How Do They Calculate the Mass of the Earth?

image001In science, there could be a wealth of observations, experiments and knowledge behind the simplest fact. For instance, the mass of the earth is 5.972 × 1024 kg. But where does that number come from? That simple question opens a can of worms for those with a phobia of math or physics. But for the rest of us, it reveals a pleasant sequence of questions, each opening up a new topic, like a set of Russian dolls.

At first the problem seems deceivingly simple. If you know the circumference of the Earth, you know its radius from which you can get calculate its volume. Then with an estimate of average density, you can obtain its mass. Begin with an educated guess:  an average of the densities of the common rock granite and iron gives a value (5250 kg/m3, almost 95% accurate, by fluke). The circumference of 40 000 km attributed to another clever but lucky measurement by Eratosthenes, when converted to a volume and then multiplied by the density yields an Earth-mass of 5.67 X 1024 kg, off by 12%.
But what are the more accurate methods of measuring the mass of the earth (me) and its radius (re)? Its mass can be obtained from a formula based on two experimentally determined constants, g = the gravitational acceleration on Earth and G = gravitational constant. image015

Of course my questions begs other questions: from where does the expression originate?How do you get a better value for the radius and how do you get G and g?
Calculating the Radius of the Earth

image005Jean-Félix Picard, a contemporary of Sir Isaac Newton, first measured the size of the Earth to a reasonable degree of accuracy in a survey conducted in 1669–70. He measured one degree of latitude along the Paris Meridian using triangulation along thirteen triangles that stretched from Paris to a clock tower near Amiens. From his measurements, one degree of latitude was equivalent to 110.46 km. A full circle or 360 degrees produced a circumference of 39765.60 km, corresponding to a terrestrial radius of 6328.9 km. That was not a fixed value because as they realized in the 17th century, our planet is not a perfect sphere.

The value of G

Newton realized that any pair of objects, whether planets or metal spheres,  attract one another with a force proportional to the product of their masses. Before we get to what inspired the relationship, the more general formula that led to  image015

is the Law of Universal Gravitation:image025

( a nice mnemonic for it is , G-memory!) where F is the force of attraction between the mass a planet, in this case Earth, and that of an object, mo.  Since the weight (mog) of that object is the same force we can equate the two expressions, cancel  mand solve for me.




from Wikipedia article on  Cavendish’s experiment

Rather than focusing on planets whose masses were unknown, Cavendish used two large balls of lead and their attraction to small balls connected in a torsion balance. The force between the large balls (M) in the diagram caused a small angle Θ  to be formed between their connecting rod’s position and the original location when the large and small balls were separated by a distance r . He also measured the period of oscillation, T. image020

The small balls’ moment of inertia, I,  is mL2/2, where m = mass of the small ball and L is the length of the torsion balance beam.  He also used two equivalent expressions ( image023) for the turning force(torque) of the torsion wire where k = the wire’s torsion constant. When these expressions are substituted into image026 we obtain




which gave  Cavendish 6.75 x 10-11 N m2/kg2, a value only off by 1.2 % from the currently accepted 6.67 x 10-11 N m2/kg2. The latter was also obtained from a torsion balance using gold, platinum, and glass small spheres at the U. S. National Bureau of Standards from 1925 to 1928.

But what about g, the value for gravitational acceleration?

This involves a much simpler experiment and mathematics. When an object is dropped from rest, since its initial velocity is zero, its final velocity will be the product of g and time, t.

v = gt

Its potential energy, mgh, with respect to the surface will be all converted to kinetic energy, 0.5mv2.

So mgh = 0.5mv2.

g = v2 /(2h), but if we substitute gt for v from the previous expression:

g = 2h/ t2  

All we need is a free fall apparatus which measures the time taken for a steel ball to fall from a height, h. Using several data points, height is plotted against the square of time

( h = (g/2) t2  ) and the slope of the line is g/2.  Gravitational acceleration varies on different locations on earth, but the average value is 9.81 m/s

The Source of…image026
From the measurements of other astronomers, Kepler realized that the relative ratio  of any two planets’ periods, t1 and t2 , was related to the ratio of their orbital radii, r1 and r2 , by what is dubbed Kepler’s 3rd law:image030
image028For example Earth and Venus’ orbital periods are 224.7 and 365.2 days long. We are 1 astronomical unit from the sun while Venus is 0.723 times as far, a fact that can be obtained trigonometrically from observing the planet at its position of greatest elongation, in other words when it’s at the vertex of a 90o angle between us and the sun. The sine of the observed angle, 46.3o , is the ratio of the opposite side (Venus-sun relative distance) to the hypotenuse(sun-earth distance). If we use a hypotenuse of 1, we obtain a value of 0.723.
(224.7/365.25)2  = 0.378 = (0.723)3 = 0.378.
Another way of expressing Kepler’s 3rd Law is that the square of any planetary period, T, is some constant, C, multiplied by cube of its orbital distance, R:   T2 = CR3.
When an object like a planet moves at constant speed, its direction constantly changes. So its velocity, a vector quantity, is changing or accelerating.  The centripetal force towards the sun is given by F = mv2 /R. We’ll derive that shortly.  In one period, the planet will cover a circumference of 2πR in time T, so its velocity, v = distance/time = 2πR/T. Substituting that into the centripetal force expression, we get

F =  image033.

Since T2 = CR3 ,   F =  image034 , and this made Newton cognizant of the inverse square relationship between force and orbital distance. He also realized that 4π2 /C represented the product of a universal constant, G, and a second mass attracted to the first. Hence, image026
Why is Centripetal Force Equal to mv2 /R?


drawing by the author

In the smallest of time intervals, the velocity of an object moving in a circle changes because its direction changes. But its speed is constant. So if in time Δt, an object goes from v1 to vf, their absolute values are the same. If you draw the velocity vectors as tangents to the circle, each one makes a 90 degree angle with the radius. Extending the vectors (see dotted lines) we get a quadrilateral ABCD, where

180 – Φ +90 + 90 + Θ = 360. Thus Φ = Θ!
In a small time interval, Δt, the arc DB will be a straight segment forming a triangle. The distance covered in time Δt, will equal v Δt = DB.  Since the triangle is isosceles, just like DEF and since both triangles’ equal sides form the same angle, they are of the same shape and the sides are proportional. image038Since Δv/Δt = a = acceleration, substituting and cross- multiplying we get a = v2/r. Multiplying both sides by the mass, m,
F = ma = mv2/r.






Cancer Prevention

There’s a valuable book out there on cancer prevention written by Richard Beliveau, a biochemist who specializes in cancer prevention and by Denis Gingras, an oncology researcher, both of whom work at UQAM. They start with the following tantalizing question:

Why do many spontaneously formed cancerous lesions remain latent in one person and yet develop into cancer in another? Factors beyond our control, like aging and heredity are often cited as the main agents determining cancer risk, but their influence is in fact much weaker than might be thought.

To back this thesis they cite evidence that in every age group, including in people over 75, esophageal cancer rates have jumped 3 to 12 times between the 1975-79 period and between 2000-2004.

But there are a couple of things that irk me about the book.

(1) If there isn’t strong evidence that age plays a major role in cancer development, why aren’t the authors criticizing organizations like the American Cancer Society and Center for Disease Control for consistently presenting age-adjusted cancer data to the public?


from CDH

Given that heredity and age are not the most fundamental variables in cancer formation, why then will the number of new cases possibly rise by about 70% over the next 2 decades according to the World Health Organization?  The authors focus on individual lifestyle choices, and in their defense, tobacco smoke is indeed one of the main causes of cancer, accounting for 30% of worldwide cases. In the same manner that production of goods has shifted overseas where standards are lower and therefore inexpensive, tobacco companies have found new nicotine users in a global market. Consequently, tobacco consumption is still increasing worldwide even though the number of smokers has gone down in several Western countries. Of the 4000 compounds and elements in tobacco smoke that include radioactive polonium 210, 70 are known carcinogens, and those in group I  include benzene, benzopyrene, NNK, arsenic, chromium and four others.  As a result of this, smoking increases the risk of lung cancer by a factor of 25 and also plays a role in other cancers.

Also central to the authors’ thesis is that along with tobacco smoke, other lifestyle-roles including exposure to UV, poor diet, obesity, alcohol and physical activity supposedly account for 75% of cancers.

(2) This brings us to a second flaw in the book: “Chance” accounts for the rest of the causes, where “pollution” is clumped with inflammatory diseases, infections, defective genes, cell division and spontaneous DNA damage, factors that are beyond the individual’s control. Beliveau and Gingras make no mention of the 2010 President’s Cancer Panel report which stated that :

  • There’s a growing body of evidence linking environmental exposures to cancer.
  • There’s much work ahead to identify the many existing but unrecognized environmental carcinogens and eliminate those that are known from our workplaces, schools, and homes.
  • The true burden of environmentally induced cancer has been grossly underestimated. With nearly 80,000 chemicals on the market in the United States, many of which are used by millions of Americans in their daily lives and are understudied and largely unregulated, exposure to potential environmental carcinogens is widespread.
  • The public remains unaware of many common environmental carcinogens such as naturally occurring radon and manufacturing and combustion by-products such as formaldehyde and benzene.

In fact benzene, the group I carcinogen in tobacco smoke, is not only a combustion byproduct, but we expose ourselves to benzene each time we gas up at the fuel pump. At best, 0.62% of the volume of gasoline is benzene.

What I do like about the UQAM researchers’work is that they not only present epidemiological statistics, but they provide the chemical basis of cancer-preventative strategies. For instance:

  • Depending on the amount of consumption, red meat and processed meat increase the colon cancer rate by 10 to 50 % due to PhIp, a heterocyclic amine formed during cooking and due to nitrosamines formed from the addition of nitrites, respectively.
  • curcumin

    Curcumin,C21H20O6, an anti-cancer compound in an Indian spice.

    Although other  variables are involved, India’s cancer rate is only about 1/4 of that of Europe and the U.S. partly because of their six to 12 times rate of spice consumption. Turmeric has over 200 polyphenolic compounds, including curcumin, which inhibits cancer cell growth and blocks angiogenesis. Angiogenesis, the process where new blood vessels form from old ones, is used by some cancer growths. Curcumin also has anti inflammatory properties and it induces apoptosis, in this context, the death of cancer cells in a tumor. Although curcumin on its own is poorly absorbed in the intestine due to conversion by the enzyme UDP glucorryl transferase, the latter is inhibited by a component of another common spice used by Indians, piperidine in pepper.

In a nutshell then, many cancers can be prevented but not only from making lifestyle changes but by pressuring our governments to further limit our exposure to industrial carcinogens.

Why Do Cacti Taste Bitter in the Afternoon?

sedumWhile cycling to work one morning, I spotted a little hogweed(purslane) growing next to sedum in a rock garden. It wasn’t a coincidence. Each has evolved its own biochemical strategies to deal with high temperatures and low humidity.  The hogweed’s biochemistry is also one of the reasons why a lawn left to its own devices will soon cease to be a monoculture.

Unlike sedum, the majority of land plants are like Kentucky bluegrass, which is the common grass found in lawns. They are C3 plants. During photosynthesis, an enzyme known as RuBisCO catalyzes the capture of CO2 to produce three-carbon molecules. Derivatives of the molecules remain within the cycle, and one is used as a building block to produce sugars.

Unfortunately, high temperatures make it easier for RuBisCo to catalyze the reaction between the five-carbon compound and oxygen, instead of CO2.  With one less carbon, in what’s known as photorespiration, a three and a two-carbon compound (2PG) are produced. The latter is not totally wasted because a metabolized version leaves the chloroplast, eventually goes to mitochondria and works its way back into photosynthesis in the form of carbon dioxide. But the process wastes time and energy. Worse, when there is little moisture available and flabby guard cells cause a plant’s pores to close, carbon dioxide does not enter, and the rate of wasteful photorespiration increases.


C4 plants use specialization of cells to fix carbon. This is ecologically vital because the efficiency of such plants makes them more water-efficient.

So what is the strategy that makes a C4 plant so tough? They have two types of chloroplast-bearing cells: one type(mesophyll) near the leaf surface where the oxygen-avoiding PEP carboxylase enzyme produces a four-carbon compound(oxoloacetate), and a second type(bundle-sheath) found deeper and below the surface where oxygen levels are low. There, CO2 is released from the 4-carbon compound, and it’s taken care of by a now less-distracted RuBisCo. Having more specialization requires a larger investment on the part of the C4 plant, but it pays off because it avoids the losses of photorespiration. Also, if hogweed can photosynthesize more efficiently, their pores don’t have to be as fully open as often, and less water is lost to evaporation.

Water is needed in photosynthesis to return electrons to chlorophyll molecules after they are excited. When losing electrons, water splits into oxygen and into H+. The subsequent proton gradient across chloroplast membranes then provides the energy required for ATP synthesis. Without ATP a plant’s (or animal’s) metabolism shuts down.

camSource of image

The sedum’s strategy is known as CAM photosynthesis. It is similar to C4, but it fixes its carbon dioxide at night so that its pores(stomata) can remain closed during the day, saving even more water. CAM plants have the same two carbon-fixing steps as C4 plants, but without the differentiation of cell types. Instead, they have both carbon dioxide-fixing enzymes within the same cell.  CAM plants’ PEP carboxylase is only active at night when pores open up, letting in carbon dioxide and storing it. The Rubisco is only active diurnally when pores are closed and oxygen cannot get in, avoiding photorespiration.

Note that the production of oxoloacetate from the fixing of CO₂ at night eventually leads to the production of malic acid in their vacuoles.

photo (4)

A saguaro cactus from the McDowell Sonoran Desert Reserve in Scottsdale, Arizona. Like all cacti, it uses CAM photosynthesis.

CAM stands for Crassulacean Acid Metabolism:  Crassulaceae is the plant family to which sedums belong; and acid, because of the malic acid formed.

This metabolic pathway is also found among another drought-tolerant family, the cacti. After a night of accumulating malic acid, it explains why they taste more sour in the morning  and later after converting malic acid to malate, the rise in pH explains why they become bitter-tasting in the afternoon.