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UNIVERSAL GRAVITATIONBY : Indah Cahaya Pramesti Luluk Nur Farida12The Gravitational FieldThe gravitational field is when two or more particle interact with another one when they were not in contact with each other. Its caused by thr gravitational field.When a particle of mass m is placed at a point where the gravitational field is g, the particle experiences a force Fg = mg.In other words :

30In 1687 Newton published his work on the law of gravity in his treatise Mathematical Principles of Natural Philosophy. Newtonts law of universal gravitation states that :

m 1 : mass 1m 2 : mass 2r : distancewhere G is a constant called universal gravitational constant. Its value in SI unit is :

Every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

4Quick Quiz 13.1The Moon remains in its orbit around the Earth wather than falling becauseIt is outside of the of the gravitational influence of the EarthIt is in balance with the gravitational forces from the sun and other planetsThe net force on the Moon is zeroNone of theseAll of these5Answer :(d).None of these because The gravitational force exerted by the Earth on the Moon provides a net force that causes the Moons centripetal acceleration. 6Quick Quiz 13.2A planet has two moons of equal mass. Moon 1 is in a circular orbit of radius r. Moon 2 is in a circular orbit of radius 2r. The magnitude of the gravitational force exerted by the planet on the moon 2 isFour times as large as that on moonTwice as large as that on moonEqual to that on moonHalf as large as that on moon 1One fourth as large as that on moon 1 7Answer :(e). One fourth as large as that on moon 1 because the gravitational force follows an inverse-square behavior, so doubling the distance causes the force to be one fourth as large.8Measuring of Gravitational ConstantThe universal gravitational constant G was measured in an important experiment by Henry Cavendish (17311810) in 1798. The Cavendish apparatus consists of two small spheres, each of mass m, fixed to the ends of a light horizontal rod suspended by a fine fiber or thin metal wire. When two large spheres, each of mass M, are placed near the smaller ones, the attractive force between smaller and larger spheres causes the rod to rotate and twist the wire suspension to a new equilibrium orientation. The angle of rotation is measured by the deflection of a light beam reflected from a mirror attached to the vertical suspension. The deflection of the light beam is an effective technique for amplifying the motion. The experiment is carefully repeated with different masses at various separations. In addition to providing a value. for G, the results show experimentally that the force is attractive, proportional to the product mM, and inversely proportional to the square of the distance r.9

10So . The equation is :

Then, subituting again with :

Where g is the value of free fall acceleration at the altitude h. So the last equation is :

Free Fall Acceleration and the Gravitational ForceFrom Newtons law we have equation :

Subsituting with So :

Now consider if object of mass m located a distance h above the Earths surface or a distance r from the Earths center.

11Quick Quiz 13.3Superman stands on top of a very tall mountain and throws a baseball horizontally with a speed such that the baseball goes into a circular orbit around the Earth. While the baseball is in orbit, the acceleration of the ball Four times as large as that on moona. depends on how fast the baseball is thrown b. is zero because the ball doesnt fall to the groundc. is slightly less than 9.80 m/s2 d. is equal to 9.80 m/s2.12Answer :(c).is slightly less than 9.80 m/s2 because an object in orbit is simply falling while it moves around the Earth. The acceleration of the object is that due to gravity. Because the object was launched from a very tall mountain, the value for g is slightly less than that at the surface.13Keplers Law and the Motion of PlanetKeplers complete analysis of planetary motion is summarized in three statements known as Keplers laws:14

1. Keplers First LawMayor axis : the The longest distance perihelionthrough the center between points on the ellipse (2a)PSemimajor axis : The distance aMinor axis : the shortest distance through Othe center between points on the ellipse (2b)Semiminor axis : The distance baphelionF1 and F2 : focus , where F1 is Sun and P is planet theres nothing in F2 C : Central distance ellips (O) and Focus (F1 and F2), where C is number who doesnt has dimension number. Its value range 0-1 its called eccentricityPerihelion : The nearest point from the SunAphelion : The Farest point from the SunSo, Keplers first law is a direct result of the inverse square of separation distance

These are the allowed object that are bound to the gravitational force center. The object include planets, asteroids, and comet that move repeatedly around the sun, as well as moon orbiting planet. There also be unbound objects, such as meteorids from deep space that might pass by the sun once and then never return.

1. Keplers Second LawMPvrMMF

torque

Always to central object

?rMMdr

dAarea with broom r from time dt r

qdr

= konstan From time interval in same position r wipe off the same area toohSo, Keplers second law can be shown to be a consequence of angular momentum conservation as follows.We can conclude that the radius vector from the Sun to any planet sweeps out equal areas in equal times.

1. Keplers Third LawWe use Newtons second law for a particle in uniform circular motion

The orbital speed Mpv of the planet is , so the equation become :

Where Ks is a constant given by :

This equation is also valid for elliptical orbit if we replace r with the length a of the semimajor axis :

This table is a collection of useful planetary data. The last column verifies that the ratio is constant. The small variation in the values in this coloumn are due to uncertainties in the data measured for the periods and semimajor axes of the planet.

Quick Quiz 13.4Pluto, the farthest planet from the Sun, has an orbital period that isa. greater than a year b. less than a year c. equal to a year.24Answer :(a). greater than a year . Because Keplers third law ,which applies to all the planets, tells us that the period of a planet is proportional to a3/2. Because Pluto is farther from the Sun than the Earth, it has a longer period. The Suns gravitational field is much weaker at Pluto than it is at the Earth. Thus, this planet experiences much less centripetal acceleration than the Earth does, and it has a correspondingly longer period.25Quick Quiz 13.5An asteroid is in a highly eccentric elliptical orbit around the Sun. The period of the asteroids orbit is 90 days. Which of the following statements is true about the possibility of a collision between this asteroid and the Earth? a. There is no possible danger of a collisionb. There is a possibility of a collision c. There is not enough information to determine whether there is danger of a collision.26Answer :(a). There is a possibility of a collision because from Keplers third law and the given period, the major axis of the asteroid can be calculated. It is found to be 1.2 # 1011 m. Because this is smaller than the EarthSun distance, the asteroid cannot possibly collide with the Earth.27Quick Quiz 13.6A satellite moves in an elliptical orbit about the Earth such that, at perigee and apogee positions, its distances from the Earths center are respectively D and 4D. The relationship between the speeds at these two positions is a. vp ! Va b. vp ! 4va c. va ! 4vp d. vp ! 2va e. va ! 2vp.28Answer :(b). b. vp ! 4va because from conservation of angular momentum, mvprp ! mvara, so that vp ! (ra/rp)va ! (4D/D)va ! 4va.29As an example of how the field concept works, consider an object of mass m near the Earths surface. Because the gravitational force acting on the object has a magnitude, the field at a distance r from the center of the Earth is

Where is a unit vector pointing radially outward from the Earth and the negative sign indicates that the field points toward the center of the Earth, as illustrated in figure.This equation shows that E may be positive, negative, zero, depending on the value of . We can easily establish that for the system consisting of an object of mass m moving in a circular orbit about an object of mass

Newtons second law applied to the object of mass m gives

Multiplying both sides by r and dividing by 2 gives

Substituting this into Equation @,we obtain

(circular orbits)

The expression for E for elliptical orbits is the same as circular orbits with replaced by the semimajor exis length

we see that the both the total energy and the total angular momentum of a gravitationally bound, two-object system are constant of the motion.

Quick Quiz 13.7A comet moves in an elliptical orbit around the Sun.Which point in its orbit (perihelion or aphelion) represents the highest value ofthe speed of the comet (b) the potential energy of the cometSun system (c) The kinetic energy of the comet (d) the total energy of the cometSun system?Answer :

(a) Perihelion. Because of conservation of angular momentum,the speed of the comet is highest at its closestposition to the Sun. (b) Aphelion. The potential energyof the cometSun system is highest when the comet is atits farthest distance from the Sun. (c) Perihelion. The kineticenergy is highest at the point at which the speed ofthe comet is highest. (d) All points. The total energy ofthe system is the same regardless of where the comet is inits orbit.

Escape SpeedSuppose an object of mass m is projected vertically upward from the Earths surface with an initial speed We can use energy considerations to find the minimum value of the initial speed needed to allow the object to move infinitely far away from the Earth.

At the surface of the Earth, and When the object reaches its maximum altitude, and Because the total energy of the system is constant. solving for gives

Therefore, if the initial speed is known, this expression can be used to calculate the maximum altitude h because we know thatWe are now in a position to calculate escape speed, which is the minimum speed the object must have at the Earths surface in order to approach an infinite separation distance from the Earth.

And taking , we obtain

Note that this expression for is independent of the mass of the object.

Black HolesAn even more unusual star death may occur when the core has a mass greater than about three solar masses. The collapse may continue until the star becomes a vey small object in space, commonly reffered to as a black hole.

Is called Schwarzschild radiusAlthough light from a black hole cannot escape, light from events taking place near the black hole should be visible.For example, it is possible for a binary star system to consist of one normal star and one black hole. Material surrounding the ordinary star can be pulled into the black hole, forming an accretion disk around the black hole.

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