as one of the Founding Fathers of the United States. He was an author, printer, political theorist, politician, postmaster, scientist, inventor, statesman, and diplomat. He invented the following: The lightning rod Bifocals The Franklin stove A carriage odometer The glass “armonica” (a popular musical instrument of the day) The first public lending library in America Franklin even created the first fire department in Pennsylvania. He was also a leading newspaperman and printer in Philadelphia (the major city of the colonies at that time). He became wealthy publishing Poor Richard’s Almanack and The Pennsylvania Gazette. He played a large role in the creation of the University of Pennsylvania and was elected the first president of the American Philosophical Society. He became a national hero when he headed the effort to have Parliament repeal the unpopular Stamp Act. As a scientist, Franklin is famous for his work with electricity. The idea that lightning is electricity may seem pretty clear today, but in Franklin’s day, the largest manmade sparks were only an inch or so long. No one knows whether he really performed his most famous experiment — tying a key to a kite string and flying it during a thunderstorm to see whether it could draw sparks from the key, indicating that lightning was electricity. However, Franklin did write about how someone could carry out such an experiment, saying that flying the kite before the storm actually started would be important, or else you’d risk getting electrocuted. Charles-Augustin de Coulomb Charles-Augustin de Coulomb (1736–1806) is best known for developing Coulomb’s law, which defines the electrostatic force of attraction or repulsion between charges. In fact, the MKS unit of charge, the coulomb (C), was named after him. Coulomb originally came to prominence with his long-titled work Recherches théoriques et expérimentales sur la force de torsion et sur l’élasticité des fils de métal (“Theoretical and experimental research on the force of torsion and the elasticity of metal wire”). Throughout his life, Coulomb conducted research in many fields, but his work in electrostatics was what brought him true fame. He showed that electrostatic attraction and repulsion varied inversely as the square of the distance between the charges. There was still a lot of work to be done, though — Coulomb thought electric “fluids” were responsible for the charges. Amedeo Avogadro Amedeo Avogadro (1776–1856) is best known for Avogadro’s number, approximately 6.022 × 1023 — the number of molecules contained within a mole. He started practicing as a lawyer after getting his doctorate. In 1800, he started studying mathematics and physics and became so interested (who wouldn’t be?) that he turned to it as his new career. Avogadro was a pioneer of physics on the microscopic level with Avogadro’s hypothesis, which says that “equal volumes of all gases under the same conditions of temperature contain the same number of molecules.” Unfortunately, acceptance of the hypothesis was slow because of opposition from other scientists and a general confusion between molecules and atoms. Fifty years later in the Karlsruhe Congress, Stanislao Cannizzaro was able to get general agreement on Avogadro’s hypothesis. When Johann Josef Loschmidt calculated Avogadro’s number for the first time in 1865, Loschmidt happily called it Loschmidt’s number. But the general scientific community, in deference to the guy who first suggested that such a number existed, renamed it Avogadro’s number. Nicolas Léonard Sadi Carnot Nicolas Léonard Sadi Carnot (1796–1832) was a French physicist and military engineer. In 1824, he published his work Reflections on the Motive Power of Fire, which gave the theoretical description of heat engines, now called the Carnot cycle. That work laid the theoretical foundations for the second law of thermodynamics. Some people call Carnot the father of thermodynamics because he came up with concepts such as Carnot efficiency, the Carnot theorem, the Carnot heat engine, and others. William Thomson (Lord Kelvin) William Thomson (1824–1907) did important work in analyzing electricity mathematically and formulating the first and second laws of thermodynamics. Like many physicists of his day, he had many interests, starting off as an electric telegraph engineer and inventor, which made him famous — and rich. With enough money to do what he wanted, Thomson turned to physics, naturally. Physicists remember Thomson for developing the absolute zero scale of temperature, which bears his name to this day — the Kelvin scale. Already a knight, he became a nobleman in recognition of his achievements in thermodynamics. He’s also almost as well known for his work on developing a maritime compass as on the laws of thermodynamics. Queen Victoria knighted him as Lord Kelvin for his work on the transatlantic telegraph. Albert Einstein Perhaps the most well-known physicist in the popular mind is Albert Einstein (1879–1955). Einstein, whose name has become synonymous with genius, made many contributions to physics, including the following: The special and general theories of relativity The founding of relativistic cosmology The explanation of the perihelion precession of Mercury, which is the gradual rotation of the axis of the elliptical orbit of the planet The prediction of the deflection of light by gravity (gravitational lensing) The first fluctuation dissipation theorem, which explained the Brownian motion of molecules, which is the random jittery motion of small particles suspended in a fluid, which is caused by collisions with the molecules of the fluid The photon theory Wave-particle duality The quantum theory of atomic motion in solids Einstein was the scientist who, on the eve of World War II, alerted President Franklin D. Roosevelt that Germany could be creating an atomic bomb. As a result of that warning, Roosevelt created the top-secret Manhattan Project, which led to the development of the atomic bomb. In 1921, Einstein won the big one, the Nobel Prize, “for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.” Einstein was affected by that absent-mindedness that scientists who habitually spend all their time thinking about their studies can suffer from. He’s said to have painted his front door red so he could tell which house was his. People joke that he once asked a child, “Little girl, do you know where I live?” And the little girl answered, “Yes, Daddy. I’ll take you home.” Amalie Emmy Noether Amalie Emmy Noether (1882–1935) was a German mathematician and physicist. She was certified to teach English and French but decided to study mathematics. She became interested in physics when the famous mathematicians David Hilbert and Felix Klein asked for her help in understanding Albert Einstein’s theory of general relativity. Noether’s most important contribution to physics was Noether’s theorem, which says that for every (continuous) symmetry, there’s an associated conserved quantity. Noether’s theorem reveals surprising relationships between seemingly unrelated concepts, like time and energy. Because you’ll get the same result if you do your favorite physics experiment today, tomorrow, or 97 years from now, Noether’s theorem says that energy is conserved.

Article / Updated 03-26-2016

In physics, the moment of inertia measures how resistant an object is to changes in its rotational motion about a particular axis. Here are some of the most common moments of inertia: Solid cylinder or disk of radius r rotating about its axis of symmetry: Hollow cylinder of radius r rotating about its axis of symmetry: Solid sphere of radius r rotating about its center: Hollow sphere of radius r rotating about its center: Thin rod of length r rotating about its middle: Thin rod of length r rotating about one end: