Posts Tagged ‘mass of the Earth’

The Sun’s Gravitational Force

Monday, August 3rd, 2015

      Last time we discovered that Earth zips around the sun at the mind boggling speed of 29,680 meters per second.   This is the final bit of information required to calculate Fg, the gravitational force exerted upon Earth by its sun, as set out in Newton’s equation on the subject and derived from his Second Law of Motion.   We’ll calculate that quantity today.

      Newton’s formula that we’ll be working with is,

Fg = [m ×  v2] ÷  r

where Earth’s speed, or orbital velocity, is the v in the equation.   The other variables, m and r, have previously been determined in this blog series.   For a refresher see Centripital Force Makes the Earth Go Round, What is Earth’s Mass, and Calculating the Distance to the Sun.   Earth’s mass, m, is valued at 5.96 × 1024 kilograms, while r is Johannes Kepler’s astronomical unit, equal to about 149,000,000,000 meters.

      Inserting these numerical values into Newton’s equation to determine the sun’s gravitational force acting upon Earth we arrive at,

Fg = [(5.96 × 1024 kilograms) × (29,680 meters per second)2] ÷ 149,000,000,000 meters

Fg = 3.52 × 1022 kilogram • meter per second2

      This metric unit of force, kilogram • meter per second2, represents kilograms multiplied by meters, and their product divided by seconds squared.   It’s known in scientific circles as the Newton, in honor of Sir Isaac Newton, widely recognized as one of the greatest scientists of all time and a key figure in the scientific revolution that began over three centuries ago.   Therefore the sun’s gravitational force acting upon Earth is typically referred to as,

Fg = 3.52 × 1022 Newtons

      Here in the US where we like to use English units such as feet and pounds, the Newton is said to equal 0.225 pounds of force.   Therefore in English units the sun’s gravitational force is expressed as,

Fg = (3.52 × 1022 Newtons) × (0.225 pounds of force per Newton)

Fg = 7.93 × 1021 pounds

      That’s scientific notation for 7,930,000,000,000,000,000,000 pounds!   That’s the amount of force exerted by the sun’s gravitational pull on Earth.   Seems about right — right?

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      Now that we know Fg, we have everything we need to calculate the mass of the sun, which in turn enables us to determine the mass and gravity of other planets in our solar system.   We’ll calculate the sun’s mass next time.


Centripetal Force

Friday, July 3rd, 2015

      Have you ever wondered how Earth keeps its steady orbit around its life sustaining sun, or what prevents it from breaking away and flying off willy-nilly into the universe?   It’s more than just simple gravity, it’s the physics behind centripetal force, the topic we’ll be exploring today.

      We’ve been working our way towards a full discussion on gravity in this long blog series, navigating subjects such as the behavior of falling objects, the acceleration of gravity, the masses of Earth and the sun, and the optical measurement of cosmic distances.   We’ve now come full circle from my opening blog on the subject, Gravity and the Mass of the Sun.

      In that blog an equation was introduced as a means to calculate the mass of the sun, and in that equation is the variable we’ll be working towards solving today, Fg, the sun’s gravitational force upon the Earth.   Here again is that equation,

M = (Fg × r2) ÷ (m × G)

      Gravity, mass, and distance all come into play in forming the structure of our universe, and the variables in this equation reflect that:  M, the mass of the sun, r the distance between Earth and the sun, m the Earth’s mass, and G the universal gravitational constant.   With the exception of Fg, all variables in this equation have already been solved for in previous blogs in this series.   For a refresher go to, Calculating the Distance to the Sun, What is Earth’s Mass? and Newton’s Law of Gravitation and the Universal Gravitational Constant.

      As there is no direct means to measure the cosmic quantity, Fg, we’re left to an indirect method for its computation.   The indirect method is based on the phenomenon of centripetal force, Fc something most children become acquainted with when they experience the thrill of twirling an object attached to a string, say a rubber ball, above their heads.   See Figure 1.

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Figure 1

      As the ball is twirled, the string becomes taut.   The energy exerted upon it by the child’s hand, coupled with the ball’s mass and traveling speed/velocity, v, make the ball want to move off in a straight trajectory into space, like a launched projectile.   But the string it’s attached to prevents it from doing so, forcing the ball to instead travel a circular path around the center point of rotation.   The taut string and the ball’s circular path are evidence of centripetal force, Fc, at work.

      Next time we’ll employ Isaac Newton’s Second Law of Motion to the centripetal force phenomenon to see how the sun’s gravitational force keeps Earth in a stable circular orbit around the sun.


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!

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      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.

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      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.