Looking for a Tutor Near You?

Post Learning Requirement »
x

Choose Country Code

x

Direction

x

Ask a Question

x

x
x
For better view click here
x
Hire a Tutor
DISCOVER

Gravitation

Loading...

Published in: Mathematics
3,792 Views

Physics of ICSE and CBSE board for Class 9 and 10 students.

Aritra D / Kolkata

20 years of teaching experience

Qualification: M.A (Rabindra Bharati University - [RBU], Kolkata - 2005)

Teaches: English, Mathematics, Physics, BA Tuition, LIC, NDA, SSB Exam, SSC Exams, NET, SET, TET, UGC Net

Contact this Tutor
  1. Gravitation Learning Objectives: Newtonian Gravitatlon and the Laws of Kepler Kepler's Laws The Law of Orbits Orbit EccentTicity Law ofPUi0ds Sir Isaac Newton: The Universal Law of Gravitation Newton's Laws and Kepler's Laws The Centre of Mass for a Binary System Newton's Modification of Kepler's Third Law Two Limiting Cases Weight and the Gravitational Force Mass and Weight Inverse Square Law Variafion of 'g' due to Altitude, Latitude and Depth Geostationary Satellites Derivafion of geostationary altitude Weightlessness in Artificial Satellites: Numericals Newtonian Gravitation and the Laws of Kepler Kepler's Laws Johannes Kepler, working with data painstakingly collected by Tycho Brahe without the aid of a telescope, developed three laws which described the motion of the planets across the sky. 1. The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus. 2. The Law of Areas: A line that connects a planet to the sun sweeps out equal areas in equal times. 3. The Law of Periods: The square of the period of any planet is proportional to the cube of the semi-major axis of Its orbit. Tyco Brahe, born in 1546, three years after the death of Copernicus, was a Danish physicist. He was helped by King Frederik Il to build state-of-the-art observatories from which he made numerous accurate astronomical observations. His observations disproved Aristotelian cosmology. He formulated the Tychonic System, much different from Aristotle's, but unlike that of Copernicus, which suggested that the earth remained fixed while the sun and the moon circled around it. The orbits of the planets, however, had the sun as their centre. King Fredrik's death brought his son, King Kristian to the throne who discontinued the patronage given to Tycho. So the astronomer was forced to seek the patronage of Emperor Rudolf Il and moved to Prague, leaving behind all his instruments. But, this loss was of Tycho's was compensated by the gain of a young assistant named Johannes Kepler. Kepler came from a humble family. His father was a quarrelsome person and his mother was suspected of being a witch. However, he was a gifted student and became professor of astronomy at the University of Graz. His first book, Mysterium Cosmographicum, published in 1596, however adhered to the Copernician cosmology and concluded that the Sun was the centre of the universe.
  2. Kepler's laws were derived for orbits around the sun, but they apply to satellite orbits as well. un The Law of Orbits All planets move in elliptical orbits, with the sun at one focus. Elliptical Orbit of a Planet This is of Kepler's first law. The elliptical shape of the orbit is a result of the Inverse square force of gravity. The eccentricity of the ellipse is greatly exaggerated here. Orbit EccentTicity The eccentricity of an ellipse can be defined as the ratio of the distance p e 0.7 between the foci to the major axis of the ellipse. The eccentricity is zero for a circle. Of the planetary orbits, only Pluto has a large eccentricity. Lam ofPUi0ds Ill. The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semimajor axes: 82 1 R 23
  3. In this equation P represents the period of revolufion for a planet (in some other references the period is denoted as "T") and R represents the length of its semi-major axis. The subscripts "1" and "2" distinguish quantifies for planet 1 and 2 respectively. The periods for the two planets are assumed to be in the same fime units and the lengths of the semi-major axes for the two planets are assumed to be in the same distance units. Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit. Thus, we find that Mercury, the innermost planet, takes only 88 days to orbit the Sun but the outermost planet (Pluto) requires 248 years to do the same. Sir Isaac Newton: The Universal Law of Gravitation There is a popular story that Newton was sitting under an apple tree, an apple fell on his head, and he suddenly thought of the Universal Law of Gravitafion. As in all such legends, this is almost certainly not true in its details, but the story contains elements of what actually happened. What Really Happened with the Apple? Probably the more correct version of the story is that Newton observed In an apple falling from a tree, that it was accelerated, since Its velocity changed from zero while hanging on the tree, while moving towards the ground. Thus, by Newton's 2nd Law there must have been a force that acted on the apple to cause the acceleration towards the ground. Sir Isaac Newton was born posthumously on Christmas Day, 1642 and when he was three years old, his mother re-married, choosing a prosperous but unsympathetic clergyman for her husband. The young Newton was never welcome in his step-father's family. Throughout his life, he was a loner, though it is rather peculiar for a person of his intelligence and striking good looks. However, the few good friends he had, lasted a lifetime. He was also a voracious reader, having read the works of Galileo and Descartes. During his Cambridge days, where he remained until 1696 and where all his major work was done, he lived an austere and devout life. But, his brilliance allowed him to recognise the brilliance of other young scientists like Edmond Halley. Newton died in 1727. That observation was followed by Newton's ffuly brilliant insight: if the force of gravity reached the top of the highest tree, then it could have reached even further, maybe to the orbit of the moon also. Then, the orbit of the Moon about the Earth could be a consequence of the gravitational force, because the acceleration due to gravity could change the velocity of the Moon in just such a way that it followed an orbit around the earth. Suppose we fire a cannon horizontally from a high mountain; the projectile will eventually fall to earth because of the gravitational force directed towards the centre of the Earth and the associated accelerafion. (Remember, accelerafion is a change in velocity and that velocity is a vector, so it has both a magnitude and a direction. Thus, accelerafion occurs if either or both the magnitude and the direction of the velocity change.)
  4. But if the muzzle velocity for the cannon IS Increased, the projectile will travel further and further before returning to earth. Finally, Newton reasoned that if the cannon projected the cannon ball with exactly the right velocity, the projectile would travel completely around the Earth, always falling in the gravitatlonal field but never reaching the Earth, which is curving away at the same rate that the projectile falls. That is, the cannon ball mould have been Put into orbit around the EaMh. Newton concluded that the orbit of the Moon was of exactly the same nature: the Moon continuously "fell" in its path around the Earth because of the acceleration due to gravity, thus producing its orbit. By such reasoning, Newton came to the conclusion that any two objects in the Universe exert gravitational attraction on each other, with the force having a universal form: Fg Gnum2/r2 where G 6.7 X 10 Il Nm2kg -2 G is the universal gravitational constant. It is basically a conversion factor to adjust the number and units so they come out to the correct value. This is a universal constant so it is true of separation between the centre of masses of each object; FG IS the force of attraction between the two objects. The constant of proportionality G is known as the universal gratitational constant. It IS termed "universal constant" because it is thought to be the same at all places and at all times, and thus universally characterizes the intrinsic strength of the gravitational force. Important Concepts 1. The direction of the force is not given by this formula since there are actually two forces equal in size but opposite in direction. This formula calculates them both. 2. The formula is an inverse square law for radius of separation (notice the r2 on the bottom of the equation). This means that if you double the separation you quarter the force, or if you cut the separation in half you quadruple the force of attraction. 3. If you double a single mass, you double the force. If you cut one of the masses in half, you cut the force in half. But if you double both masses you would quadruple the force. Common Misconceptions 1. "There is no gravity in space." If there were no gravity in space, the space shuttle would not be able to orbit the Earth, the moon would not orbit the Earth, and the Earth would not orbit the Sun. The reason we tend to think of there being no gravity in space is that we have seen movies of the astronauts being "weightless". They aren't actually weightless, they are still being pulled down by gravity but they and the space shuttle are In a constant state of freefall around the Earth. So they seem to be weightless as a result of the falling just as you would seem weightless if you were In an elevator when the cable broke. 2. "G and g are the same." G is the universal gravitational constant and g IS the acceleration due to the force of gravity and its value of 9.8 ms-2 down IS only true on this planet. It is not a universal constant. 3. "g is gravity." g is the effect of the force of gravity, but is not gravity. Gravity is a force, g is an accelerafion caused by gravity.
  5. We now come to the great synthesis of dynamics and astronomy accomplished by Newton: the Laws of Kepler for planetary motlon may be derived from Newton's Law of Gravitatlon. Furthermore, Newton's Laws provide corrections to Kepler's Laws that turn out to be observable, and Newton's Law of Gravitation will be found to describe the motions of all objects in the heavens, not just the planets. Newton's Laws and Kepler's Laws We now outline how Kepler's Laws are implied by those of Newton, and use Newton's Laws to supply corrections to Kepler's Laws. 1. 2. 3. 4. Thus, Since the planets move on ellipses (Kepler's 1st Law), they are continually accelerating. This implies that a force is acting continuously on the planets. Because the planet-Sun line sweeps out equal areas in equal times (Kepler's 2nd Law), it is possible to show that the force must be directed toward the Sun from the planet. From Kepler's 1st Law the orbit is an ellipse with the Sun at one focus; from Newton's laws it can be shown that this means that the magnitude of the force must vary as one over the square of the distance between the planet and the Sun. Kepler's 3rd Law and Newton's 3rd Law imply that the force must be proportional to the product of the masses for the planet and the Sun. Kepler's laws and Newton's laws taken together imply that the force which holds the planets In their orbits by continuously changing the planet's velocity so that it follows an elliptical path is (1) directed toward the Sun from the planet, (2) is proportional to the product of masses for the Sun and the planet, and (3) is inversely proportional to the square of the planet-Sun separation. This is precisely the form of the gravitational force, with the universal gravitational constant G as the constant of proportionality. Thus, Newton's laws of motion, with a gravitational force used In the 2nd Law, imply Kepler's Laws, and the planets obey the same laws of motion as objects on the surface of the Earth. Thus, Kepler's elliptical orbitals are but one example of the possible orbits in a gravitational field. Only ellipses (and their special case, the circle) lead to bound orbits; the others are associated with one-time gravitational encounters. For a given central force, increasing the velocity causes the orbit to change from a circle to an ellipse to a parabola to a hyperbola, with the changes occurring at certain critical velocities. For example, if the speed of the Earth (which is in a nearly circular gravitational orbit) were Increased by about a factor of 1.4, the orbit would change into a parabola and the Earth would leave the Solar System. The Centre of Mass for a Binary System If you think about it a moment, it may seem a little strange that in Kepler's Laws the Sun is fixed at a point in space and the planet revolves around it. Why is the Sun privileged? Kepler had rather mystical ideas about the Sun, endowing it with almost god-like qualities that justified its special place. However Newton, largely as a corollary of his 3rd Law, demonstrated that the
  6. situation actually was more symmetrical than Kepler Imagined and that the Sun does not occupy a privileged position; in the process he modified Kepler's 3rd Law. Consider the diagram shown to the right. We may define a point called the centre of mass between two objects through the equations midi = m2d2 where, R is the total separation between the centres of the two objects. The centre of mass is familiar to Center of Mass ml anyone who has ever played on a see-saw. The fulcrum point at which the see-saw will exactly balance two people sitting on either end is the centre of mass for the two persons sitting on the see-saw. Newton's Modification of Kepler's Third Law Because for every action there IS an equal and opposite reaction, Newton realized that in the planet-Sun system the planet does not orbit around a stationary Sun. Instead, Newton proposed that both the planet and the Sun orbited around the common centre of mass for the planet-Sun system. He then modified Kepler's 3rd Law to read, 3 2 3 where P is the planetary orbital per10d and the other quantifies have the meanings described above, with the Sun as one mass and the planet as the other mass. (As in the earlier discussion of Kepler's 3rd Law, this form of the equation assumes that masses are measured in solar masses, times in Earth years, and distances in astronomical units.) Notice the symmetry of this equation: since the masses are added on the left side and the distances are added on the right side, it doesn't matter whether the Sun is labeled with 1 and the planet with 2, or Lice-versa. One obtains the same result in either case. Now notice what happens In Newton's new equation if one of the masses (either 1 or 2; remember the symmetry) is very large compared with the other. In particular, suppose the Sun is labeled as mass 1, and its mass is much larger than the mass for any of the planets. Then the sum of the two masses is always approximately equal to the mass of the Sun, and if we take ratios of Kepler's 3rd Law for two different planets the masses cancel from the ratio and we are left with the original form of Kepler's 3rd Law: 1 2 Thus Kepler's 3rd Law is approximately valid because the Sun is much more massive than any of the planets and therefore Newton's correction IS small. The data Kepler had access to were not good enough to show this small effect. However, detailed observations made after Kepler show that Newton's modified form of Kepler's 3rd Law is in better accord with the data than Kepler's original form. Two Limiting Cases
  7. We can gain further insight by considering the position of the centre of mass In two limits. First consider the example just addressed, where one mass is much larger than the other. Then, we see that the centre of mass for the s stem essenfiall coincides with the centre of the massive object: This is the situation in the Solar System: the Sun is so massive compared to any of the planets that the centre of mass for a Sun-planet pair is always very near the centre of the Sun. Thus, for all practical purposes the Sun IS almost (but not quite) motionless at the centre of mass for the system, as Kepler originally thought. However, now consider the other limiting case where the two masses are equal to each other. Then it IS easy to see that the centre of mass lies equidistant from the two masses and if they are gravitationally bound to each other, each mass orbits the common centre of mass for the system I in midwa between them: This situatlon occurs commonly with binary stars (two stars bound gravitationally to each other so that they revolve around their common centre of mass). In many binary star systems the masses of the two stars are similar and Newton's correction to Kepler's 3rd Law is very large. By making one mass much larger than the other in this interactive animation you can illustrate the Ideas discussed above and recover Kepler's original form of his 3rd Law where a less massive object appears to revolve around a massive object fixed at one focus of an ellipse. These limiting cases for the location of the centre of mass are perhaps familiar from our afore mentioned playground experience. If persons of equal weight are on a see-saw, the fulcrum must be placed in the middle to balance, but if one person weighs much more than the other person, the fulcrum must be placed close to the heavier person to achieve balance. Weight and the Gravitational Force We have seen that in the Universal Law of Gravitafion the crucial quantity is mass. In popular language mass and weight are often used to mean the same thing; in reality they are related but quite different things. What we commonly call weight is really just the gratitationalforce exerted on an object of a certain mass. Thus, the weight of an object of mass m at the surface of the Earth is obtained by multiplying the mass m by the acceleration due to gravity, g, at the surface of the Earth. The acceleration due to gravity IS approximately the product of the universal gravitational constant G and the mass of the Earth M, divided by the radius of the Earth, r, squared. (We assume the Earth to be spherical and neglect the radius of the object relafive to the radius of the Earth in this discussion.) The -2 measured gravitational acceleration at the Earth's surface is found to be about 980 cms Mass and Weight Mass IS a measure of how much matter an object contains, but weight is a measure of the gravitational force exerted on that material in a gravitational field; thus, mass and weight are proportional to each other, with the acceleration due to gravity as the proportionality constant. It follows that mass is constant for an object (actually this IS not quite true as proved by Einstein's
  8. Relativity Theory), but weight depends on the locatlon of the object. For example, if we transported the preceding object of mass m to the surface of the Moon, the gravitational acceleration would change because the radius and mass of the Moon both differ from those of the Earth. Thus, our object has mass m both on the surface of the Earth and on the surface of the Moon, but it will weigh much less on the surface of the Moon because the gravitational acceleration there is a factor of 6 less than at the surface of the Earth. For most objects you get near every day, the force of attraction is so incredibly small that you would never notice the force. Gravity is a very weak force, so between common objects like you and your pencil, the force of attracfion is very small because your mass and the mass of your pencil are small. We only get noticeable amounts of gravity when at least one object is very massive... like a planet. The force of attraction between you and the planet Earth is a noticeable force! We call the force of attractlon between you and the Earth, your weight. Weight is another name for the force of gravity pulling down on you or anything else. Inverse Square Law The inverse square law proposed by Newton suggests that the force of gravity acting between any two objects is inversely proportional to the square of the separation distance between the object's centres. Altering the separation distance (r) results in an alteration in the force of gravity acting between the objects. Since the two quantities are inversely proportional, an increase in one quantity results in a decrease in the value of the other quantity. That is, an increase in the separation distance causes a decrease in the force of gravity and a decrease in the separation distance causes an Increase in the force of gravity. Furthermore, the factor by which the force of gravity is changed is the square of the factor by which the separation distance is changed. So if the separation distance is doubled (increased by a factor of 2), then the force of gravity is decreased by a factor of four (2 raised to the second power). And if the separation distance (r) is tripled (increased by a factor of 3), then the force of gravity is decreased by a factor of nine (3 raised to the second power). Thinking of the force- distance relationship in this way involves using a mathematical relationship as a guide to thinking about how an alteration In one variable effects the other variable. Equations can be more than merely recipes for algebraic problem-solving; they can be "guides to thinking." The proportionalities expressed by Newton's universal law of gravitation IS represented graphically by the following illustration. Observe how the force of gravity is directly proportional to the product of the two masses and inversely proportional to the square of the distance of separation.
  9. of Mas on a force of attnctwith a force of @ 2F 1'@) attnctwith a force of attmctwith a force of of Distanæ on attract with a force of attnctwith a fome of attnctwith a force of attnctwith a force of In the above figure, the figure on the left hand side indicates the effect of "mass" if the diatnce between the two objects remains fixed at a given value "d". The right hand figure shows the effect of changing the distance while keeping the mass constant, and the last part of it shows the effect of changing both the distance and the mass. Check your understanding of the inverse square law as a guide to thinking by answering the following questions below. 1. Suppose two objects attract each other with a force of 16 units (like 16 N or 16 1b). If the distance between the two objects IS doubled, what is the new force of attraction between the two obj ects? Answer: If the distance is increased by a factor of 2, then distance squared will increase by a factor of 4. Thus, the inverse square law implies that the force will be "1/4" of the original 16 units. Therefore, the force of gravity becomes 4 units. 2. Suppose the distance In question 1 is tripled. What happens to the forces between the two objects? Answer: Again using inverse square law, we get distance squared to go up by a factor of 9. The force decreases by a factor of 9 and becomes 1.78 units. 3. If you wanted to make a profit in buying gold by weight at one altitude and selling it at another altitude for the same price per weight, should you buy or sell at the higher altitude location? Answer: To make a profit, buy at a high altitude and sell at a low altitude. 4. What would happen to your weight if the mass of the Earth somehow increased by 100 0? Answer: Your weight is nothing but force of gravity between the earth and you (as an object with a mass m). As shown in the above graph, changing one of the masses results in change in force of gravity. In this case, if the earth' mass goes up by 100 0, then the force of gravity on you, or your weight, will increase by the same amount, that is 100 0.
  10. 5. The planet Jupiter is more than 300 fimes as massive as Earth, so it might seem that a body on the surface of Jupiter would weigh 300 times as much as on Earth. But it so happens a body would scarcely weigh three times as much on the surface of Jupiter as it would on the surface of the Earth. Explain why this is so. Answer: The effect of greater mass of Jupiter is partly offset by its larger radius which is about 10 times the radius of the earth. This means the object is fimes farther from the centre of the Jupiter compared to the earth. Inverse of the distance brings a factor of 100 to the denominator and as a result, the force increases by a factor of 300 due to the mass, but decreases by a factor of 100 due to the distance squared. The net effect is that the force increases 3 times. We know that the force of gravity varies from place to place on the surface of earth. There are tvvo reasons behind this variation: 1. The shape of the earth. 2. The rotation of the earth. The earth is not a perfect sphere, but bulges at equator. Therefore if a body is taken from pole to equator its distance from the centre of the earth will change. Consequently, the gravitational force also varies. Variation of 'g' due to Altitude, Latitude and Depth (i)Variation of 'g' with altitude: Let M be the mass of the earth and R be the radius of the earth. Consider a body of mass m is situated on the earth's surface. Let g be the acceleration due to gravity on the earth's surface. The acceleration due to gravlty g at the surface of the earth is GM The acceleration due to gravity Ohat height h from earth's surface is GM (R + 102 (Since r = R + h Dividing (3) by (2) we get I.e This relation holds for any height. If , then using bionomial theorem -2 Or (1 From equation (4) and (5) it is obvious that the value of g decreases with Increase of height from the surface of the earth.
  11. From this equation it is obvious that accelerafion due to gravity goes on decreasing as altitude of body from the earth's surface increases. (ii) Variation of 'g' due to depth: Let M be the mass of the earth and R be the radius of the earth. The acceleration due to gravity on the surface of the earth is given by, Assuming earth to be a sphere of uniform density P,we have mass of earth The acceleration due to gravity at earth's surface G(-rR3p) 4 GM c;pR 3 4 RR3p 3 If body is taken to a depth x below earth's surface ( e.g. in a mine ) , the body will be attracted only by the mass ( M') of earth which is enclosed in a sphere of radius ( R h) given by Effect of depth on g 4 — x)3p 3 The value of acceleration due to gravity at depth x is given by Dividing (7) by (6) , we get Thus the value of g decreases with increase of depth below earth's surface. This decrease is due to reducfion in mass of attracting sphere of earth. The above considerafion shows dial the value of g is maximum at the surface of the earth and it goes on decreasing (i) with increase in depth below the earth's surface. (ii) with increase in the height above the earth's surface. The variafion of accelerafion due to gravity according to the depth and the height
  12. from the earth's surface can be expressed with help of following graph. Gravitational potentlal energy The gravitational potential energy or self-energy of the system of two masses ml and nu IS given by Gnu where r is the distance between the masses. Gravitational potential energy has two familiar expressions U = mgh (at height h from earth's surface near surface of earth, assuming zero P.E at earth's surface) and (assuming zero P.E. at infinity where r centre of earth usually r = R + h) Geostationary Satellites distance of body of mass m from A geostationary orbit is a circular orbit directly above the Earth's equator approximately 35,786 km above ground. Any point on the equator plane revolves about the Earth in the same direction and with the same period as the Earth's rotation. The period of the satellite is one day or approximately 24 hours. To find the speed of the satellite in orbit we use Newton's law of gravity and his second law of motlon along with that we know about centripetal acceleration. The inward and outward forces on the satellite must equal each another (by Newton's first law of motion). F centripetal cen trifugal By Newton's second law of motion: where: ms Mass of satellite; ag Gravitational accelerafion; ac The centripetal acceleration provided by Earth's gravity: a where: Centrifugal accelerafion Me Mass of Earth in kilograms (5.9742 x 1024 kg); G Gravitational constant (6.6742 x 10-11 N m2 kg-2 - 6.6742 x 10 Il m3s2kgl) Magnitudes of the centrifugal acceleration derived from orbital motion: where; o Angular velocity In radians per second; r - Orbital radius In meters as measured from the Earth's centre of mass.
  13. a a From the relationship F centripetal cen trifugal r We note that the mass of the satellite, ms, appears on both sides, geostationary orbit is independent of the mass of the satellite. n,fe.G r r r (Orbital radius) = Earth's equatorial radius + Height of the satellite above the Earth surface 6,378 km + 35,780 km 42,158 km 4.2158 x 107 m 4:2158x107m v = 3.0754 x 103 ms-I Speed of the satellite is 3.0754 x 103 m/ s Orbit is a word we hear quite often. Every time the Space Shuttle lifts off the launch pad, we hear lt. Every time the Space Shuttle meets up with the International Space Station (ISS), we hear lt. And every time a rocket launches a payload, we hear the word "orbit." It's a widely used term, but do you know what an orbit really is? An orbit IS a regular, repeating path that one object In space takes around another one. An object In an orbit is called a satellite. A satellite can be natural, like the Earth or the Moon. It can also be man-made, like the Space Shuttle or the ISS. In our solar system, the Earth and the eight other planets orbit the Sun. Most of the objects orbiting the Sun move along or close to an imaginary flat surface. This imaginary surface is called the ecliptic plane. Many planets also have moons. These moons orbit around them. Orbits are elliptical in shape, this means they are similar to an oval. For the planets, the orbits are almost round. The orbits of comets have a different shape. They are highly eccentric or squashed." Satellites that orbit the Earth are not always the same distance from the Earth. Sometimes they are closer, and at other times they are farther away. The closest point a satellite comes to the Earth is called its perigee. The farthest point is the apogee. The time it takes a satellite to make one full orbit is called its period. The inclinafion is the angle the orbital plane makes when compared with the Earth's equator. An object in motion will stay in motion unless something pushes or pulls on it. This is Isaac Newton's First Law of Motion. Without gravity, an Earth orbiting satellite would go off into space along a straight line. With gravity, it is pulled back toward the Earth. There is a constant tug-of-war between the satellites tendency to move In a straight line, or momentum, and the tug of gravity pulling it back.
  14. An object's momentum and the force of gravity have to be balanced for an orbit to happen. If the forward momentum of one object is too great, it will speed past the other one and not enter Into orbit. If momentum is too small, the object will be pulled into the other one and crash. When these forces are balanced, the object IS always falling into the planet, but because It's moving sideways fast enough, it never hits the planet. Escape velocity is the speed an object must go to break free from a planet's gravity and enter into orbit. Escape velocity depends on the mass of the planet. Each planet has a different escape velocity. The object's distance from the planet's centre is also important. The escape velocity from the Earth is about 11.3 kilometers (7 miles) per second. Orbital velocity is the speed needed to stay in orbit. At an altitude of 242 kilometers (150 miles), this is about 17,000 miles per hour. This is just a little less than full escape velocity. Low-Earth Orbit (LEO) is restricted to the first 100 to 200 miles of space. LEO is the easiest orbit to get to and stay in. This is where the Shuttle and ISS conduct their operations. One complete orbit in LEO takes about 90 minutes. Satellites that seem to be attached to some location on Earth are in Geosynchronous Earth Orbit (GEO). These satellites orbit about 23,000 miles above the equator and complete one revolution around the Earth precisely every 24 hours. Satellites headed for GEO first go to an elliptical orbit with an apogee about 23,000 miles. Firing the rocket engines at apogee then makes the orbit round. Geosynchronous orbits are also called geostationary. Any satellite with an orbital path going over or near the poles maintains a polar orbit. Polar orbits are usually in low-Earth orbit. They remain in place while the Earth passes under. This means that eventually, the entire Earth's surface passes under a satellite In polar orbit. When a meteorite enters our atmosphere and becomes a "shooting star," it IS no longer In an orbit. Some space probes, like Voyager, have reached escape velocity and broken away from the pull of the Sun's gravity. These probes are leaving the solar system. They are not In orbit around a planet or the Sun. Derivafion of geostationary altitude In any circular orbit, the centripetal accelerafion required to maintain the orbit is provided by the gravitational force on the satellite. To calculate the geostationary orbit altitude, one begins with this equivalence, and uses the fact that the orbital period is one sidereal day. By Newton's second law of motion, we can replace the forces F with the mass m of the object multiplied by the accelerafion felt by the object due to that force: mac = mg We note that the mass of the satellite m appears on both sides geostationary orbit is independent of the mass of the satellite. So calculating the altitude simplifies Into calculating the point where the magnitudes of the centripetal acceleration required for orbital motion and the gravitational acceleration provided by Earth's gravity are equal. The centripetal acceleration's magnitude is: lacl L' r where is the angular speed, and r is the orbital radius as measured from the Earth's centre of mass. The magnitude of the gravitational acceleration is:
  15. lg 2 where M is the mass of Earth, 5.9736 >< 1024 kg, and G is the gravitational constant, 6.67428 ± 0.00067 x 10-11 m3 kg-I s-2. Equating the frvvo accelerations Ives: GM 2 GM 3 2 The product GM is known with much greater accuracy than either factor; it is known as the geocentric ravitational constant = 398,600.4418 ± 0.0008 km3 s-2: 3 The angular speed is found by dividing the angle travelled in one revolution (3600 2P rad) by the orbital period (the time it takes to make one full revolution: one sidereal day, or 86,164.09054 seconds) .[3] This gives: 2m rad 86 164 s 7.2921 x 10-5 rad/s The resulting orbital radius is 42,164 kilometres (26,199 mi). Subtracting the Earth's equatorial radius, 6,378 kilometres (3,963 mi), gives the altitude of 35,786 kilometres (22,236 mi). Orbital speed (how fast the satellite is moving through space) is calculated by multiplying the angular speed by the orbital radius: v 3.0746 km/s 11068 km/h 6877.8 mph. ADVANTAGES OF GEOSTATIONARY ORBITS Make repeated observations over a DISADVANTAGES GEOSTA TIONARY ORBITS given area (constant view area) Get high temporal resolufion data. GOES E and W can you a temporal resolution of 1 minute! Hence, GOES E and W can effectively monitor the severe weather environment and track severe storms and hurricanes In real time. Weightlessness in Artificial Satellites: Due to the high orbit, the spatial resolufion of the data is not as great as for the polar orbiting satellites Poor spatial resolution in the Polar Regions (parallax). A astronaut in a satellite experiences weightlessness. The reason is that there act two forces on the astronaut. (i) Gravitational pull Where 2 mass of earth, m = mass of astronaut, r= distance between satellite from earth's centre (ii) Centrifugal force: As astronaut is in rotating frame; the astronaut experiences a centrifuge force whose direction is away from the centre of earth and magnitude given by, •mv20
  16. where vo is orbital speed of satellite. Net force on astronaut: F=fy-fc 2 GAIem me; o 2 Also the condition of circular motion of satellite is 2 2 Using this equation, equation (1) gives i.e. the net force on astronaut is zero the astronaut experiences weightlessness In artificial satellite. Remark: The weight of a person/ body is zero in artificial satellite, but it is not zero in a natural satellite, because a natural satellite has its own g. The value of g on moon is one -sixth that on earth. Numericals: 1. Kepler's third law states that if r is the mean distance of a planet from the Sun, and T is its orbital period, then r3/T2 const. Show that the force acting on a planet is Inversely proportional to the square of the distance. For the purpose of simplicity let us assume that the orbit of a planet is circular. Then the centripetal force acting on the planet is F = mv2/r where v is the orbital velocity. Since v = ro expression as F = m (2ar/T)2/r = 41t2mr/T2 But T20c r3 or Kr3 (Kelpler's 3rd law) 2ar/T, where T is the period, we can rewrite the above where K is a constant of proportionality. Hence F 41t2mr/T2 = 41t2mr/ Kr3 = 4112m/Kr2 = 47t2m/K x 1/ r2 As 4112m/K is a constant value for a planet, 1/r2. 2. The magnitude of force befrvveen two masses placed at a certain distance is F. What happens to F if (i) the distance IS doubled without any change In masses, (ii) the distance remains the same but each mass is doubled, (iii) the distance is doubled and each mass is also doubled? Let ml be the mass of the first body, m2 be the mass of the second body and R be the distance between them. F — Gnum2/R2 (where G — gravitation constant)
  17. i. The distance R IS doubled to 2r. So, F — F - Gmlm2/4R2 4F = Gnum2/R2. The magnitude of force F decreases 4 times. ii. Each mass is doubled without any change in the distance between them. So, F — G2m12 m2/R2 = 4 Gnum2/R2 The magnitude of the force F Increases 4 times. iii. If each mass IS doubled and the distance is also doubled, Then, F - G2m12m2/4R2 Gnum2/R2 So, the magnitude of the force F remains the same. 3. An aircraft flies at a height of 10 km. Calculate the value of g at that altitude. Take the radius -2 of the earth as 6400 km and the value of G on the surface of the earth as 9.8 ms We know that, -2 gh = g/[l+ 2(h/R)] = 9.8/11 + 200/6400)] = 9.77 ms 4. At what depth would the value of g be 800 0 of what it is on the surface of the earth? We know that, gd = g[l- (d/R)] Now, gd — 80g/100 . o.8g - or, d/R = 0.2 Or d = 0.2R o.8g 5. The mass of the earth is 5.97 >< 1024 kg and its radius is 6371 km. Calculate the escape velocity from the earth. From the formula foe escape velocity, 2GM 107 -1 2 x 6.67 x 5.97 x— 11.3 kms 6.371 6. Suppose the earth shrunk suddenly to one-fourth its radius without any change in its mass. What would be the escape velocity then? 1 We know that vesc so, if R 4Rl
  18. vesc 4R1 or, 2 vesc 2 RI RI Thus, the escape velocity is doubled.

Need a Tutor or Coaching Class?

Post an enquiry and get instant responses from qualified and experienced tutors.

Post Requirement

Related Study Notes

Upload Study Notes

If you have your own Study Notes which you think can benefit others, please upload on LearnPick. For each approved Study Note you will get 100 Credit Points and 100 Activity Score which will increase your profile visibility.

Upload Now
Home > Study Notes > Gravitation