where G is the gravitational constant, R is the radius of the orbit, M is the massive of the bigger object, choose the Earth, around which the smaller object orbits.Note that the formula does not rely on the mass of the smaller object.(This last truth follows indigenous the Newtonian theory, and is related to the experimental inference by Galileo that two objects of different mass dropped indigenous the same height loss to planet in the very same time.)We have the right to use this formula to calculation how fast the moon move inits orbit about the Earth. Plugging in the Earth"s fixed of M=6 x 1024 kg, the radius of the moon"s orbit the R=3.84 x 108 meters, and gravitational constant G= 6.67 x 10-11 Newton meter2 / kg2,the size of the moon"s velocity is then 1020 meter/s. This is about 2278 miles per hour. (Hey therefore if its moving so fast, 5 times fasterthan jet airplanes, why carry out jets seem to move much faster on the skies than the moon?You"ll need to answer this concern on your own..)Since the circular velocity varies inversely v the square source of R, things in a smallerorbit has quicker speed due to the fact that the gravity is stronger.The exact same calculation because that R=6578 km above the center of theEarth tells united state that a satellite must relocate at speed of 17,400 milesper hour (=7790 m/s). Therefore rockets need to move very fast.The rocket has to get up, and also then revolve to suggest in a circularorbit at the exactly speed. Yet once it is at the speed, that willstay in orbit without succeeding rocket propulsion.Since the orbital velocity the a satellite counts on the distancefrom the facility of the Earth, the furthermore out, the longer the periodof the orbit. Close to the planet the orbital duration is about 1.5 hours.If one goes the end to about 42,000 km (26,000) miles, the orbital periodis 24 hours. Hence the satellite would certainly be in GEOSYNCHRONOUS ORBIT.Imagine launching a satellite eastward above the Earth"s equatorin geosynchronous orbit: then the satellite will continue to be over the exact same spoton the planet at every times in that is orbit.

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## Open and also Closed Orbits

Orbits which close ~ above themselves prefer circular or ellipticalorbits are referred to as CLOSED ORBITS.The thing in together orbits always return to the same place in the orbitperiodically. In circular orbits, the speed of the thing remainsthe same everywhere in the orbit. In elliptical orbits the speedis quicker as the object moves to the closer part of that orbit andthen slows together the thing proceeds come the farther component of the orbit.There are likewise orbits called ESCAPE ORBITS or open up ORBITS.In these orbits the object never returns and goes off to largedistances.To watch thisImagine a cannon shooting off of a mountain on the surfaceof the earth (fig 5-13). There room 5 qualitatively various possibilities depending on how quick the cannonballtravels with respect come the one velocity initially.If it travels at the one velocity, the orbit will certainly be circular.If it travel much much less than circular velocity the cannon ball willdrop to the surface. If it travels a small bit much less than circular velocity,it will form an elliptical orbit through the cannonat APOGEE (the farthest suggest in the orbit). If the sphere travels a small bit much faster than one velocity, thenit will form an elliptical orbit through the cannon at the PERIGEE (thenearest point in the orbit).If the sphere travels in ~ the to escape VELOCITY (see below), thenthe orbit will be OPEN and will it is in a parabola.If the round travels faster than the escape VELOCITY,the orbit with be OPEN and also be a hyperbola.

## Escape Velocity

The to escape VELOCITY from an object like the planet is provided by Ves = (2GM/R)1/2where R is the radius of the launch allude for the object.When the launch suggest is ~ above the Earth"s surface, thenR would certainly be the radius that the Earth.This escape velocity is the an essential velocity an item must haveto subsequently coast to infinity once shot upward from within a gravitational field. That is, if a rocket is shot native the Earth and also consumes all of its fuel to accelerate to this velocity, climate evenafter the rocket is no longer burning fuel, it will coastto infinity and the Earth"s heaviness cannot traction the rocket back toEarth. Plugging in for the Earth"s mass, radius, and for G,we obtain 11.2 km/s for the escape velocity for things launchedfrom the Earth"s surface. This is around 25,000 miles per hour!I dare you to shot to escape. (So much for Aristotle"s worries aboutbirds flying turn off the Earth"s surface-- no chance in the visibility of gravity.)

## Weight and the Gravitational Force

We have seen the in the Universal legislation of Gravitation the an essential quantity ismass. In well-known language mass and weight are frequently used to average the samething; in fact they room related but quite various things. What wecommonly contact weight is really just the gravitational force exerted top top anobject of a certain mass. We can illustrate by selecting the earth as one of the two massesin the ahead illustration that the legislation of Gravitation: Thus, the load of things of mass m in ~ the surface ar of the earth is obtainedby multiply the mass m through the acceleration because of gravity, g, at the surfaceof the Earth. The acceleration due to gravity is roughly the product ofthe global gravitational continuous G and the fixed of the earth M, divided by the radius that the Earth, r, squared. (We i think the earth to bespherical and neglect the radius the the object relative to the radius of theEarth in this discussion.) The measure gravitational acceleration at theEarth"s surface ar is found to be around 9.8 m/second2.

## Mass and also Weight

Mass is a measure of just how much product is in one object, but weight is a measure up ofthe gravitational pressure exerted top top that product in a gravitational field;thus, mass and weight room proportional to every other, with the acceleration dueto gravity together the proportionality constant.It complies with thatmass is continuous for an object (actually this is not fairly true, however wewill save that surprise for our later conversation of the Relativity Theory), but weight relies on the place of the object.For example, if us transported the coming before object of mass m come the surface ofthe moon, the gravitational acceleration would readjust because the radius andmass that the Moon both different from those of the Earth. Thus, ours objecthas fixed mboth ~ above the surface of the Earth and also on the surface ar of the Moon, but itwill weigh much less on the surface of the Moonbecause the gravitational acceleration over there is a aspect of 6less than at the surfaceof the Earth.

## Newton"s source of Kepler"s Laws

Notice the magnitude that the one velocity of things in orbitis likewise equal to thethe one of an orbit split by the duration of the orbit.That is:

## Vcirc = (GM/R)1/2 = (2 pi R)/ p

where pi=3.1415 and also P is the orbital period.The last equality follows from just noting that the magnitudeof velocity is a measure of distance traveled per unit time.The distance traveled is the one of the orbit (= 2 pi R), and also the period P is the moment it take away to travel this distance.But look: now magically we have actually recovered Kepler"s third law!:Rearranging the latter equality we have

## (GM) P2 = 4 pi2 R3

the point being the P2 is proportional to R3. The is Kepler"s 3rdlaw, now falling directly from Newton"s theory.

## Newton"s translate of Kepler"s Laws

Because for every action there is one equaland the contrary reaction, Newton realized that in the planet-Sun system the planetdoes not orbit approximately a stationary Sun. Instead, Newton proposed that both theplanet and the sun orbited roughly the common facility of mass for the planet-Sunsystem. He climate modified Kepler"s third Law so that the massM provided is now the amount of the fixed of the sunlight plus the planet.Instead of making use of M1 and also M2 as above, let us use Ms and also Mp. Then wehave M= ms + Mp and also so

## G(Ms + Mp) P2 = 4 pi2 R3

But notification what happens in Newton"s brand-new equation if the massof the sunlight is much larger than the fixed for any kind of of the planets(which is constantly the case). Climate the sum of the 2 masses is always approximately equal to the massive of theSun and also so are back to

## G ms P2 = 4 pi2 R3

for planet-Sun systems.If we take ratios of Kepler"s 3rd Law for two different planets the sun massthen cancels from the ratio and also we room left v theoriginal form of Kepler"s third Law: Thus Kepler"s 3rd Law is roughly valid since the sun is much moremassive than any of the planets and also thereforeNewton"s correction is small.

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The data Kepler had accessibility to were not goodenough to show this small effect. However,detailed observations made after ~ Kepler show that Newton"s modified form ofKepler"s 3rd Law is in better accord v the data than Kepler"s original form.