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# UA PTYS 206 - Apples and Planets

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Apples and PlanetsList of SymbolsNewton’s LawsNewton and GravityFalling Apples and Orbiting PlanetsNewton’s cannonballSlide 7Gravitational Force: UnitsWhat are the units of G?Newton’s Law of GravityFalling Apples: Gravity on EarthNewton Explains GalileoPlanetary motion is more complicated, but governed by the same laws. First, we need to consider the acceleration of orbiting bodiesCircular AccelerationReal Life Example A Circular Race TrackOrbiting Planets ContinuedPowerPoint PresentationStep 1: Calculate the VelocityStep 2: The Velocity is related to the semi-major axis and periodStep 3: Relate the Period to the Orbital RadiusHow Does This Relate to Kepler’s Third Law?Newton’s form of Kepler’s Third LawDo Newton and Kepler Agree?Using Newton’s Form of Kepler’s Third Law: Example 1Using Newton’s Form of Kepler’s Third Law: Example 2Using Newton’s Form of Kepler’s Third Law: Example 3Using Newton’s Form of Kepler’s Third Law: Example 4Apples and PlanetsApples and PlanetsPTYS206-228 Feb 2008QuickTime™ and a decompressorare needed to see this picture.QuickTime™ and a decompressorare needed to see this picture.List of SymbolsList of Symbols•F, force•a, acceleration (not semi-major axis in this lecture)•v, velocity•M, mass of Sun•m, mass of planet•d, general distance•r,radius of circle, semi-major axis of orbit•R, radius of EarthNewton’s LawsNewton devised a uniform and systematic method for describing motion, which we today refer to as the Science of Mechanics. It remains the basic description of motion, requiring correction only at very high velocities and very small distances.Newton summarized his theory in 3 laws:1. An object remains at rest or continues in uniform motion unless acted upon by a force.2. Force is equal to mass x acceleration (F=ma)3. For every action there is an equal and opposite reaction.Newton and GravityLink for animationCambridge was closed because of the Plague. As the story goes, Newton was sitting under the apple tree outside his farmhouse (shown right) and while watching the apples fall he realized that the force that made the apples fall also made the planets orbit the sun. Using his newly invented Calculus, Newton was able to show that Kepler’s 3 laws of planetary motion followed directly from this hypothesis.Falling Apples and Orbiting PlanetsFalling Apples and Orbiting PlanetsQuickTime™ and a decompressorare needed to see this picture.SplatQuickTime™ and a decompressorare needed to see this picture.What do these have in common?Newton’s cannonball From PrincipiaApples and PlanetsApples and PlanetsWe will know analyze the motion of terrestrial falling bodies and orbiting planets in more detail. We will analyze both phenomenon in the same way and show that Newton’s theory explains both. The plan is to combine Newton’s second law with Newton’s law of gravitation to determine the acceleration.The interesting thing here is that we are applying laws determined for motion on Earth to the motion of heavenly bodies. What an audacious idea!Gravitational Force: UnitsGravitational Force: UnitsAccording to Newton’s 2nd law, Force=mass x accelerationThe units must also match. Units of mass = kilogramsUnits of acceleration = meters/sec2Unit of force must be kilograms-meters/sec2 = kg m s-2 (shorthand)We define a new unit to make notation more simple. Let’s call it a Newton. From the definition we can see that1 Newton = 1 kg m s-2 From now on we measure force in Newtons.What are the units of G?What are the units of G?Newton’s law of gravitationF = GMm/d2Let’s solve for G (multiply by d2, divide by Mm)G = Fd2/MmExamine the unitsFd2/Mm has units of N m2/kg2 or N m2 kg-2 Or, expressing Newtons in kg, m, and s (1 N = 1 kg m s-2)Fd2/Mm has units of N m2 kg-2 = (kg ms-2)m2 kg-2= m3 s-2 kg-1G has units of m3 s-2 kg-1Numerically, G = 6.6710-11 m3 s-2 kg-1Newton’s Law of GravityNewton’s Law of Gravity•All bodies exert a gravitational force on each other.•The force is proportional to the product of their masses and inversely proportional to the square of their separation.F = GMm/d2 where m is mass of one object, M is the mass of the other, and d is their separation.•G is known as the constant of universal gravitation.Newton’s Second LawNewton’s Second LawForce = mass x accelerationF = maFalling Apples: Gravity on EarthFalling Apples: Gravity on Earth F = m a = G m M / R2F = m a = G m M / R2 (cancel the m’s) a = G M / R2 where: G = 6.67x10-11 m3kg-1s-2 M = 5.97x1024 kg On Earth’s surface: R = 6371 kmThus: a = G M / R2 = 9.82 m s-210 m s-2 a on Earth is sometimes called g.•QuickTime™ and a decompressorare needed to see this picture.The separation, d, is the distance between the centers of the objects.Newton Explains GalileoThe acceleration does not depend on m!Bodies fall at the same rate regardless of mass.Newton’s 2nd Law: F = maNewton’s law of gravity: F = GMm/d2The separation d is the distance between the falling body and the center of the Earth d=RF = GMm/R2Set forces equal ma = GMm/R2Cancel m on both sides of the equationa = GM/R2Planetary motion is more complicated, but governed by the same laws.First, we need to consider the acceleration of orbiting bodiesCircular AccelerationCircular AccelerationAcceleration is any change in speed or direction of motion. Circular motion is accelerated motion because direction is changing. For circular motion: a = v2/rReal Life ExampleReal Life ExampleA Circular Race TrackA Circular Race TrackQuickTime™ and a decompressorare needed to see this picture.AccelerationOrbiting Planets ContinuedOrbiting Planets ContinuedSo, orbiting planets are accelerating. This must be caused by a force. Let’s assume that the force is gravity. We should be able to calculate the force and acceleration using Newton’s second law and Newton’s law of gravity.QuickTime™ and a decompressorare needed to see this picture.Orbits come in a varietyof shapes (eccentricities).In order to keep the mathsimple, we will considerin this lecture only circularorbits. All of our results also apply to ellipticalorbits, but we will not derive them that way.Step 1: Calculate the VelocityStep 1: Calculate the VelocityWe take as given that acceleration and velocity in circular motion are related bya = v2/rAccording to Newton’s 2nd lawF = ma = mv2/rAccording to Newton’s law of gravityF = GMm/r2Equating the expressions for force we

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