Potensial energy

Potential Energy

Potential energy is the energy associated with the configuration of a system of objects that exert forces on each other.

Potential energy exists whenever an object which has mass has a position within a force field.

Potential Energy of a System

The work done by an external agent on the system of the book and the Earth as the book is lifted from a height ya to a height yb is equal to mgyb = mgya.

While the book was at the highest point, the energy of the system had

the potential to become kinetic energy, but did not do so until the book was allowed to

fall. Thus, we call the energy storage mechanism before we release the book potential

energy. We will find that a potential energy can only be associated with specific types of

forces. In this particular case, we are discussing gravitational potential energy.

we can identify the quantity mgy as the gravitational potential energy Ug :

Ug ≡ mgy                 

Using our definition of gravitational potential energy, Equation 8.1 can now be rewritten as

W = ∆Ug                 

The Isolated System-Conservation of Mechanical Energy

As the book falls from yb to ya, the work done by the gravitational force on the book is :

W on book=(mg) . ∆r = (- mgĵ) . [(yb-ya)ĵ] = mgyb-mgya        

The work done on the book is equal to the change in the kinetic energy of the book:

W on book = ∆K book

    It can be written as:

∆Kbook = mgyb-mgya

We define the sum of kinetic and potential energies as mechanical energy:

 E mech = K + Ug

   we can write the general form of the definition for mechanical energy without a subscript on U:

 E mech = K + U

Isolated System

An isolated system is one for which there are no energy transfers across the boundary. The energy in such a system is conserved—the sum of the kinetic and potential energies remains constant.

Elastic Potential Energy

The elastic potential energy function associated with the block–spring system is defined by:

The elastic potential energy of the system can be thought of as the energy stored in the deformed spring (one that is either compressed or stretched from its equilibrium position).

Conservative Forces

Conservative forces have these two equivalent properties:

1. The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle.

2. The work done by a conservative force on a particle moving through any closed path is zero. (A closed path is one in which the beginning and end points are identical.)

Nonconservative Forces

Nonconservative forces acting within a system cause a change in the mechanical energy Emech of the system.

8.4      Changes in Mechanical  for Nonconservative  Forces

Ø  if the forces acting on objects within a system are conservative, then the mechanical energy of the system is conserved.

Ø  if some of the forces acting on objects within the system are not conservative, then the mechanical energy of the system changes. 

Consider the book sliding across the surface in the preceding section. As the book
moves through a distance d, the only force that does work on it is the force of kinetic friction. This force causes a decrease in the kinetic energy of the book. This decrease was calculated in Chapter 7, leading to 7.20, which we repeat here:

∆K   = -f_kd

                                      Changes kinetic energy = amount by which the       mechanical energy  of the changes because of the force of kinetic friction

o   In general, if a friction force acts within a system,

o   if the book moves on an incline that is not frictionless, there is a change in both the kinetic energy and the gravitational potential energy of the book–Earth system.

>> where ∆U is the change in all forms of potential energy

v  Change in mechanical energy of a system due to friction within the system

8.5  Relationship Between Conservative Forces and Potential Energy

The work done on a member of a system by a conservative force between the members does not depend on the path taken by the moving member. The work depends only on the initial and final coordinates.

A potential energy function U such that the work done by a conservative

force equals the decrease in the potential energy of the system,

The work done by a conservative force acting between members of a system equals the negative of the change in the potential energy associated with that force when the configurationof the system changes, where the change in the potential energy is defined as    

                                           

                                                    

o   If the point of application of the force undergoes an infinitesimal displacement
dx, we can express the infinitesimal change in the potential energy of the system dU as

o   Therefore, the conservative force is related to the potential energy function through

o   The x component of a conservative force acting on an object within a system equals the negative derivative of the potential energy of the system with respect to x

8.6 Energy Diagrams and Equilibrium of a System

The motion of a sy stem can often be understood qualitatively through a graph of its potential energy versus the position of a member of the system.

         

UNIVERSAL GRAVITATION

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.

Universal Gravitation

Chapter 13

Universal Gravitation

13.1 Newton’s Law of Universal Gravitation
You may have heard the legend that Newton was struck on the head by a falling apple while napping under a tree. This alleged accident supposedly prompted him to imagine that perhaps all objects in the Universe were attracted to each other in the same way the apple was attracted to the Earth. Newton analyzed astronomical data on the motion of the Moon around the Earth. From that analysis, he made the bold assertion that the force law governing the motion of planets was the same as the force law that attracted a falling apple to the Earth. This was the first time that “earthly” and “heavenly”
motions were unified. We shall look at the mathematical details of Newton’s analysis in this section.
In 1687 Newton published his work on the law of gravity in his treatise Mathematical
Principles of Natural Philosophy. Newton’s law of universal gravitation states that

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

If the particles have masses m1 and m2 and are separated by a distance r, the magnitude of this gravitational force is

Fg=G (m_1 m_2)/r2
(13.1)
with the statement:
F is greater than the gravitational force between two point masses
G is the gravitational constant = 6.67 x 10-11 Nm2 / kg2
m1 is the mass of the first point
m2 is the mass of the second point
r is the distance between the two mass points
Where G is a constant called the universal gravitational constant, which has been measured experimentally. Its value in SI units are:

G=6,673 x 10-11N.m2 / kg2
( 13.2)

The form of the force law given by Equation 13.1 is often referred to as an inverse square law because the magnitude of the force varies as the inverse square of the separation of the particles.
the magnitude of the force
exerted by the Earth on a particle of mass m near the Earth’s surface is
Fg=G (M_E m)/(R_E^2 )
(13.3)
where ME is the Earth’s mass and RE its radius. This force is directed toward the center of the Earth

Quick Quiz 13.1 The Moon remains in its orbit around the Earth rather
than falling to the Earth because
(a) it is outside of the gravitational influence of the
Earth
(b) it is in balance with the gravitational forces from the Sun and other planets
(c) the net force on the Moon is zero
(d) none of these
(e) all of these.
Answer :
(d). The gravitational force exerted by the Earth on the
Moon provides a net force that causes the Moon’s centripetal
acceleration.

13.2 Measuring the gravitational constan
experimentally that the force is attractive. The universal gravitational constant G was measured in an important experiment by Henry Cavendish (1731–1810) 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. The experiment is carefully repeated with different masses at various separations.
For get the value of G, the results show e, proportional to the product mM, and inversely proportional to the square of the distance r.

Quick Quiz 13.2 A 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 moon 2 is
(a) four times as large as that on moon 1
(b) twice as large as that on moon 1
(c) equal to that on moon 1
(d) half as large as that on moon 1
(e) one fourth as large as that on moon 1.

Answer

(e). The gravitational force follows an inverse-square behavior,
so doubling the distance causes the force to be
one fourth as large.




13.3 Free Fall Acceleration and the Gravitational Force
Because the magnitude of the force acting on a freely falling object of mass m near the Earth’s surface is given by Equation 13.4, we can equate mg to this force to obtain
mg=G (M_E m)/(R_E^2 )
g=G (M_E )/(R_E^2 )
Now consider an object of mass m located a distance h above the Earth’s surface or
a distance r from the Earth’s center, where r=RE + h. The magnitude of the gravitational force acting on this object is
Fg=G (M_E m)/r^2 =G 〖GM〗_E/(R_E+ h )2
The magnitude of the gravitational force acting on the object at this position is also
Fg = mg, where g is the value of the free-fall acceleration at the altitude h. Substituting this expression for Fg into the last equation shows that g is
g=(〖GM〗_E )/r^2 =〖GM〗_E/(R_E+ h )2

Thus, it follows that g decreases with increasing altitude. Because the weight of an object is
mg, we see that as r ∞ its weight approaches zero.

Quick Quiz 13.3 Superman 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
(a) depends on how fast the baseball is thrown
(b) is zero because the ball does not fall to the ground
(c) is slightly less than 9.80 m/s2
(d) is equal to 9.80 m/s2.

Answer
(c). 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.

Motion in two dimensions

MOTION IN TWO DIMENSIONS
 
In this chapter we explore the kinematics of particles moving in two dimensions. We begin by studying in more detail the nature of the position vector, velocity and acceleration.

DISPLACEMENT, VELOCITY AND ACCELERATION VECTORS

 Displacement

Displacement is a vector and the particle displacement is the difference between final and initial position. We now define the displacement vector for the particle Figure 4.1 as the difference between the final position vector and the initial position vector.

Direction Δr is shown in Figure 4.1. As we see from the picture, the amount less than the distance traveled along the curved path followed by the particles.

VELOCITY VECTOR
average speed of particles during a time interval Δt as the particle displacement divided by time interval

 Instantaneous velocity v is defined as the limit of an average speed of Δr / Δt as close to zero:

ACCELERATION VECTOR
Knowing the speed at points allows us to determine the average acceleration of the particle as it moves is defined as the change in instantaneous velocity vector v divided by the interval of time during which the changes occur

When the average acceleration of particles changes during different time intervals, it is useful to determine the instantaneous acceleration. Instantaneous acceleration is defined as the limiting value of the ratio close to zero

In other words, instantaneous acceleration equal to the derivative of the velocity vector with respect to time.

TWO DIMENSIONAL MOTION WITH CONSTANT VELOCITY

Two-dimensional motion during acceleration remained constant in both magnitude and direction. It would be useful to analyze some common types of motion.
The vector position of a particle moving in the xy plane:

                                                          r = x iˆ + y jˆ

• Where x, y and r change with time as a particle moving
• i and j remain constant

velocity vector as a function of time
If the position vector is known, the particle velocity can be obtained from equations 4.3 and 4.6, which gives

 

Hence, substituting from equation 2.9 into the equation 4.7

Graphical representation Equation 4.8

Vf is generally not along the direction of either Vi or because this is the relationship between the amount of expression vector. We can write in component form

 The position vector as a function of time X and y coordinates of a particle moving with constant acceleration

Substituting this expression into the equation r = 4.6 x i + y j which gives

This equation tells us that the position vector is the vector sum of the original position displacement arising from the initial particle velocity and displacement resulting from particle acceleration constant

Graphical representation Equation 4.9

Rf generally not along the direction of Vi or expression vector. We can write them in component form

PROJECTILE MOTION

Anyone who has watched a baseball moving projectile motion has been observed. The ball moves in a curved path, and motion are simple to analyze if we make two assumptions: (1) free-fall acceleration g is constant over the range of motion and directed downward and
(2) the effect of air resistance is negligible.dengan this assumption, we find that the path of projectiles, which we call the track, always a parabola.
Therefore, x and y components of the initial velocity is: