Science Articles

Cheat Sheet / Updated 05-08-2023

Why is Neuroscience important? The most complex structure in the world is the 3-pound mass of cells within your skull called the brain. The brain consists of about 100 billion neurons, which is about the same number as all the stars in our Milky Way galaxy and the number of galaxies in the known universe. It also contains about a trillion glial cells, which contribute to the proper function of neurons. Like any complex machine, the brain contains a lot of parts, each of which has subparts, which themselves have subparts, all the way down to the “nuts and bolts” — the neurons and glia. In this Cheat Sheet, you find information on the key parts of the brain and the role and function of the cells that make up the nervous system.

Article / Updated 05-04-2023

Recombinant DNA technology can be controversial. People, including scientists, worry about the ethical, legal, and environmental consequences of altering the DNA code of organisms: Genetically modified organisms (GMOs) that contain genes from a different organism are currently used in agriculture, but some people are concerned about the following potential impacts on wild organisms and on small farms: Genetically modified plants may interbreed with wild species, transferring genes for pesticide resistance to weeds. Crop plants that are engineered to make toxins intended to kill agricultural pests can also impact populations of other insects. Small farmers may not be able to afford genetically modified crop plants, putting them at a disadvantage to larger corporate farms. Genetic testing of fetuses allows the early detection of genetic disease, but some people worry that genetic testing will be taken to extremes, leading to a society where only “perfect” people are allowed to survive. Genetic testing of adults allows people to learn whether they have inherited diseases that run in their family, but some people worry that one day insurance companies will use genetic profiles of people to make decisions about who to insure. Parents of children with life-threatening diseases that can be treated with bone marrow transplants are using genetic testing to conceive children that can provide stem cells for their sick siblings. The umbilical cord is an excellent source of these stem cells, so the new babies aren’t harmed, but people worry that this may lead to an extreme future scenario where babies are born to serve as bone marrow or organ donors for existing people. Human hormones like insulin and human growth hormone are produced by bacteria through recombinant DNA technology and used to treat diseases like diabetes and pituitary dwarfism. However, some people seek hormones like human growth hormone for cosmetic reasons (for example, so that their children can be a little taller). People question whether it’s ethical for parents to make these choices for their children and whether too much emphasis is being placed on certain physical traits in society. Making useful proteins through genetic engineering Scientists use the bacterium E. coli as a little cellular factory to produce human proteins for treatment of diseases. To get E. coli to produce human proteins, cDNA copies of human genes are put into plasmid vectors and then the vectors are introduced into E. coli. The bacterium transcribes and translates the human gene, producing a human protein that is identical to the protein made by healthy human cells. Several human proteins are currently produced by this method, including the following: Human insulin for treatment of diabetes Human growth hormone for treatment of pituitary dwarfism Tumor necrosis factor, taxol, and interleukin-2 for treatment of cancer Epidermal growth factor for treatment of burns and ulcers Searching for disease genes Some people carry the potential for future disease in their genes. Genetic screening allows people to discover whether they’re carrying recessive alleles for genetic diseases, allowing them to choose whether or not to have children. Also, diseases that show up later in life, such as Alzheimer’s and Huntington’s disease, can be detected early, to seek the earliest possible treatment. In order to screen for a particular genetic disease, scientists must first discover the gene that causes the disease and study the normal and disease-causing sequences. Scientists have identified the genes for several genetic diseases, including cystic fibrosis, sickle-cell anemia, Huntington’s disease, an inherited form of Alzheimer’s, and an inherited form of breast cancer. Once the gene for a genetic disease has been identified, doctors can screen people to determine whether they have normal or disease-causing alleles. In order to screen a person for a particular gene, scientists amplify the genes linked to the disease using PCR. Then, scientists screen the genes for disease alleles: Scientists can copy and sequence a specific gene. If you have risk for a genetic disease, perhaps because people in your family suffer from the disease, scientists can use PCR to make amplify your copies of the gene associated with that disease. They use DNA sequencing to read the code of your genes, then compare your code to known codes for normal and disease-causing alleles of the gene. You might find out that you don’t have any disease-causing alleles, or that you’re a carrier who has one disease and one normal allele, or that you have two copies of the disease-causing form. Scientists can sequence your genome. If a specific gene isn’t identified as causing a problem, a doctor may order genome sequencing. A sample of all of your DNA will be cut into pieces, then sequenced using next-generation sequencing methods. The code from your DNA will be compared to reference human genomes to look for variations in your code that might be associated with disease. Building a “better” plant with genetic engineering Many important crop plants contain recombinant genes. These transgenic plants, which are a type of genetically modified organism (GMO), provide labor-saving advantages to farmers who can afford them: Transgenic plants that contain genes for herbicide resistance require less physical weed control. Farmers can spray crop plants that are resistant to a particular herbicide with that herbicide to control weeds. Weed plants will be killed, but the modified crop plants will not. Transgenic plants that contain genes for insect toxins will be less damaged by grazing insects. The crop plants use the introduced gene to produce insect toxins that kill insects that graze on the plants. Scientists often use the bacterium Agrobacterium tumefaciens to modify plant genomes. In nature, this soil bacterium slips a piece of its DNA into plant cells, resulting in crown gall disease. Scientists studying this disease discovered that Agrobacterium tumefaciens contains a small circle of DNA they named the Ti plasmid (Ti for tumor-inducing), which contains the genes necessary for the bacterium to transfer a section of its DNA into plant cells. When this bacterium receives the right signals, it takes a piece of DNA from the Ti plasmid and sends it into plant cells where it integrates into the plant genome. In the case of crown gall disease, the bacterial DNA causes production of plant hormones that produce a tumor-like growth (see the following figure). In the case of genetic engineering, scientists replace the disease-causing genes with the genes they want to introduce into the plant. Another potential benefit of transgenic plants is that certain crop plants may be altered to become more nutritious. For example, scientists are currently working on developing a strain of golden rice that may help combat Vitamin A deficiency in people around the world. Vitamin A deficiency can cause blindness and increase susceptibility to infectious diseases. Golden rice is being engineered to contain the genes necessary for the rice plants to produce beta-carotene. When people eat golden rice, their bodies will use beta-carotene to make Vitamin A. Rice is a staple food for half of the world’s people, so golden rice has great potential for fighting Vitamin A deficiency! Fixing a broken gene with gene therapy The ultimate cure for a genetic disease would be if scientists could replace the defective genes. As soon as recombinant DNA technology became available, scientists started wondering if they could use this technology to create cures for genetic diseases. After all, if scientists can transfer genes successfully into bacteria and plants, perhaps they can also transfer them into people that have defective disease-causing alleles (see the following figure). By introducing a copy of the normal allele into affected cells, the cells could be made to function normally, eliminating the effects of the disease. The introduction of a gene in order to cure a genetic disease is called gene therapy. Gene therapy for humans is being studied, and clinical trials have occurred for some diseases, but this type of treatment is far from being perfected. Many barriers to successful human gene therapy still need to be overcome: Scientists must discover safe vectors that can transfer genes into human cells. One possible vector is viruses that naturally attack human cells and introduce their DNA. Viral DNA is removed and replaced with therapeutic genes that contain the normal allele sequence. The viruses are allowed to infect human cells, thus introducing the therapeutic genes. Following are several safety issues associated with the use of viruses as vectors in gene therapy: Viruses that have been altered may recombine with existing viruses to recreate a disease-causing strain. Viruses that have been altered so that they can’t directly cause disease may still cause a severe allergic reaction that is potentially life threatening. Viruses that introduce genes into human cells may interrupt the function of normal genes. Scientists must develop methods for introducing therapeutic genes into populations of target cells. Humans are multicellular and have complex tissues. Genetic diseases can affect entire populations of cells. If gene therapy is to cure these diseases, the therapeutic genes must be introduced into all of the affected cells. Stem cells that produce target populations of cells need to be identified. If therapeutic genes are introduced into cells that have a limited lifespan in the body, then gene therapy will need to be repeated at regular intervals to maintain populations of healthy cells. On the other hand, if stem cells could be repaired with normal alleles, then they would continuously produce new populations of healthy cells, and the cure would be permanent. Because of the challenges of successfully treating people with genes delivered with vectors, many scientists are turning their attention to the newer technology of genome editing.

Article / Updated 05-03-2023

In physics, you can examine how much potential and kinetic energy is stored in a spring when you compress or stretch it. The work you do compressing or stretching the spring must go into the energy stored in the spring. That energy is called elastic potential energy and is equal to the force, F, times the distance, s: W = Fs As you stretch or compress a spring, the force varies, but it varies in a linear way (because in Hooke’s law, force is proportional to the displacement). The distance (or displacement), s, is just the difference in position, xf – xi, and the average force is (1/2)(Ff + Fi). Therefore, you can rewrite the equation as follows: Hooke’s law says that F = –kx. Therefore, you can substitute –kxf and –kxi for Ff and Fi: Distributing and simplifying the equation gives you the equation for work in terms of the spring constant and position: The work done on the spring changes the potential energy stored in the spring. Here’s how you give that potential energy, or the elastic potential energy: For example, suppose a spring is elastic and has a spring constant, k, of and you compress the spring by 10.0 centimeters. You store the following amount of energy in it: You can also note that when you let the spring go with a mass on the end of it, the mechanical energy (the sum of potential and kinetic energy) is conserved: PE1 + KE1 = PE2 + KE2 When you compress the spring 10.0 centimeters, you know that you have of energy stored up. When the moving mass reaches the equilibrium point and no force from the spring is acting on the mass, you have maximum velocity and therefore maximum kinetic energy — at that point, the kinetic energy is by the conservation of mechanical energy.

Article / Updated 05-03-2023

Acid-base reactions and their associated calculations play a primary role in many chemical, biological, and environmental systems. Whether you’re determining hydrogen ion concentration, [H+]; hydroxide ion concentration, [OH˗]; pH; or pOH, an equation and a calculator are important tools to have in your toolbox. Following are some handy formulas for solving acid/base problems. Calculating hydrogen or hydroxide ion concentration The following equation allows you to calculate the hydrogen ion concentration, [H+], at 25°C if you know the hydroxide ion concentration, [OH–]; you can also find [OH–] if you know [H+]. Just divide 1 × 10–14 by the concentration given, and you get the concentration that you need. Tip: To use scientific notation on your calculator, use the EE or EXP key (followed by the exponent) rather than the × 10^ keys. Calculating hydrogen or hydroxide ion concentration from the pH or pOH Be familiar with how to solve for [H+] or [OH–] when given the pH or pOH (or vice versa). Use the following formulas: Many scientific and graphing calculators differ in how they handle inputting values and taking logarithms, so know the proper keystroke order for your calculator. Be sure to review your calculator manual or look online. Calculating pH when given the pOH Calculating pH when you know the pOH (or vice versa) is probably the easiest of the acid-base calculations. Here’s the formula: pH + pOH = 14 Simply subtract the given value from 14 (keeping significant digits in mind) to get the value that you need. Doing titration calculations with a 1:1 acid-to-base ratio When you’re given titration calculations where the acid and base are reacting in a 1:1 ratio according to the balanced equation, the following equation offers a quick and easy way to solve for either the concentration of one of the substances or the volume necessary to complete the titration: MAVA = MBVB If the acid and base aren’t reacting in a 1:1 ratio, use stoichiometry (or dimensional analysis) to solve for your unknown quantity. By the way, stoichiometry works for the 1:1 ratio questions, too; it just takes one or two more steps. Remember: Keep track of your units! Cancel what you need to get rid of and make sure that you still have the units you need in your final answer.

Article / Updated 05-03-2023

The hyperbolic functions are certain combinations of the exponential functions ex and e–x. These functions occur often enough in differential equations and engineering that they’re typically introduced in a Calculus course. Some of the real-life applications of these functions relate to the study of electric transmission and suspension cables.

Article / Updated 04-27-2023

General relativity was Einstein’s theory of gravity, published in 1915, which extended special relativity to take into account non-inertial frames of reference — areas that are accelerating with respect to each other. General relativity takes the form of field equations, describing the curvature of space-time and the distribution of matter throughout space-time. The effects of matter and space-time on each other are what we perceive as gravity. Einstein immediately realized that his theory of special relativity worked only when an object moved in a straight line at a constant speed. What about when one of the spaceships accelerated or traveled in a curve? Einstein came to realize the principle of equivalence, and it states that an accelerated system is completely physically equivalent to a system inside a gravitational field. As Einstein later related the discovery, he was sitting in a chair thinking about the problem when he realized that if someone fell from the roof of a house, he wouldn’t feel his own weight. This suddenly gave him an understanding of the equivalence principle. As with most of Einstein’s major insights, he introduced the idea as a thought experiment. If a group of scientists were in an accelerating spaceship and performed a series of experiments, they would get exactly the same results as if sitting still on a planet whose gravity provided that same acceleration, as shown in this figure. Einstein’s brilliance was that after he realized an idea applied to reality, he applied it uniformly to every physics situation he could think of. For example, if a beam of light entered an accelerating spaceship, then the beam would appear to curve slightly, as in the left picture of the following figure. The beam is trying to go straight, but the ship is accelerating, so the path, as viewed inside the ship, would be a curve. By the principle of equivalence, this meant that gravity should also bend light, as shown in the right picture of the figure above. When Einstein first realized this in 1907, he had no way to calculate the effect, other than to predict that it would probably be very small. Ultimately, though, this exact effect would be the one used to give general relativity its strongest support.

Article / Updated 04-20-2023

Conversations about the cause of global warming typically focus on the big offenders — the worst industries, dirtiest factories, and scoff-law nations. There’s nothing wrong with that. But everyone plays a role in climate change. Each of us uses energy — specifically, fossil fuels — on a daily basis: Electricity: From the moment the alarm sounds in the morning until you shut off the computer or TV at night, you’re connected to an electrical grid, often fueled by coal or oil. Transportation: Everyone needs energy to move from here to there — a steady supply of gas for your car or diesel fuel for the bus you ride to work. Food: Most of humanity’s food travels great distances before it arrives in homes, a journey it undertakes thanks to greenhouse gas (GHG)-producing fuels. Typically, in talking about climate change, big business and government tend to point the finger at the individual. Asking citizens “What are you prepared to do?” even when it’s hard and expensive to make personal changes without a major structural shift further up the food chain, becomes an excuse for inaction by those with the real power to make change. The COVID pandemic gave a real-life lesson in how much the individual is really to blame. For all of 2020, citizens everywhere around the world drove a whole lot less and flew hardly at all. You would have thought that GHGs would have dropped and in a big way. They dropped — but only by 6.4 percent, according to Nature. And because of emissions in previous years, the concentration of GHGs in the atmosphere continued to rise, smashing through previous records to 412.5 parts per million (ppm) — more than at any time in more than 3 million years. Still, individual choices do matter. The most powerful individual change you can make is political by letting your elected officials know you demand they take the climate emergency seriously. And the personal choices you make are important. They send a signal. They keep your own sense of personal choice and personal power intact. How transportation impacts global warming About 24 percent of all GHG emissions come from moving people and goods, according to the World Resources Institute. In the United States, the proportion is higher; closer to one-third of emissions come from transportation — 29 percent. In Canada, it’s 25 percent. Almost all transportation — about 95 percent, according to the Intergovernmental Panel on Climate Change (IPCC) — runs on oil-based fuels, such as diesel and gas. This explains why transportation accounts for such a large portion of overall emissions (electrification of transport is rising fast around the world, but it’s still only about 2.2 percent of all vehicles). Cars and diesel trucks are the top two offenders, but ships, airplanes, trains, and buses play a part, too. We discuss the culprits of driving and flying in the following sections. The figure below shows the breakdown of how each mode of transportation contributes to GHG emissions. Driving emissions Whether you need to run errands or drive the kids to soccer practice, cars, minivans, and SUVs are useful and often perceived as necessary. Most households in industrialized countries own at least one car because most cities and housing developments are built around road infrastructure — making it difficult to survive without one. However, that internal combustion engine now has serious competition — hybrid cars that generate energy to a battery to reduce fuel use and electronic vehicles (EVs), vehicles that run 100 percent on electricity. These technologies promise to make the internal combustion obsolete, just as that technology disrupted the horse and buggy. In November 2021, Hertz, a major U.S. car rental company, placed an order for 100,000 Tesla EVs — that’s how fast things are changing. Despite the pandemic, global sales of EVs increased by 43 percent in 2020. Still, even as sales of EVs and hybrids ramp up, in 2020 only 1 in 250 cars on the road is electric. The majority of people in the developing world still don’t have access to a personal vehicle — but that’s quickly changing. China is soaring ahead in private car ownership, which jumped from 45 million cars in 2009 to more than 225 million in 2019! Total EV sales in China were 1.3 million, an increase of 8 percent compared to 2019. The 2019 sales of EVs in China amounted to 41 percent of all EVs sold worldwide. Flying emissions Planes burn fuel similar to kerosene, which gives off more emissions than the gasoline in your family car. The 2018 IPCC special report on 1.5 degrees found that aviation has grown to 14 percent of transport sector global carbon dioxide emissions. Not only does flying emit a lot of GHGs, it emits them in the atmosphere in a more damaging way. The warming impact of the exhaust from air travel is far worse than the same volume of GHGs emitted on the ground. People made about 38.9 billion flights in 2019, and that dropped to less than 17 billion in 2020 due to the COVID pandemic. Still, it’s shocking that in 2007, globally, there were 4 billion individual plane trips for business or pleasure. And, even though flights dropped dramatically in 2020, they have started rising once again. The people of China are flying more and exploring their own country by air and rail. They’ve also increased what is called outbound tourism — that is Chinese tourists exploring other countries. In 2003, 20 million tourists from China explored the world. By 2019, that figure was up to 155 million! China now tops the charts for outbound tourism. Concern about the climate crisis is leading some countries to reduce flying. Impressively, the French government decided post-COVID that no domestic flights would be allowed where train travel was available. Increasingly, climate-aware travelers work to eliminate air travel altogether. The pandemic allowed many organizations to experience meetings — big and small — using online video technologies. The cost of flights and the wasted time traveling have likely made a permanent change in how employers see the practicality of video conferencing. This could increase the trend to staying home and avoiding flights wherever you can. If you have to fly, buying carbon offsets is a worthwhile option. Energy uses around the house In Canada and the United States, the floor space of the average home has continued to grow while family size is shrinking. House size has real implications for the climate crisis. The bigger the home, the more energy required to heat, cool, and light it. Fewer people are occupying — and heating and cooling — more space. When it comes to energy use in your home, you can think about it in two ways: Direct energy: This term refers to the energy you use, which comes from gas or fuel oil that you consume directly — such as the oil-fed heaters or propane gas stoves you may have in your home. Indirect energy: This term refers to how some other energy — oil, hydroelectric, wind, or nuclear power — is used to produce the direct energy. For example, natural gas is used to heat the oil sands enough so that the otherwise solid bitumen flows and can be extracted. How your electricity is produced affects your individual GHG emissions. The energy that people use in their homes accounts for about 25 percent of GHG emissions around the world. Most of the fossil fuel energy you use directly goes toward heating your home. You use most of your electricity to power your lighting and appliances. See the figure below for a complete breakdown of the percentages of GHG emissions produced from heating, lighting, and other energy uses. Your energy use may be very different than the average home. For example, you may not have an air conditioner. When talking about climate change, scientists mean changes to the global climate systems. But people can also talk about “climate control” in their homes. Modern air conditioning can make the indoors feel like winter in a sweltering summer. Controlling the climate in your home can also impact the global climate system. Controlling the climate in your home Homes in the United States create 150 million metric tons of carbon dioxide every year from heating and cooling alone for 333 million people, according to the U.S. Department of Energy. That’s a full 2020 percent of U.S. GHG emissions. Check out the following sections for how heating and cooling your home plays a role in global warming. Heating Heating takes either direct energy or electricity, depending on whether you have an oil or gas furnace or electrical baseboard heaters. Other types of home heating, such as wood stoves or gas fireplaces, also create emissions. The U.N Food and Agricultural Organization notes that burning wood for home heating (and in some countries, cooking, as well) accounts for about 6 percent of energy use in the world. Burning wood adds to GHG emissions both through the carbon dioxide released during burning and through deforestation. Older furnaces emit more GHG than newer models. These old clunkers guzzle fossil fuels, but unfortunately, many homeowners cling to them, worried about the expense of buying a new unit. In reality, these homeowners can save money if they buy a new energy-efficient furnace, which would save them significantly on energy costs — and be less costly to the planet, too. Cooling Electricity used to be used only for keeping the lights on. Now, it’s what keeps people cool all summer long. In fact, the largest share of home electricity use now goes directly to air conditioning. And in places such as south central Canada, the greater share of power demand has recently shifted from winter to summer. With more 86 degrees F (30 degrees C) days every summer — thanks to global warming — the demand for air conditioning goes up annually. Only industrialized countries used air conditioning, for the most part, until now. Recent news reports show that sales of home air conditioners have tripled in the last ten years in China. As countries such as China and India move to catch up to industrialized countries, residents are starting to widely use luxuries such as air conditioning. Add warming temperatures into the mix, and you can see a growing air-conditioning trend and a growing demand for electricity to meet that desire. Traditionally, most Europeans never considered air conditioning. But because killer heatwaves have ravaged Europe in recent years, this perspective is changing. For example, the U.K. had to consider new labor laws — in the past, laws ensured a legal minimum temperature so workers could stay warm enough. Because of intense summer heat, they’ve also had to consider legal maximum temperatures! Perhaps the most surprising area to need air conditioning is in Canada’s far north. Buildings in the Northwest Territories and the Yukon are now being built with air conditioning. The average high temperature in the summer in those territories ranges from 70 to 80 degrees F (in the 20s C), but has been warming up recently and has reached the 90s F (about 30 degrees C). Your food choices and global warming Like a warm home in freezing weather, food is a necessity, not a luxury. But sadly, when people sought to make food more accessible and more convenient and to offer a greater variety, they often did so without considering the environmental toll their innovations might have. Much of the food that people buy at the grocery store uses a lot of energy to get there — and creates a lot of GHG emissions as a result. Here are some of the key offenders: Frozen food: Whether you’re talking refrigerated or frozen, these foods burn energy when they’re made, while they’re being transported, and even when they’re sitting in a freezer or cooler in the grocery store (or in your home). The most-energy-used-per-serving prize goes to freeze-dried coffee. Processed and packaged food: Moving these foods through the production line takes energy, as does making the packaging (not to mention the emissions that come from all that packaging when it ends up in a landfill). Food from afar: Elizabeth never even saw a kiwi until she was about 18 years old. Her daughter started asking for them for her school lunch in first grade. You may enjoy strawberries and mangoes in the dead of winter, when you can’t pick fresh fruit right in your backyard, but moving exotic fruits and veggies around the world by plane, ship, and truck has a real cost in energy. Could people afford them if companies factored in the cost to the climate? And why should your apple be more well-traveled than you? Meat products: Feeding livestock takes an average of 10 pounds of grain — grain that plays a large role in agricultural emissions — to produce 1 pound of meat. Also, when people eat more meat, more land is needed to raise livestock, which often means clearing forests and losing trees that breathe in our carbon dioxide.

Article / Updated 04-18-2023

Although pain is a necessary function for preventing damage to the body, in some cases, pain itself becomes disabling. Chronic pain can occur in disease conditions such as cancer, in which case the normal function of pain that forces you to rest, protect, or not use some injured part of the body until it heals is simply inappropriate in a disease state in which destruction is occurring from the cancer all over the body that cannot be healed from rest. Pain can also arise from psychological factors or from factors that cannot be medically identified and are assumed to be psychological. Examples include some types of chronic pain and depression. Pain from both medically identified sources and that which is psychological (or cognitive) appears to activate a brain area called the anterior cingulate cortex. The anterior cingulate cortex is the anterior portion of an area of the mesocortex, just above the corpus callosum. The anterior cingulate appears to be a high-level cortical monitoring center. It tends to be activated by pain, anticipation of pain, and failure in goal-seeking activity. Its function seems to be to arbitrate between taking different strategies in response to experience. At a low level, after placing your hand on a hot stove burner, it may make you cautious when you're around the stove. At a higher level, getting reprimanded for sending a flaming email at work may make you wary of doing so again. Considerable individual differences with respect to pain tolerance exist, just as there can be differences in tolerance in different situations for a particular person. Men are reported to be less tolerant of chronic pain than women, though they are more tolerant of acute pain. Pain tolerance generally increases with age, based on tests for pain tolerance such as the total time one can stand to have one's arm immersed in ice water. It is not clear whether the increase in tolerance with age is based on psychological or physical factors. Athletic training and strong motivation to obtain some goal can significantly reduce the disabling effects of pain. Suggestions that different cultures or ethnic groups have intrinsically different pain thresholds — in other words, there's a physiological difference among cultures about pain tolerance — have almost always been shown to be the effect of at what point the perceived stimulus is reported as painful or unbearably painful, not whether the pain itself is perceived. Cultures that encourage expression of emotions in general tend to be associated with lower pain-reporting tolerance.

Article / Updated 04-14-2023

General relativity was Einstein’s theory of gravity, published in 1915, which extended special relativity to take into account non-inertial frames of reference — areas that are accelerating with respect to each other. General relativity takes the form of field equations, describing the curvature of space-time and the distribution of matter throughout space-time. The effects of matter and space-time on each other are what we perceive as gravity. The theory of the space-time continuum already existed, but under general relativity Einstein was able to describe gravity as the bending of space-time geometry. Einstein defined a set of field equations, which represented the way that gravity behaved in response to matter in space-time. These field equations could be used to represent the geometry of space-time that was at the heart of the theory of general relativity. As Einstein developed his general theory of relativity, he had to refine the accepted notion of the space-time continuum into a more precise mathematical framework. He also introduced another principle, the principle of covariance. This principle states that the laws of physics must take the same form in all coordinate systems. In other words, all space-time coordinates are treated the same by the laws of physics — in the form of Einstein’s field equations. This is similar to the relativity principle, which states that the laws of physics are the same for all observers moving at constant speeds. In fact, after general relativity was developed, it was clear that the principles of special relativity were a special case. Einstein’s basic principle was that no matter where you are — Toledo, Mount Everest, Jupiter, or the Andromeda galaxy — the same laws apply. This time, though, the laws were the field equations, and your motion could very definitely impact what solutions came out of the field equations. Applying the principle of covariance meant that the space-time coordinates in a gravitational field had to work exactly the same way as the space-time coordinates on a spaceship that was accelerating. If you’re accelerating through empty space (where the space-time field is flat, as in the left picture of this figure), the geometry of space-time would appear to curve. This meant that if there’s an object with mass generating a gravitational field, it had to curve the space-time field as well (as shown in the right picture of the figure). Without matter, space-time is flat (left), but it curves when matter is present (right). In other words, Einstein had succeeded in explaining the Newtonian mystery of where gravity came from! Gravity resulted from massive objects bending space-time geometry itself. Because space-time curved, the objects moving through space would follow the “straightest” path along the curve, which explains the motion of the planets. They follow a curved path around the sun because the sun bends space-time around it. Again, you can think of this by analogy. If you’re flying by plane on Earth, you follow a path that curves around the Earth. In fact, if you take a flat map and draw a straight line between the start and end points of a trip, that would not be the shortest path to follow. The shortest path is actually the one formed by a “great circle” that you’d get if you cut the Earth directly in half, with both points along the outside of the cut. Traveling from New York City to northern Australia involves flying up along southern Canada and Alaska — nowhere close to a straight line on the flat maps we’re used to. Similarly, the planets in the solar system follow the shortest paths — those that require the least amount of energy — and that results in the motion we observe. In 1911, Einstein had done enough work on general relativity to predict how much the light should curve in this situation, which should be visible to astronomers during an eclipse. When he published his complete theory of general relativity in 1915, Einstein had corrected a couple of errors and in 1919, an expedition set out to observe the deflection of light by the sun during an eclipse, in to the west African island of Principe. The expedition leader was British astronomer Arthur Eddington, a strong supporter of Einstein. Eddington returned to England with the pictures he needed, and his calculations showed that the deflection of light precisely matched Einstein’s predictions. General relativity had made a prediction that matched observation. Albert Einstein had successfully created a theory that explained the gravitational forces of the universe and had done so by applying a handful of basic principles. To the degree possible, the work had been confirmed, and most of the physics world agreed with it. Almost overnight, Einstein’s name became world famous. In 1921, Einstein traveled through the United States to a media circus that probably wasn’t matched until the Beatlemania of the 1960s.

Article / Updated 04-14-2023

A conversion factor uses your knowledge of the relationships between units to convert from one unit to another. For example, if you know that there are 2.54 centimeters in every inch (or 2.2 pounds in every kilogram or 101.3 kilopascals in every atmosphere), then converting between those units becomes simple algebra. It is important to know some common conversions of temperature, size, and pressure as well as metric prefixes. Conversion factor table The following table includes some useful conversion factors. Using conversion factors example The following example shows how to use a basic conversion factor to fix non-SI units. Dr. Geekmajor absentmindedly measures the mass of a sample to be 0.75 lb and records his measurement in his lab notebook. His astute lab assistant, who wants to save the doctor some embarrassment, knows that there are 2.2 lbs in every kilogram. The assistant quickly converts the doctor’s measurement to SI units. What does she get? The answer is 0.34 kg. Let’s try another example. A chemistry student, daydreaming during lab, suddenly looks down to find that he’s measured the volume of his sample to be 1.5 cubic inches. What does he get when he converts this quantity to cubic centimeters? The answer is 25 cm3. Rookie chemists often mistakenly assume that if there are 2.54 centimeters in every inch, then there are 2.54 cubic centimeters in every cubic inch. No! Although this assumption seems logical at first glance, it leads to catastrophically wrong answers. Remember that cubic units are units of volume and that the formula for volume is Imagine 1 cubic inch as a cube with 1-inch sides. The cube’s volume is Now consider the dimensions of the cube in centimeters: Calculate the volume using these measurements, and you get This volume is much greater than 2.54 cm3! To convert units of area or volume using length measurements, square or cube everything in your conversion factor, not just the units, and everything works out just fine.