Key Ideas:Law that Falling bodies (Galileo) all falling bodies experience the exact same gravitational accelerationLaw of global Gravitation (Newton) gravity is one attractive force between all pairs of massive objects Gravitational pressure is proportional to the masses, and inversely proportional to the square that the distance between them.NOTE: This and the adhering to lecture are probably the most mathematical the allthe lectures that will be offered in this class. Ns encourage you every toread these notes in development and try to monitor the debates in them. Inwill do it much easier to monitor along throughout lecture.
The regulation of falling BodiesPrior come his telescopic work, Galileo perform fundamentalresearch top top motion.Explored the rate of falling body by dropping differentweights, or sliding them under inclined planes.Law of fallout’s BodiesIn the lack of air, heavy objects and light objects fall at the same, constant rate the acceleration.
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Universal common GravitationIsaac Newton, in his Principia, formulated the legislation ofUniversal common Gravitation:Gravity is an Attractive
Gravitational ForceThe pressure of heaviness between any type of two objects depends just upon:The masses of the two objects:More massive objects exert a stronger the gravitational force.The distance in between them:The force gets stronger together the 2 objects move closer together.The force gets weaker together the two objects relocate farther apart.It go not count on the shapes, colors, or compositions of theobjects.
The law of universal GravitationThe force of gravitational attraction between any kind of two substantial bodies is proportional to their masses and also inversely proportional come the square the the distance in between their centers.The pressure of heaviness is an example of an train station Square law ForceStated mathematically, the force of gravity in between two massivebodies is:
Where:F = force because of gravity.M1 = fixed of the first bodyM2 = massive of the second bodyd = distance in between their centers.G = Gravitational pressure Constant
The Gravitational pressure ConstantThe pressure constant, G, is a number which provides the dimension of thegravitational coupling between two huge objects.G is an extremely small, in metric units:G=6.7x10-11 Newtons meter2 / kilogram2The Newton is the metric unit that force:4.41 Newtons = 1 poundG needs to be measured experimentally18.1.
The loss of an Apple.Stand on the Earth and also drop an apple.What is the pressure of the earth on the apple?F = GMearth Mapple/Rearth2What is the apple"s acceleration (Newton"s 2nd Law of Motion):aapple = F/Mapple = GMearth/Rearth2 = 9.8 meters/sec2Note that the fixed of the to apologize (Mapple) had separated out ofthe equation. This way that the acceleration as result of gravity isindependent the the massive of the apple, similar to Galileo had shownearlier.
Equal and Opposite ReactionsBut, Newton"s 3rd Law of motion states the all forces come in equal however opposite pairsWhat pressure does the the apple apply in return upon the Earth?F = GMearth Mapple/Rearth2How lot does the planet accelerate towards the apple?aearth = F/Mearth = GMapple/Rearth2This can be rewritten to offer the acceleration of the planet in termsof the acceleration of the apple in the direction of the earth asaearth = aapple x (Mapple/Mearth)where aapple=9.8 meters/sec2, and the ratio of the fixed of the apple to the mass of the planet is very little number.For a typical 200g apple, this works out come be about 10-25 meters/sec2, a really tiny acceleration.
The mass of the EarthWe can straight measure the acceleration of gravity at the surface ofthe earth by dropping objects and also timing their loss (e.g., favor was doneby Galileo). Us finda = 9.8 meters/sec2We can likewise measure the radius that the earth using geometry (Eratosthenes):Rearth=6378 kilometers = 6,378,000 metersCombining these together using Newton"s formula for the GravitationalForce enables us to estimate the mass of the Earth, as follows:
This is an example of among the powerful implications of Newton"sLaw of Gravity: It gives us a way to usage the motions of objects underthe influence of their common gravitation to measure up the masses ofplanets, stars, galaxies, etc.
The Orbit of the MoonFalling apples room one thing, yet what around the Moon?What keeps the Moon in orbit roughly the Earth?The legislation of Inertia (Newton"sFirst legislation of Motion) predicts:If there were no gravitational force acting in between the Moon and the Earth, the Moon would take trip in a straight line at a consistent speed.But, of food the Moon really moves follow me a bent path:It is deflected from a straight-line course by the force the gravity.This reasons the Moon to autumn a little bit towards the planet at the sametime at it moves to one side.
The autumn of the MoonHow far does the Moon fall approximately the planet in 1 second?Newton computed this. In stimulate to continue to be in the orbit, the Moon must fall by 0.00136 meter (about 1.4 mm) every second.Call this quantity xmoon, the deflection that the orbiting Moon in 1 second.How far does one apple autumn on the Earth throughout the very first second?Newton also knew this (he might measure that directly), the falls4.9 meters in the an initial 1 second.Call this quantity xapple, the deflection of afalling apologize in the 1 2nd of motion.Newton additionally knew that: Moon is around 60 planet Radii native the Earth.Summarizing the numbers:The Moon: distance that the Moon drops towards planet in 1 second: xmoon = 0.00136 meters The street of the Moon native the facility of the Earth: dmoon = 60 Rearth Acceleration of the Moon: amoon = GMearth/dmoon2 = GMearth / (60Rearth)2The Apple: distance the an Apple falls on planet in 1 second: xapple = 4.9 meters The street of the Apple native the center of the Earth: dapple = 1 Rearth Acceleration of the Apple: aapple = GMearth/dapple2 = GMearth/Rearth2The ratio of the deflections of the Apple and the Moon in 1 2nd is ratio of their accelerations:Putting every the details we have together, we acquire the following:
This predicts the the deflection the the Moon in 1 second necessary tokeep that in orbit roughly the earth should be 1/3600th thedeflection of an apple throughout the an initial second of its fall to the Earth.Observations vs. PredictionIs this right?Previously we found from observations that the deflections the theMoon and apple in 1 2nd are:xmoon = 0.00136 meter xapple = 4.9 metersGravity predicts thatxapple/3600 = 4.9 meters/3600 = 0.00136 meters!!The commitment is essentially perfect!This demonstrates that the same law of gravity applies to both theapple and the Moon! Both feeling the gravity of the earth in the form of aforce the gets weaker together the square that their distance from the centerof the Earth.
So why does the Moon orbit the Earth?If the Moon is falling a little towards the Earth, similar to an appledropped on the surface, why does the Moon travel around the planet in anorbit instead of falling onto it?The way to price this question is to very first consider what wouldhappen if there was no heaviness acting:Question:How far would the Moon take trip in a straight line in 1 2nd if there to be no heaviness acting?Answer: About 1000 meters.At the exact same time, the Moon"s movement along this straight-line pathwould likewise cause that to relocate away native the Earth.
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Question: How far away native the earth would the Moon move in 1second if no gravity were acting?Answer:About 0.00136 meters!In ring numbers, the quantity the Moon drops towards the earth due togravity is just enough to counter the straight-line route it would certainly take ifgravity were not acting to direction it. This balance effectively closesthe loop.We have because of this reached a frighten conclusion:The Moon is yes, really perpetually fall around the Earth!This is a entirely different means of looking in ~ an "orbit" underthe affect of gravity.While at first sight the autumn of an apple and also the orbit the the Moonappear to be two totally different phenomena, viewed in light ofNewton"s regulations of motion, they space in fact different manifestations ofthe same thing! The fall of the Moon about the earth is the very same kindof activity as the autumn of an apple to the Earth. Both are defined bythe exact same three laws of motion, and both feel a gravitational forcedescribed through the same, universal force law.