## Posts Tagged ‘Henry Cavendish’

### Gravity and the Mass of the Sun

Friday, December 12th, 2014
 As a young school boy I found it hard to believe that scientists were able to compute the mass of our sun.  After all, a galactic-sized measuring device does not exist.  But where there’s a will, there’s a way, and by the 18th Century scientists had it all figured out, thanks to the work of others before them.  Newton’s two formulas concerning gravity were key to later scientific discoveries, and we’ll be working with them again today to derive a third formula, bringing us a step closer to determining our sun’s mass.       Newton’s Second Law of Motion allows us to compute the force of gravity, Fg, acting upon the Earth, which has a mass of m. It is, Fg = m × g                     (1)       Newton’s Universal Law of Gravitation allows us to solve for g, the sun’s acceleration of gravity value, g = (G × M) ÷  r2              (2) where, M is the mass of the sun, r is the distance between the sun and Earth, and G is the universal gravitational constant.       You will note that g is a common factor between the two equations, and we’ll use that fact to combine them.  We’ll do so by substituting the right side of equation (2) for the g in equation (1) to get, Fg = m × [(G × M) ÷  r2] then, using algebra to rearrange terms, we’ll set up the combined equation to solve for M, the sun’s mass: M = (Fg × r2) ÷ (m × G)           (3)       At this point in the process we know some values for factors in equation (3), but not others.  Thanks to Henry Cavendish’s work we know the value of m, the Earth’s mass, and G, the universal gravitational constant.  What we don’t yet know is Earth’s distance to the sun, r, and the gravitational attractive force, Fg, that exists between them.       Next time we’ll introduce some key scientists whose work contributed to a method for computing the distance of our planet Earth to its sun. _______________________________________

### The Force of Gravity

Thursday, November 20th, 2014
 Last time we saw how Henry Cavendish built upon the work of scientists before him to calculate Earth’s mass and its acceleration of gravity factor, as well as the universal gravitational constant.   These values, together with the force of gravity value, Fg, which we’ll introduce today, moved scientists one step further towards being able to discover the mass and gravity of any heavenly body in the universe.       According to Newton’s Second Law of Motion, the force of gravity, Fg, acting upon any object is equal to the object’s mass, m, times the acceleration of gravity factor, g, or, Fg = m × g       So what is Fg?  It’s a force at play way up there, in the outer reaches of the galaxy, as well as back home.   It keeps the moon in orbit around the Earth and the Earth orbiting around the sun.   In the same way, Fg keeps us anchored to Earth, and if we were to calculate it, it would be calculated as the force of our body’s mass under the influence of Earth’s gravity.   It’s common to refer to this force as weight, but it’s not quite so simple.       Using the metric system, the unit of measurement most often used for scientific analyses, weight is determined by multiplying our body’s mass in kilograms by the Earth’s acceleration of gravity factor of 9.8 meters per second per second, or 9.8 meters per second squared.       For example, suppose your mass is 100 kilograms.   Your weight on Earth would be: Weight = Fg = m × g = (100 kg) × (9.8 m/sec2) = 980 kg · m/sec2 = 980 Newtons      Newtons?   That’s right.   It’s easier than saying kilogram · meter per second per second.   It’s also a way to pay homage to the man himself.       In the English system of measurement things are perhaps even more confusing.   Your weight is found by multiplying the mass of your body measured in slugs by the Earth’s acceleration of gravity factor of 32 feet per second per second.   Slugs is British English speak for pounds · second squared per foot.   We normally refer to weight in units of  pounds, and in engineering circles it’s pounds force.       For example, suppose your mass is 6 slugs, or 6 pounds · second squared per foot.   Your weight on Earth would be: Weight = Fg = m × g = (6 Lbs · sec2/ft) × (32.2 ft/sec2)= 193.2 Lbs       To avoid any confusion, you could just step on the bathroom scale.       Next time we’ll see how the force of gravity is influenced by an inverse proportionality phenomenon. _______________________________________

### What is Earth’s Mass?

Friday, November 7th, 2014
 Last time we learned how Henry Cavendish used Christiaan Huygens’ work with pendulums to determine the value of g, the acceleration of gravity factor for Earth, to be 32.3 ft/sec2, or 9.8 m/sec2.    From there Cavendish was able to go on and arrive at values for other factors in Isaac Newton’s gravity formula, namely G, the universal gravitational constant, and M, Earth’s mass.   Today, we’ll discuss how Cavendish was able to calculate the Earth’s mass.       Newton’s formula for gravity, once again, is: M = (g × R2) ÷ G where M stands for the mass of the heavenly body being quantified.   For our case today M will represent the mass of Earth, which was originally quantified in slugs, a British unit of measurement.   Today the measurement unit of choice in most parts of the world is the kilogram, which is the metric equivalent of a slug.       With regard to the other variables in Newton’s gravity formula, namely, R and G, their values had previously been determined.   Eratosthenes’ measurement of shadows cast by the sun on Earth’s surface had revealed Earth’s radius, R, to be 6,371 kilometers, or 6,371,000 meters.   And Cavendish’s experiments led him to conclude that the universal gravitational constant, G, was 6.67 × 10-11 cubic meters per kilogram-second squared.   Plugging these values into Newton’s equation, we calculate Earth’s mass to be: M = ((9.8 m/sec2) × (6,371,000 m)2) ÷ (6.67 × 10-11 m3/kg-sec2) M = 5.96 × 1024 kilograms       Incidentally, 5.96 × 1024 is scientific notation, or mathematical shorthand, for the number 5,960,000,000,000,000,000,000,000.   That’s a whole lot of zeros!       Calculating the mass of Earth was an impressive accomplishment.   Now that its value was known, scientists would be able to calculate the mass and acceleration of gravity for any heavenly body in the universe.   We’ll see how that’s done next time. _______________________________________

### Huygens’ Use of Pendulums

Tuesday, October 21st, 2014
 Last time we learned that Henry Cavendish determined a value for G, the universal gravitational constant, fast on his way to determining a quantity he was determined to find, the Earth’s mass.   Today we’ll see how the previous work of Christiaan Huygens, a contemporary of Isaac Newton’s, helped him get there.       First Cavendish used algebra to rearrange terms in Newton’s gravitational formula so as to solve for M, Earth’s mass.   Rearranged, Newton’s formula becomes, M = (g × R2) ÷  G       But in order to solve for M, Cavendish first needed to know Earth’s acceleration of gravity, g.   To aid him in this calculation he referred back to the work of Christiaan Huygens, a Dutch mathematician from Newton’s time.       Huygens was eager to devise a formula capable of predicting clock pendulums’ motions on ships, his goal being to invent a timepiece accurate enough to make navigating ships easier.   He hypothesized that a key factor in predicting a pendulum’s movement was an unknown constant, the acceleration of gravity factor, g, which Newton had previously posited existed.   Through meticulous observation, Huygens came to realize that the time it took for pendulums to complete one swing back and forth was dependent not only on the length of the pendulum, but also this unknown quantity.       In order for Huygens’ computations to work, the value of g had to be a constant, meaning, its value could not vary between computations; g‘s value was in fact a fudge factor, a phantom he would assign a specific numerical value.   Huygens’ needed it in order to make his hypothesis work, a practice commonly use by scientists, even today.   Determining a value for g would allow Huygens to successfully relate the length of the pendulum to the timing of its swing and to create a mathematical relationship between them.       Huygens ultimately determined g’s value to be a whopping 32.2 feet per second per second, or 32.2 ft/sec2.    We’ll see how he did it next time. _______________________________________

### How Big is the Earth?

Wednesday, October 8th, 2014