The chapter is divided into two fascinating and captivating sections. One speaks about Kepler's Laws of Planetary
Motion, of course, Universal Gravitation, and Weighing the Earth. The other touches on Motion of Planets and Satellites,
Weight and Weightlessness, The Gravitation Field, and Einstein's Theory of Gravity.
8.1 Motion In The Heavens and On Earth
This section starts off on Kepler's Laws of Planetary Motion. These laws came about when he assisted the Danish
astonomer Tycho Brahe. Kepler took Brahe's data and did a careful and accurate mathematical analysis, he discovered
three laws. These laws clearly state the planets move in elliptical orbits. They like to sweep out or move in equal areas
of equal time, and the ratio of the square of the periods of any two planets is equivalent to the ratio of the cube of
their distances to the sun. Here's an equation to show this law:
(Ta/Tb)2 = (ra/rb)3.
Next, it speaks about Newton's Law of Universal Gravitation. He had used mathematical facts and equations to
demonstrate that if the passage way of a planet were an ellipse,in agreement with Kepler's first law, then the net force F,
on the planet must vary differently with the square of the distance between the planet and the sun. Here's the equation
and the explanation of the variables here, known as the inverse square law:
F=(porportional to) 1/d2
Then, he later wrote about the falling apple which made him think about the problem
of the motion of planets. He realized that the apple descended because Earth attracted it.
He believed that the apple attracts the earth and that the force of the attraction is
porportional to the mass of the earth. In essence, the same force of attraction acted between
any two masses
F=G m1m2/d2
Newton used his inverse square law into universal gravitation. He utilized the symbol
Mp for the mass of the planet, Ms for the masss of the sun, and rps for the radius of the
planet's orbit. Then he used his second law of motion, F=m*a, with F the gravitational force
and the a the centripedal acceleration. That F= Mpa.
The equation is: G= MsMp/rps2 =
.
He then rearranged the equation in the form: Tp2=(4*Pi2/GM
s)rps3
At the end of the section, it describes how Cavendish was the first to equalize the
gravitational attraction between two bodies on earth. Here are the equations:
F= GMem/r2 g= GMe/r2.
He then later changed it to: Me= gre2/G.
8.2 Using the Law of Universal Gravitation
The second part of this chapter begins to discuss about motion of planets and satellites.
It states that satellites in a circular orbit accelerate towards planet Earth at a rate
equivalent to the acceleration of gravitiy at it's orbital radius. For instance, a satellite
in an orbit that always has a similar height above Earth moves with uniform circular motion
with its centripedal acceleration is ac= v2/r. Using Newton's second law, F=ma with the
gravitational force between Earth and the satellite. As the equation states:
Fgrav = (G* Msat * MCentral ) / r2=mv2/r.
Solving for velocity:
Using Newton's law of universal gravitation, we have shown that the time for a
satellite to circle Earth is given by: T= 2*Pi SQ.RT r3/GME.
The mass of the central body, like the sun, would replace ME in the equation and
r
is the distance from the sun to the orbiting body.
Next, weight amd weightlessness was talked about and had a few equations. For example,
due to earth's gravitation can be found by using a combination of the inverse sqaure law
and Newton's second law:
F=GMEm/d2=ma so a= GME/d2 but on the Earth's surface, this equation can be rewritten as:
GME/d2. On the earth's surface, the equation can be written: g=GM
E/RE2
thus, a= g(RE/d)2
As it goes further away from the Earth's center, the acceleration due to gravity is
decreased according to this inverse square relationship. How can someone measure weight? You can
choose to stand on a spring scale. Weight is found by the force the scale gives off in opposing
force of gravity.
Gravitation field is said to take all bodies surrounding them can signify
a collection of vectors representing the force per unit mass in all locations.
To find out what's strength of the gravitational force, a small body of mass m in the
field and measure the forc. The field's strength is defined as g, to be the force being
divided by a unit mass, F/m. This is measured in newtons per kilogram as seen here: g= F/m. The strength of the field differs inversly with the square of the distance from
the center of Earth.
Lastly, Einstein's theory was the last topic spoken about. It
describes gravitation attraction as a property of space in itself. Einstein's theory is not
fully theoretical and doesn't thoroughly explain how masses curve space, but physicists out
there today are still working and making a huge effort to comprehend the true nature of
gravity.
Main Resource: Merrill- Physics ~Principles and Problems, 1992
Universal Gravitation To see futher information on Universal Gravitation, you can look at these 4 internet links and
5 magazine/newspaper links
Momentum and Conservation is such an intriguing and impacting chapter. Ths chapter is as
well broken into two sections. The first one focuses on. Momentum and Impulse and Angular
Momentum. The next one touches on Newton's Third Law and Momentum,Internal and External Force,
and Conservation of Momentum in Two Dimensions. This will be one interesting chapter.
9.1 Impulse and Change in Momentum Momentum is the product or result of the mass and velcity of a body.
The equation that represents momentum is p=mv. The unit for momentum is kilogram*meters
(kg*m/s). Refering to Newton's First Law, if there is no net force on a body, the velocity
is constant. If there is only a single object that is isolated, it means that the mass can not
change. If the velocity is also constant so is the momentum, the product of mass and velocity.
Basically, it states, if a there is a single body that has no net force acting on it, the
momentum is constant and conserved.
Newton's second law depicts the process of how the velocity of a body is altered by a
force acting on it as seen in this equation: F= m*s= m (delta)v/(delta)t.
Next, you multiply both sides of the equation: F(delta)t = m(delta)v.
On the left side of this equation, the product of the net force and time over which
it reacts to is known as the impulse. An impulse is a vector in the direction of the
known force. Impulses is measured in the units of newtons*seconds (N*s). For example, if
the mass of an object is uniform, a change in he velocity will occur in a change in the momentum.
(Delta)p=m(delta)v.
Then, the impulse is given to an objct is equal to the change in the momentum:
F(delta)=(Delta)p.
This main equation is recognized as the impulse-momentum theorum. The exactness
between the impulse and changing momentums is another way of writing Newton's second law. It's
quite frequent that the force is not uniform during the time it's exerted, meaning that the
average force is used in the impulse-momentum theorum.
A large change in momentum happens only when there is a large impulse. In addition,
a large impulse can result from either a large force acting over a short time, or a smaller force
acting over time. Next, his section talks about angular momentum. Angular Momentum
is the quantity of angular motion that the same or in common links to linear momentum. If there
is no torque on an object that when he angular momentum is constant. A perfect example of
angular momentum is a huge rotating mass in the air to form a massive and destructive hurricane.
This occurs because the air possesses a large amount of angular momentum. Another example
is angular momentum of planets around the sun is constant, meaning that when a planet's
distances from the sun is bigger, the velocity is smaller.
9.2 The Conservation of Momentum
On the next and final section of this chapter, it begins to talk about
Newton's third law and momentum. To talk about and describe this topic, we are first going to
compare it to a bat and a baseball. A ball that's hit is involved with some type of collision. In
this collision, the momentum of the ball is changed as a result of a given impulse by the bat.
The bat released a force on the ball. By Newton's third law, the ball has exerted a force on the
ball of equal magnitude on the opposite direction. This law is another way to express the statement
of the law of conservation. Here are one its known equations. The first, is the new momentum of
the first object:p'A= pA+(delta)p. The next one shows change of
momentum in the second object:p'B= pB+([-delta]p).
This happens in the first equation because the first object is moving with momentum pA and
the second object with pB. They both collide and the second object releases force
+F on the first object. During that collision, the first object exerts a force on the
second object. According to Newton's third law, this first object exerts on the second force has
an equal amount in magnitude but opposite in direction to the force on the second object exerted
on the first object. The momentum of the second object lessens while the momentum of the first
object increases. The momentum lost by the second object equals to the momentum gained by the
first object. The whole system consisting of two objects, the net change in the initial momentum
of the system: PA+PB= P'A+P'B. The total momentum
before the collision is the same as the total momentum after the collision. The momentum of the
system has not been altered; it's conserved.
The next topic that this section focuses on is the Law of Momentum. The law states:
The momentum of any closed, isolated system does not change. It doesn't matter how
many objects are in the system but it's necessary that there are no objects entering or leaving the
system and that there is no net external force on the system. This is an equation for initial
momentum: PA+PB+PB= mvB.
After the collision of the two objects that have some velocity: v'A=v'B
=v' . Due to the masses that are equal: p'A=p'B=mv'.
The final momentum of the system is: p'A+p'B=2mv'.
The Law of Conservation of Momentum states:
pA+pB=p'A
mvB=2mv
vB=2v'
or v'=1/2vB
.
Internal forces are forces between objects within a system. The total momentum
of a system is conserved or remained the same only when it's closed and isolated from external
forces. An external forceis when a force is not isolated and the momentum of the
system doesn't remain the same.
At the end of this section and chapter, it states that conservation of momentum is looked
at into two dimensions. At first, that momentum was looked at into one dimension only. Now, it's
looked at into two dimensions due to the laws of conservation of momentum and vertical and
horizontal components.
Main Resoures:Merrill Physics ~ Principles and Problems
1992 Chapter 9- Momentum and Its Conservation Here are 4 other internet sources and 5 newspaper and magazine resources to look more in depth
on Momentum and Its Conservation:
Internet Links:
infodemophysicsphysics Newspaper and Magazine Links:
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