Universal Gravitation

Kepler’s Three Laws of Planetary Motion Tycho Brahe ( ) – Danish astronomer who dedicated much of his life to accurately collecting astronomical data Built the finest observatory at the time on the island of Hven and charted the movements of the planets and stars for 20 years

Kepler’s Three Laws of Planetary Motion In 1597 Johannes Kepler( ) became an assistant of Brahe in Prague Kepler attempted to use Geometry and Mathematics to describe a heliocentric (sun-centered) system that would agree with Brahe’s Data Result: Kepler’s Three Laws of Planetary Motion

Kepler’s Three Laws of Planetary Motion 1.The path of the planets around the sun are ellipses with the sun at one of the focus points Consequence: The planets traveling in their orbits are sometimes closer to the sun and other times further away Note: The orbits are only slightly elliptical. They are close to circular.

Kepler’s Three Laws of Planetary Motion 2.The planets sweep out equal areas in equal time intervals no matter how close or far away a planet is from the sun Consequence: Planets speed up as they move closer to the sun and slow down as they move further away

Kepler’s Three Laws of Planetary Motion 3.The ratio of the squares of the Periods of any two planets going around the sun is equal to the ratio of the cubes of their Average Distances away from the sun. Consequence: Planets that are farther away from the sun have a longer Period of Revolution

Equation for Kepler’s Third Law Kepler’s 3 rd Law can be written in an equation form. (T a ) 2 = (R a ) 3 (T b ) 2 (R b ) 3 Kepler’s 3 rd law applies to any two objects going around a third central object! (2 planets around the sun or 2 moons around a planet)

Astronomical Units (A.U.) The average distance between the Earth and the Sun is d E-S = 1.5 x m To avoid using large numbers and exponents in calculations scientists have developed the Astronomical Unit (A.U.) 1 A.U. = 1.5 x m –d E-S = 1A.U.

History of Universal Gravitation Greeks (Aristotle and Ptolemy) – Geocentric Model Copernicus – Heliocentric Model Galileo – Discoveries with his telescope Kepler – Three Laws of Planetary Motion

History of Universal Gravitation After Galileo and Kepler, the heliocentric model gained acceptance but scientists still did not know what causes the motion of the planets! The stage was set in 1666 for Issac Newton to use his concept of a force and his Three Laws of Motion to explain the force behind planetary motion

Universal Gravitation Newton named this force Gravity He suggested Gravity is a force of attraction between any two masses and that each mass pulls on the other with an equal and opposite force

Universal Gravitation The Force of Gravity is directly proportional to the mass of each object F  m The Force of Gravity is inversely proportional to the square of the distance between the center of the masses F  1/d 2 (Inverse Square Law)

Universal Gravitation Equation Newton used his Law of Gravity (Universal Gravitation) to make the following equation: F g = G m 1 m 2 d 2 Where “G” is the Universal Gravitational Constant! G = 6.67 x Nm 2 /kg 2

Universal Gravitation Equation Note: In using the Universal Gravitation Equation, One mass(m 1 ) doubles - F g is doubled Both masses(m 1, m 2 ) double– F g is quadrupled (multiplied by 4) Distance(d) doubles – F g is divided by 4 (2 2 = 4) Note: d has to be measured from center to center! (Not center to SURFACE!)

Useful Information for Universal Gravitation Calculations G = 6.67 x Nm 2 /kg 2 M E = 5.96 x kg (Mass of Earth) R E = 6.37 x 10 6 m (Radius of Earth) M S = 2.0 x kg (Mass of Sun) d E-S = 1.5 x m = 1A.U. (Distance Earth to Sun) d E-M = 3.9 x 10 8 m (Distance Earth to Moon) Table 7-1 Page 173

Using Universal Gravitation Calculate the Period of the Moon – Newton’s Proof of the validity of his Universal Gravitational Equation Universal Gravitation to Kepler’s 3 rd law Weighing the Earth

Henry Cavendish ( ) was the first to develop equipment sensitive enough to detect the gravitational force between two masses on the earth. From his experiments, he verified an actual experimental value for the Universal Gravitational Constant G = 6.67 x Nm 2 /kg 2

Universal Gravitation and Acceleration Due to Gravity Show why g = -9.8 m/s 2 What affects the acceleration due to gravity (g)? Answer: 1) Mass of the planet. 2) Distance from the center of the object to the center of the planet

Universal Gravitation and Acceleration Due to Gravity The acceleration due to gravity (g) changes if you go to another planet or moon or as you move away from the surface of Earth g = 9.8 m/s 2 at the surface of the Earth g will be greater on a larger planet (Jupiter g J = 25 m/s 2 ) g will be smaller on a smaller planet (Mercury g M = 3.78 m/s 2 )

Universal Gravitation and Acceleration Due to Gravity The Acceleration Due to Gravity Away from the Surface of the Earth (g / ) can be found with the Equation: g / = g (R E ) 2 (d) 2 Note: d has to be measured from center to center! (Not center to SURFACE!)

Satellites After Newton formulated his Theory of Universal Gravitation, A question was presented to him regarding his theory. If Gravity is the same force that pulls an apple to the ground from a tree and also keeps the Moon in orbit, what keeps the moon from crashing into the Earth like an apple?

Newton’s Thought Experiment Newton answered the question with a thought experiment (Because you really cannot do this experiment with air resistance). Answer: The Moon is falling, but since it has a horizontal speed tangent to the orbital path, the Moon misses the Earth as it falls and is falling around the Earth!

Newton’s Thought Experiment Newton’s thought experiment explains how any satellite stays in orbit! Satellites are difficult to put into orbit but easy to get back to Earth. Just slow the satellite down! Note: A satellite is any object (Natural or Man-made) traveling around another object

Satellites The Moon is a satellite of the Earth! The Earth is a satellite of the Sun! Jupiter’s Moons are satellites of Jupiter! We put up man-made satellites around the Earth and other planets! Most of the man-made satellites going around the Earth are Geosynchronous (have a period, T = 24 hours)

Satellites There are 3 Equations we can derive for Satellites: 1.v S = 2  r/t or 2  d/t 2.v S = [Square Root (Gm E /d)] 3.v S = R E [Square Root (g/d)] Note: d has to be measured from center to center! (Not center to SURFACE!)

Weight and Weightlessness Why do the astronauts in the space shuttle experience weightlessness (zero gravity)? Microgravity – Is the situation when objects experience the illusion of weightlessness because they are all falling at the same rate Astronauts orbiting the earth are falling around the earth at a rate of “ g / ”!

Gravitational Field Gravity is not a contact force. Therefore, masses produce a Gravitational Field! Michael Faraday first suggested field strength with magnets. It was later applied to gravity g measures the gravitational field strength acting on an object g = F g /m

Two Types of Masses Inertial Mass (m i ) – The mass related to the inertia that an object possess. Calculated with Newton’s Second Law. F = m i a Gravitational Mass (m g ) – The mass related to the gravitational force between the earth and the object. Measured with a balance. Does m i = m g ???? Newton said yes! Principle of Equivalence

Einstein’s Theory of Gravity Einstein looked at gravity differently! Einstein suggested that instead of gravity being a force, he said that gravity is an effect of space itself. Mass puts a dent in space and tries to pull in everything in to this dent. (Mattress Example) Suggested how large masses can deflect light and the existence of Black Holes!