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----------------------------- 2220 - Laws of Motion
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- The laws of motion were first set in motion by Galileo (1564-1642). Before Galileo people believed that the rate of fall of an object was proportional to its weight, cannon balls and feathers. Galileo said all objects fall at the same rate regardless of their weight, if there is no air resistance.
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- He was right but it was hard for him to prove it in those days.. No vacuums were available to get away from air resistance. The time pieces were water clocks that measured to within a fraction of an hour.
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- Galileo solved these problems by running his experiments on inclined planes. That slowed things down so he could measure them. (See footnote 1)
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- Johann Kepler came along 1581 to 1630 with a mathematical mind. He used Tycho Brahe’s observation data on the orbits of the planets to develop his three laws of motion. (See footnote 2)
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- (I) The planets orbit in ellipsis with the Sun as one focus.
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- (II) Planets sweep out equal areas in equal times. When the planet is closer to the Sun it moves faster, when it is farther away it moves slower.
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- (III) Period^2 = major axis^3, or the time for one orbit squared = the distance from the Sun cubed.
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- Next came Isaac Newton (1643-1727) who at age 23 published the law of Gravitational Force = G* M*m/d^2. The force of gravity increases as the product of the masses and decreases with the square of the distance between them. (See footnote 3)
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- Isaac Newton defined the three laws of motion using mathematics:
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- (1) A body at rest remains at rest. A body in motion remains in motion with constant velocity. Unless acted upon by an outside forces things keep moving at a constant speed forever.
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- (2) Force = mass * acceleration, or F = m*a
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- (3) Gravity Force = G* M*m/d^2.
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- G is the Gravitational Constant which was not determined until Henry Cavendish in 1798, 132 years later, performed his famous experiment.
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- He hung metal balls from wires and measured the force of gravity by the twist and torque in the wires as the balls were brought closer together. He plugged the data into Newton’s formula and calculated G, the Gravitational Constant. Then he plugged that answer back into Newton’s formula and calculated the mass of the Earth. (See footnote 4)
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- Mass of Earth = 6,000,000,000,000,000,000,000 tons
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- Density of Earth = 5.5 times density of water.
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- G = .00000000006674215 meters^3/kilograms*seconds^2
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- G = 1 Astronomical Unit / Year
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- Newton’s math derived the velocity needed to sustain a circular orbit
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- Circular Velocity^2 = Gravitational Constant * Mass / radius = G*M/r
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- Faster or slower velocities than this create ellipses larger or smaller than this circle.
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- Escape velocity is the minimum velocity needed to have a parabolic orbit and to escaped the gravitational pull of the Earth.
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- Escape Velocity^2 = 2*G*M/r
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- At the Earth’s surface:
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- Circular Velocity^2 = 17,700 miles/hour at Earth’s surface
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- Escape Velocity^2 = 25,041 miles/hour at Earth’s surface.
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- Two objects orbit about their center of mass. Strictly speaking the Earth does not orbit the Sun. The Earth and the Sun orbit each other about their mutual center of masses. In our case, since the Sun is so much larger, the mutual center of mass lies inside the Sun, but not at its center.
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- The semi-major axis of the Earth-Sun ellipse = R sun + R Earth
= 1 AU = 1.5*10^8 meters.
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- R Sun / R Earth = Mass Earth / Mass Sun
= 5.9742*10^24 Kilograms / 1.9891*10^30 Kilograms = 3*10^-6
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- R Sun = 3*10^-6- 4.5*10^2 = 450 Km = 280 Miles
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- R Sun = 280 miles
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- Radius of the Sun = 435,000 miles
- So the center of Earth-Sun mass is only .06% of the distance from the center of the Sun.
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- Newton’s calculations on Kepler’s II law of equal areas in equal times was translated into orbital motion is the conservation of angular momentum.
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- Which is the same as equal areas in equal times.
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- Angular momentum = mass * velocity / radius = m*v/r = constant
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- If the radius changes then the velocity must change to compensate to keep the momentum constant. If the distance becomes shorter the velocity increases. If the distance is greater the velocity decreases to keep the angular momentum constant.
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- Newton’s calculations for the III law, period^2 = radius^3 became:
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- Period^2 = 4*pi^2 * radius^3 / G* (m1 * m2)
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- G works out to be unity if the period is measured in years and the radius in Astronomical units. The AU is the average distance for Earth to Sun = 1.496*10^11 meters.
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- Period of Earth is one year = 3.156*10^8 seconds
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- Radius = 1 AU = 1.496*10^11 meters.
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- G = 6.674215 m^3/kg*sec^2
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- Mass Sun = 1.99*10^30 kilograms
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- Mass Sun = 2,000,000,000,000,000,000,000,000,000 kilograms
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- Mass Sun is 330,000 times the mass of the Earth
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- We can measure the mass of Jupiter form the orbit of one of its Galilean moons.
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- Mass of Jupiter = 300 Mass of Earth
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- Using Newton’s math we can calculate the orbit of Halley’s comet. William Herschel discovered the planet Uranus in 1781 orbiting beyond Saturn using this equation.
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- Data collected in 1840 showed astronomers that another mass, or planet, is disturbing the orbit of Uranus. Using this data astronomers predicted the existence of another , 8th planet beyond Uranus. On September 23, 1845 Neptune was discovered just where the calculations said it should be.
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- (1) Galileo built the first telescope, power 32. He sent one to Johan Kepler. Galileo discovered the 4 moons of Jupiter and claimed that this supported Copernicus’s claim that the Earth and planets orbited the Sun. Everyone else thought the Earth was the center of the Universe. Some politicians still do believe it. At the time his ideas were so contrary to the Church Authority beliefs that Galileo was charged with heresy and put under house arrest until he died in 1642.
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- (2) Johann Kepler graduated from the University of Tubingen. He studied Tycho Brahe’s data on the planets. In particular Mars, he showed that an ellipse fit the data for the Mars orbit, exactly. Kepler have 13 children so he was up most of the night anyway
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- (3) Isaac Newton born January 4, 1643 in England has been judged to be one of the greatest intellects who ever lived. At age 17 he entered Cambridge University and graduated at age 22. He made the connection between the apple falling from the tree and the moon falling around the Earth. He theorized that the gravitational force on the Moon fell off according to the square of the distance from the center of the Earth to the center of the Moon.
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- The same year he discovered the spectrum of sunlight. Using a prism he could split sunlight into the rainbow of colors and through another prism combine it back into white light again. At age 27 he became a math professor and invented the math of calculus. He proposed the light was a stream of particles.
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- He invented the parabolic mirror and the reflecting telescope. He confirmed that the planets orbited the Sun in ellipsis. He published the three laws of motion and lived to age 84, where he is buried in Westminster Abbey with the inscription: “If I have seen further than other men, it is because I stood on the shoulders of giants.”
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- (4) Henry Cavendish (1731-1810) was a Cambridge physicists who truly matched the mold of the mad scientists. He was excessively shy and absent minded. He rarely spoke. He had a total fear of women. If he set eyes on the women that kept the house and fed him he immediately fired them.
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- He inherited lots of money so he could dedicate himself to one thing, curiosity in science. He did not publish his findings. It was left to Maxwell 100 years later to rescue his notes and publish the results of his works. He discovered hydrogen, burned it and created water. He discovered argon. He measured the Gravitational Constant and calculated the mass of the Earth.
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- (5) See Review #393 Kepler says: “Cube of Radius = Square of Period”
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- December 28, 2018 559 10-12-05
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