Physics the Google Way

Contents

• BODY OF ARTICLE
• The Fundamental Constants
• Unit Conversions
• Illustrative Examples
  • How much do I weigh on the Moon? On the Sun?
  • Einstein's car
  • Electrostatic versus gravitational force
• Conclusion
• REFERENCES
• About the Author
• FIGURES
• TABLES

"Never memorize something that you can look up."—Albert Einstein

Are we smarter now than Socrates was in his time? Society as a whole certainly enjoys a higher degree of education, but humans as a species probably don't get intrinsically smarter with time. Our knowledge base, however, continues to grow at an unprecedented rate, so how then do we keep up? The printing press was one of the earliest technological advances that expanded our memory and made possible our present intellectual capacity. We are now faced with a new technological advance of the same magnitude, the Internet, but how do we use it effectively? A new tool is available on Google^TM (http://www.google.com)that allows a user not only to numerically evaluate equations but also to automatically perform unit analysis and conversion, with most of the fundamental physical constants built in. This paper describes some of its capabilities.

To get a feel for how the Google calculator works, you can start by googling¹ "2+2=" and clicking on "More about calculator," located directly below the result, to learn the basics of Google equation formatting. There you will find a link to more complete instructions on how to format equations for Google, e.g., "xn" means raise x to the nth power, "sqrt(x)" means take the square root of x, etc. Return to the search bar and google "G," as illustrated in Fig. 1. Google returns "gravitational constant = 6.67300 × 10^–11 m³ kg^–1 s^–2" at the top of the page next to the calculator icon, followed by web page results for the Google query. The results relevant to this paper will always be those at the top of the page adjacent to the calculator icon; only these can be used in equations. Google has most of the fundamental constants built in, many of which are listed in Table 1.²

Figure 1.

To illustrate the use of the constants in an equation, consider the fine structure constant in which fundamental constants of quantum mechanics, electricity and magnetism, and geometry are contained. We will actually google the reciprocal of the fine structure constant, as this has a well-known simple value. Google "(4 * pi * electric constant * hbar * c) / (elementary charge2)=" to find that it is equal to "137.035984." We could have just googled "1 /fine-structure constant" to find the answer.³ Note, we were able to confirm that our calculation not only matches the acceptable numerical value of the fine structure constant, but it is unitless as well.

The previous example illustrates the utility of the Google calculator in unit checking, a task that otherwise is only adequately described by the word tedium. Although SI units have solid footing as a scientific standard, it is sometimes useful to employ others, or variants of SI units. This is not only because many classic texts and some individual fields in physics employ units other than SI, but mainly because these are not necessarily the units in which we understand the world. In the United States, a comfortable room is one that is cooled to around 70°F, not 21.1°C or 294.3 K.

To further illustrate the advanced features of the Google calculator, we will consider several example calculations, exploiting the Google features to tailor the form of the answer we obtain.

Space travel is fun, and though the Internet won't allow us to do it from the comfort of our homes, we can get a sense of what it would be like to visit another planet without the harmful side effects of toxic atmospheres and unbearable temperatures (see Fig. 2). We can use Newton's law of gravity to determine what we would weigh on other planets. Suppose our mass is 80 kg (i.e., our bathroom scale on Earth reads 165 pounds of force). To find out what the scale would read if we took it to the Moon, google "convert(G*mass of the moon*80 kg/(radius of the moon)2) to pounds force." The answer is "29.2619238 pounds force." (You will have to adjust the number of significant figures yourself.) Replace the Moon with the Sun to find that we would need to take a better scale as our weight there would be "4,934.55214 pounds force." Try it for Mars or any other planet in the solar system by using "mass of planet name" and "radius of planet name" in Newton's law of gravity.

Figure 2.

Let's suppose you built a crazy car that converted mass into energy directly. So you put in a kilogram of matter, perhaps old cereal boxes, and the engine of this car converts it directly into energy through Einstein's famous energy/matter relation, E = mc². Maybe we have to put in an equal dose of anti-matter to get 100% conversion, but we'll assume that somehow it happens. Let's suppose that it is capable of converting one kilogram per year under constant operation. How many horsepower does this engine have? Google "convert 1 kilogram*c2 / year to horsepower" and it returns "3,819,290.09 horsepower." It makes a Ferrari look like a go-cart.

The force that the Sun exerts on the Earth can be calculated by googling "(G*mass of the earth*mass of the sun/(1 astronomical unit)2)" to find that the force binding the Earth to the Sun is "3.54296305 × 10²² newtons." Using the equation for centripetal acceleration, a = v²/r, the Earth's orbital speed can be determined. To do so google "sqrt(G*mass of the sun/(1 astronomical unit))" to learn that the Earth is hurtling through space at the rate "29,785.5982 m/s."

Coulomb's law for electrostatic forces is analogous to Newton's law of gravitation with the masses replaced by charges and the gravitational constant replaced by the Coulomb constant, which is written 1/(4_o). The constant _o is the permittivity of free space, or in Google, the "electric constant." How does the electric force between the electron and proton in the hydrogen atom compare to the gravitational force between them? To find the ratio of the electric to gravitational force in the hydrogen atom, google "((elementary charge)2 / (4 *pi*electric constant))/(G*mass of electron*mass of proton)" to find that the electric force is "2.26910384 × 10³⁹" times stronger.

Using the Bohr model of the hydrogen atom, we can compute the rotational kinetic energy of the electron were it orbiting the nucleus as in a planetary system by googling "convert 0.5 * ((elementary charge)2 / (4*pi*electric constant *(0.529189379 angstrom))) to eV." The answer is "13.6053788 electron volts," which is darn close to the measured value for the ionization energy of hydrogen. We could have just obtained the answer in joules, but hardly anybody thinks about atomic energy levels in joules.

I began this paper with a quote from Albert Einstein. Given the vast quantity of information available on any given subject in physics, this makes a lot of sense, but we all know how useful it is to memorize certain things before an exam, for example. The point is that there is an advantage to having information at your fingertips. The advanced features of the Google calculator make this possible without our having to do the memorizing.

Future generations of physicists will pass the responsibility of unit conversion, unit checking, algebra, calculus, and looking up the physical constants to computers much in the same way our generation passed off addition, subtraction, multiplication, and division. The earlier this responsibility is passed off in students' education, the sooner they can get to the forefront of physics. They will, of course, need some training in the basics, but we will eventually abandon spending years on multiplication tables, algebra, and calculus so that students can tackle introduction to string theory before they graduate from high school. They won't be able to do it because they got intrinsically smarter, but because the Google calculator bar in their heads-up-display, which is linked to the Internet through their personal WI-FI connection, will enable them to concentrate on the really important things without spending a lot of time worrying about the remedial math.

References

1. Variants of the verb "to google" are used throughout this paper. It means to type something into the Google search bar and press the search button. first citation in article
   
2. Additional resources are available at http://www.googleguide.com/calculator.html and http://google.davidward.org/. first citation in article
   
3. Of course, all results obtained in calculations of this kind must be rounded to the correct number of significant figures. first citation in article
   

David W. Ward received his B.S. in physics from the College of Charleston and his Ph.D. in physical chemistry from the Massachusetts Institute of Technology. He is a co-founder of the field of polaritonics and is currently a postdoctoral fellow at Harvard University. His research interests include single molecule spectroscopy and imaging, sub-diffraction imaging, novel optical materials, and computational electrodynamics.Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139; david@davidward.org

Full figure

Fig. 1. Type equations directly into the search bar, and if applicable, Google outputs the result adjacent to a calculator icon. Click on "More about calculator" to learn Google formatting of equations. First citation in article

Full figure

Fig. 2. The Google calculator has planetary data about our solar system built in. First citation in article

Table I. Guide to physical constants available on Google calculator. Entries are sometimes case sensitive.
Long Name	Shorthand
atomic mass units	amu or u
Astronomical unit	au
Avogadro's number	N_A
Boltzmann's constant	k
electric constant, permittivity of free space	epsilon_0
electron mass	m_e
electron volt	eV
elementary charge
Faraday constant
fine-structure constant
gravitational constant	G
magnetic constant, permeability of free space	mu_0
magnetic flux quantum
mass of the moon	m_moon
mass of the sun	m_sun
mass of [planet name]	m_mercury
molar gas constant	R
Planck's constant	h and hbar
proton mass	m_p
radius of the moon	r_moon
radius of the sun	s_moon
radius of [planet name]	r_mercury
Rydberg constant
speed of light	c
speed of sound (note: in air at sea level)
Stefan Boltzmann constant

First citation in article

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