CHAPTER 13

UNIVERSAL GRAVITATION

PART II

13.4 Kepler’s Laws and the Motion of Planet

Kepler’s complete analysis of planetary motion is summarized in three statements known as Kepler’s laws:

1. All planets move in elliptical orbits with the Sun at one focus.

2. The radius vector drawn from the Sun to a planet sweeps out equal areas in

equal time intervals.

3. The square of the orbital period of any planet is proportional to the cube of the

semimajor axis of the elliptical orbit.

KEPLER’s FIRST LAW

Mayor axis : the The longest distance perihelion through the center between points on the ellipse (2a)

Semimajor axis : The distance a

Minor axis : the shortest distance through O the center between points on the ellipse (2b)

Semiminor axis : The distance b

F1 and F2 : focus , where F1 is Sun and P is planet there’s nothing in F2

C : Central distance ellips (O) and Focus (F1 and F2), where C is number who doesn’t has dimension number. It’s value range 0-1 it’s called eccentricity

Perihelion : The nearest point from the Sun

Aphelion : The Farest point from the Sun

So, Kepler’s 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.

KEPLER’s SECOND LAW

So, Kepler’s 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.

KEPLER’s THIRD LAW

We use Newton’s 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.

1. Quick Quiz 13.4

Pluto, the farthest planet from the Sun, has an orbital period that is…

a. greater than a year

b. less than a year

c. equal to a year.

Answer :

(a). greater than a year . Because Kepler’s 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 Sun’s 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.

2. Quick Quiz 13.5

An asteroid is in a highly eccentric elliptical orbit around the Sun. The period of the asteroid’s 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 collision

b. There is a possibility of a collision

c. There is not enough information to determine whether there is danger of a collision.

Answer :

(a). There is a possibility of a collision because from Kepler’s 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 Earth–Sun distance, the asteroid cannot possibly collide with the Earth

13.5 The Gravitional Field

The gravitational field is when two or more particle interact with another one when they were not in contact with each other. It’s 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 :

As an example of how the field concept works, consider an object of mass m near the Earth’s 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

ř

Where ř 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.

13.6 Gravitational Potential Energy

13.7 Energy Considerations in Planetary and Satelite Moon

Consider an object of mass m moving with a speed v in the vicinity of a massive object of mass M, where (Equation @)

This equation shows that E may be positive, negative, zero, depending on the value of .

We can easily establish that E>0 for the system consisting of an object of mass m moving in a circular orbit about an object of mass M>>m

Newton’s 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.

A. Escape Speed

Suppose an object of mass m is projected vertically upward from the Earth’s 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.

t the surface of the Earth, v=vi and r=r1=RE

When the object reaches its maximum altitude, v=vE=0 and r=rf=rmax Because the total energy of the system is constant.

solving for v12 gives

Therefore, if the initial speed is known, this expression can be used to calculate the maximum altitude h because we know that

We are now in a position to calculate escape speed, which is the minimum speed the object must have at the Earth’s surface in order to approach an infinite separation distance from the Earth.

r max ---> ∞ And taking , vi=vesc we obtain :

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

B. Black Hole

An 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 very small object in space, commonly reffered to as a black hole.

ORIGIN OF BLACKHOLES

Black holes are created when an object can not withstand the strength of the force of gravity alone. Many objects (including the sun and the earth) will never be a black hole.

The pressure of gravity on the sun and the earth is not sufficient to exceed the atomic and nuclear power in him that nature against the pressure of gravity. But contrary to the object's mass is very large, the pressure of gravity was the one who wins. The mass of the black hole continues to grow by capturing all of the material nearby. All the material can not escape from the shackles of a black hole if it passed too close. So objects that can not keep a safe distance from the black hole will be sucked.

Although 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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