kinetics of particles 2
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Forces
The concept of force is useful because it enables the branches of mechanical science to be
brought together. For example, a knowledge of force required to accelerate a car makes it
possible to decide on the size of the engine and the transmission system. And force can be
treated as a currency between thermodynamics or electro technology or material science.
Newton's Three Laws of Motion
Newton's first law of motion states that if the vector sum of the forces acting on
an object is zero, then the object will remain at rest or remain moving at
constant velocity. If the force exerted on an object is zero, the object does not
necessarily have zero velocity. Without any forces acting on it, including
friction, an object in motion will continue to travel at constant velocity.
The Second Law
Newton's second law relates net force and acceleration. A net force on an
object will accelerate itthat is, change its velocity. The acceleration will be
proportional to the magnitude of the force and in the same direction as the
force. The proportionality constant is the mass, m, of the object.
F = ma
In theInternational System of Units (also known as SI, after the initials of
Systme International), acceleration, a, is measured in meters per second per
second. Mass is measured in kilograms; force, F, in newtons. A newton isdefined as the force necessary to impart to a mass of 1 kg an acceleration of 1
m/sec/sec; this is equivalent to about 0.2248 lb.
A massive object will require a greater force for a given acceleration than a
small, light object. What is remarkable is that mass, which is a measure of the
inertia of an object (inertia is its reluctance to change velocity), is also a
measure of the gravitational attraction that the object exerts on other objects.
It is surprising and profound that the inertial property and the gravitational
property are determined by the same thing. The implication of this
phenomenon is that it is impossible to distinguish at a point whether the pointis in a gravitational field or in an accelerated frame of reference. Einstein
made this one of the cornerstones of his general theory ofrelativity, which is
the currently accepted theory of gravitation.
Friction
Friction acts like a force applied in the direction opposite to an object's
velocity. For dry sliding friction, where no lubrication is present, the friction
force is almost independent of velocity. Also, the friction force does not depend
on the apparent area of contact between an object and the surface upon which
it slides. The actual contact areathat is, the area where the microscopic
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bumps on the object and sliding surface are actually touching each otheris
relatively small. As the object moves across the sliding surface, the tiny bumps
on the object and sliding surface collide, and force is required to move the
bumps past each other. The actual contact area depends on the perpendicular
force between the object and sliding surface. Frequently this force is just the
weight of the sliding object. If the object is pushed at an angle to thehorizontal, however, the downward vertical component of the force will, in
effect, add to the weight of the object. The friction force is proportional to the
total perpendicular force.
Where friction is present, Newton's second law is expanded to
The left side of the equation is simply the net effective force. (Acceleration will
be constant in the direction of the effective force). When an object moves
through a liquid, however, the magnitude of the friction depends on thevelocity. For most human-size objects moving in water or air (at subsonic
speeds), the resulting friction is proportional to the square of the speed.
Newton's second law then becomes
The proportionality constant, k, is characteristic of the two materials that are
sliding past each other, and depends on the area of contact between the two
surfaces and the degree of streamlining of the moving object.
The Third Law
Newton's third law of motion states that an object experiences a force because
it is interacting with some other object. The force that object 1 exerts on object
2 must be of the same magnitude but in the opposite direction as the force that
object 2 exerts on object 1. If, for example, a large adult gently shoves away a
child on a skating rink, in addition to the force the adult imparts on the child,
the child imparts an equal but oppositely directed force on the adult. Because
the mass of the adult is larger, however, the acceleration of the adult will be
smaller.
Newton's third law also requires the conservation ofmomentum, or the
product of mass and velocity. For an isolated system, with no external forces
acting on it, the momentum must remain constant. In the example of the adult
and child on the skating rink, their initial velocities are zero, and thus the
initial momentum of the system is zero. During the interaction, internal forces
are at work between adult and child, but net external forces equal zero.
Therefore, the momentum of the system must remain zero. After the adult
pushes the child away, the product of the large mass and small velocity of the
adult must equal the product of the small mass and large velocity of the child.
The momenta are equal in magnitude but opposite in direction, thus adding to
zero.
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Another conserved quantity of great importance is angular (rotational)
momentum. The angular momentum of a rotating object depends on its speed
of rotation, its mass, and the distance of the mass from the axis. When a skater
standing on a friction-free point spins faster and faster, angular momentum is
conserved despite the increasing speed. At the start of the spin, the skater's
arms are outstretched. Part of the mass is therefore at a large radius. As theskater's arms are lowered, thus decreasing their distance from the axis of
rotation, the rotational speed must increase in order to maintain constant
angular momentum.1
Newtons Laws of Motion in Summary
(1) Everything continues in its state of rest of rectilinear motion unless disturbed by a force.
(The frame of reference is ignored here!)
(2) The rate of change of momentum ( m x a ) is proportional to the force acting on the bodyand is in the direction of the force (i.e. F = ma)
(3) To each force, there is an equal and opposite reaction.
By Newtons 3rd law, when two objects collide, the sum of momentum before and after must
be equal.
I.e. m1u1 + m2u2 = m1v1 + m2v2
m1 (v1 - u1) = m2 (v2 - u2)
therefore
m
m
v u
v u
speed of mass
speed of mass
2
1
1 1
2 2
1
2
Therefore the 3rd law provides a means of measuring mass and also lead to the principle of
conservation of momentum.
1"Mechanics,"Microsoft Encarta 97 Encyclopedia. 1993-1996 Microsoft Corporation.
All rights reserved.
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SYSTEM
EQU
ILIBRIUM
(STATICS)
Forces
V
ector
spring
friction
GravitationN
ewton'slawsof
motion
Moment,M=rxF
Axialforce,shear
fo
rce
Bendingmoment,
twistingmoment
Solidmechanicsor
strengtho
fmaterials
kinetics
conservationof
momentum
secondla
w
thirdlaw&
firstlaw
thirdlaw
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Spring force
Hookes law
F = k x
therefore
F mx
therefore
kx mx 0
where kis the stiffness of spring.
Friction Force
The friction force between two dry lubricated surface is a quantity which depends on a large
number of factors. (e.g. Ra , clearance, relative velocity), however, consideration of an ideal
case known as Coulomb friction is often regarded as adequate.
The friction force is assumed to take any value
up to a maximum limiting value and
F = N
coefficient of limiting friction
In practice, varies with speed and drops
markedly when sliding begins to occur.
m
natural length
x
F
x
+ve
N
N
FF
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Gravitation, the force of attraction between all objects that tends to pull them
toward one another. It is a universalforce, affecting the largest and smallest
objects, all forms ofmatter, andenergy. Gravitation governs the motion of
astronomical bodies. It keeps the moon in orbit around the earth and keeps the
earth and the other planets of the solar system in orbit around the sun. On a
larger scale, it governs the motion of stars and slows the outward expansion ofthe entire universe because of the inward attraction of galaxies to other
galaxies. Typically the term gravitation refers to the force in general, and the
term gravity refers to the earth's gravitational pull.
Gravitation is one of the four fundamental forces of nature, along with
electromagnetism and the weak and strong nuclear forces, which hold together
the particles that make up atoms. Gravitation is by far the weakest of these
forces and, as a result, is not important in the interactions of atoms and
nuclear particles or even of moderate-sized objects, such as people or cars.
Gravitation is important only when very large objects, such as planets, areinvolved. This is true for several reasons. First, the force of gravitation
reaches great distances, while nuclear forces operate only over extremely
short distances and decrease in strength very rapidly as distance increases.
Second, gravitation is always attractive. In contrast, electromagnetic forces
between particles can be repulsive or attractive depending on whether the
particles both have a positive or negative electrical charge, or they have
opposite electrical charges (seeElectricity). These attractive and repulsive
forces tend to cancel each other out, leaving only a weak net force. Gravitation
has no repulsive force and, therefore, no such cancellation or weakening.
The gravitational attraction of objects for one another is the easiest
fundamental force to observe and was the first fundamental force to be
described with a complete mathematical theory by the English physicist and
mathematician Sir Isaac Newton. A more accurate theory called general
relativity was formulated early in the 20th century by the German-born
American physicistAlbert Einstein. Scientists recognize that even this theory is
not correct for describing how gravitation works in certain circumstances, and
they continue to search for an improved theory.
Earth's Gravitation
Gravitation plays a crucial role in most processes on the earth. The ocean
tides are caused by the gravitational attraction of the moon and the sun on the
earth and its oceans. Gravitation drives weather patterns by making cold air
sink and displace less dense warm air, forcing the warm air to rise. The
gravitational pull of the earth on all objects holds the objects to the surface of
the earth. Without it, the spin of the earth would send them floating off into
space.
The gravitational attraction of every bit of matter in the earth for every otherbit of matter amounts to an inward pull that holds the earth together against
the pressure forces tending to push it outward. Similarly, the inward pull of
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gravitation holds stars together. When a star's fuel nears depletion, the
processes producing the outward pressure weaken and the inward pull of
gravitation eventually compresses the star to a very compact size (see Star,
Black Hole).
Acceleration
If an object held near the surface of the earth is released, it will fall and
accelerate, or pick up speed, as it descends. This acceleration is caused by
gravity, the force of attraction between the object and the earth. The force of
gravity on an object is also called the object's weight. This force depends on
the object's mass, or the amount of matter in the object. The weight of an
object is equal to the mass of the object multiplied by the acceleration due to
gravity.
A bowling ball that weighs 16 lb is actually being pulled toward the earth with
a force of 16 lb. In the metric system, the bowling ball is pulled toward theearth with a force of 71 newtons (a metric unit of force abbreviated N). The
bowling ball also pulls on the earth with a force of 16 lb (71 N), but the earth
is so massive that it does not move appreciably. In order to hold the bowling
ball up and keep it from falling, a person must exert an upward force of 16 lb
(71 N) on the ball. This upward force acts to oppose the 16 lb (71 N)
downward weight force, leaving a total force of zero. The total force on an
object determines the object's acceleration.
If the pull of gravity is the only force acting on an object, then all objects,
regardless of their weight, size, or shape, will accelerate in the same manner.At the same place on the earth, the 16 lb (71 N) bowling ball and a 500 lb
(2200 N) boulder will fall with the same rate of acceleration. As each second
passes, each object will increase its downward speed by about 9.8 m/sec (32
ft/sec), resulting in an acceleration of 9.8 m/sec/sec (32 ft/sec/sec). In
principle, a rock and a feather both would fall with this acceleration if there
were no other forces acting. In practice, however, air exerts a significant
upward force on the falling feather and makes it fall more slowly.
The mass of an object does not change as it is moved from place to place, but
the acceleration due to gravity, and therefore the object's weight, will changebecause the strength of the earth's gravitational pull is not the same
everywhere. The earth's pull and the acceleration due to gravity decrease as
an object moves farther away from the center of the earth. At an altitude of
4000 miles (6400 km) above the earth's surface, for instance, the bowling ball
would weigh only about 4 lb (18 N). Because of the reduced weight force, the
rate of acceleration of the bowling ball at that altitude would be only one
quarter of the acceleration rate at the surface of the earth. The pull of gravity
on an object also changes slightly with latitude. Because the earth is not
perfectly spherical, but bulges at the equator, the pull of gravity is about 0.5percent stronger at the earth's poles than at the equator.
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Early Ideas About Gravitation
The ancient Greek philosophers developed several theories about the force
that caused objects to fall toward the earth. In the 4th century BC, the Greek
philosopherAristotle proposed that all things were made from some
combination of the four elements, earth, air, fire, and water. Objects that were
similar in nature attracted one another, and as a result, objects with moreearth in them were attracted to the earth. Fire, by contrast, was dissimilar and
therefore tended to rise from the earth. Aristotle also developed a cosmology,
that is, a theory describing the universe, that was geocentric, or earth-
centered, with the moon, sun, planets, and stars moving around the earth on
spheres. The Greek philosophers, however, did not propose a connection
between the force behind planetary motion and the force that made objects fall
toward the earth.
At the beginning of the 17th century, the Italian physicist and astronomer
Galileo discovered that all objects fall toward the earth with the sameacceleration, regardless of their weight, size, or shape, when gravity is the
only force acting on them. Galileo also had a theory about the universe, which
he based on the ideas of the Polish astronomerNicolaus Copernicus. In the
mid-16th century, Copernicus had proposed a heliocentric, or sun-centered
system, in which the planets moved in circles around the sun, and Galileo
agreed with this cosmology. However, Galileo believed that the planets moved
in circles because this motion was the natural path of a body with no forces
acting on it. Like the Greek philosophers, he saw no connection between the
force behind planetary motion and gravitation on earth.
In the late 16th and early 17th centuries the heliocentric model of the universe
gained support from observations by the Danish astronomerTycho Brahe, and
his student, the German astronomerJohannes Kepler. These observations,
made without telescopes, were accurate enough to determine that the planets
did not move in circles, as Copernicus had suggested. Kepler calculated that
the orbits had to be ellipses (slightly elongated circles). The invention of the
telescope made even more precise observations possible, and Galileo was one
of the first to use a telescope to study astronomy. In 1609 Galileo observed
that moons orbited the planet Jupiter, a fact that could not reasonably fit intoan earth-centered model of the heavens.
The new heliocentric theory changed scientists' views about the earth's place
in the universe and opened the way for new ideas about the forces behind
planetary motion. However, it was not until the late 17th century that Isaac
Newton developed a theory of gravitation that encompassed both the attraction
of objects on the earth and planetary motion.
Newton's Theory of Gravitation
To develop his theory of gravitation, Newton first had to develop the science offorces and motion calledmechanics. Newton proposed that the natural motion
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of an object is motion at a constant speed on a straight line, and that it takes a
force to slow down, speed up, or change the path of an object. Newton also
inventedcalculus, a new branch of mathematics that became an important tool
in the calculations of his theory of gravitation.
Newton proposed his law of gravitation in 1687 and stated that every particle
in the universe attracts every other particle in the universe with a force thatdepends on the product of the two particles' masses divided by the square of
the distance between them. The gravitational force between two objects can be
expressed by the following equation: F= GMm/d2 where F is the gravitational
force, G is a constant known as the universal constant of gravitation, M and m
are the masses of each object, and d is the distance between them. Newton
considered a particle to be an object with a mass that was concentrated in a
small point. If the mass of one or both particles increases, then the attraction
between the two particles increases. For instance, if the mass of one particle is
doubled, the force of attraction between the two particles is doubled. If thedistance between the particles increases, then the attraction decreases as the
square of the distance between them. Doubling the distance between two
particles, for instance, will make the force of attraction one quarter as great as
it was.
According to Newton, the force acts along a line between the two particles. In
the case of two spheres, it acts along the line between their centers. The
attraction between objects with irregular shapes is more complicated. Every
bit of matter in the irregular object attracts every bit of matter in the other
object. A simpler description is possible near the surface of the earth wherethe pull of gravity is approximately uniform in strength and direction. In this
case there is a point in an object (even an irregular object) called the center of
gravity, at which all the force of gravity can be considered to be acting.
Newton's law affects all objects in the universe, from raindrops in the sky to
the planets in the solar system. It is therefore known as the universal law of
gravitation. In order to know the strength of gravitational forces in general,
however, it became necessary to find the value of G, the universal constant of
gravitation. Scientists needed to perform an experiment, but gravitational
forces are very weak between objects found in a common laboratory and thus
hard to observe. In 1798 the English chemist and physicistHenry Cavendish
finally measured G with a very sensitive experiment in which he nearly
eliminated the effects of friction and other forces. The value he found was
6.754 x 10-11 N-m2/kg2close to the currently accepted value of 6.670 x 10-11N-m2/kg2 (a decimal point followed by 10 zeros and then the number 6670).
This value is so small that the force of gravitation between two objects with a
mass of 1 metric ton each, 1 meter from each other, is about 67 millionths of a
newton, or about 15 millionths of a pound.
Gravitation may also be described in a completely different way. A massive
object, such as the earth, may be thought of as producing a condition in space
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around it called a gravitationalfield. This field causes objects in space to
experience a force. The gravitational field around the earth, for instance,
produces a downward force on objects near the earth surface. The field
viewpoint is an alternative to the viewpoint that objects can affect each other
across distance. This way of thinking about interactions has proved to be very
important in the development of modern physics.
Planetary Motion
Newton's law of gravitation was the first theory to accurately describe the
motion of objects on the earth as well as the planetary motion that
astronomers had long observed. According to Newton's theory, the
gravitational attraction between the planets and the sun holds the planets in
elliptical orbits around the sun. The earth's moon and moons of other planets
are held in orbitby the attraction between the moons and the planets.
Newton's law led to many new discoveries, the most important of which was
the discovery of the planetNeptune. Scientists had noted unexplainablevariations in the motion of the planetUranus for many years. Using Newton's
law of gravitation, the French astronomer Urbain Leverrier and the British
astronomer John Couch each independently predicted the existence of a more
distant planet that was perturbing the orbit of Uranus. Neptune was
discovered in 1884, in an orbit close to its predicted position.
Problems with Newton's Theory
Scientists used Newton's theory of gravitation successfully for many years.
Several problems began to arise, however, involving motion that did not followthe law of gravitation or Newtonian mechanics. One problem was the observed
and unexplainable deviations in the orbit ofMercury (which could not be
caused by the gravitational pull of another orbiting body).
Another problem with Newton's theory involved reference frames, that is, the
conditions under which an observer measures the motion of an object.
According to Newtonian mechanics, two observers making measurements of
the speed of an object will measure different speeds if the observers are
moving relative to each other. A person on the ground observing a ball that is
on a train passing by will measure the speed of the ball as the same as thespeed of the train. A person on the train observing the ball, however, will
measure the ball's speed as zero. According to the traditional ideas about
space and time, then, there could not be a constant, fundamental speed in the
physical world because all speed is relative. However, near the end of the 19th
century the Scottish physicistJames Clerk Maxwell proposed a complete
theory of electric and magnetic forces that contained just such a constant,
which he called c. This constant speed was 300,000 km/sec (186,000 mi/sec)
and was the speed of electromagnetic waves, including light waves. This
feature of Maxwell's theory caused a crisis in physics because it indicated thatspeed was not always relative.
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Albert Einstein resolved this crisis in 1905 with his special theory of relativity.
An important feature of Einstein's new theory was that no particle, and even
no information, could travel faster than the fundamental speed c. In Newton's
gravitation theory, however, information about gravitation moved at infinite
speed. If a star exploded into two parts, for example, the change in
gravitational pull would be felt immediately by a planet in a distant orbitaround the exploded star. According to Einstein's theory, such forces were not
possible.
Though Newton's theory contained several flaws, it is still very practical for
use in everyday life. Even today, it is sufficiently accurate for dealing with
earth-based gravitational effects such as in geology (the study of the formation
of the earth and the processes acting on it), and for most scientific work in
astronomy. Only when examining exotic phenomena such as black holes
(points in space with a gravitational force so strong that not even light can
escape them) or in explaining the big bang (the origin of the universe) isNewton's theory inaccurate or inapplicable.
Einstein's Theory of Relativity
In 1915 Einstein formulated a new theory of gravitation that reconciled the
force of gravitation with the requirements of his theory of special relativity. He
proposed that gravitational effects move at the speed of c. He called this
theory general relativity to distinguish it from special relativity, which only
holds when there is no force of gravitation. General relativity produces
predictions very close to those of Newton's theory in most familiar situations,
such as the moon orbiting the earth. Einstein's theory differed from Newton's
theory, however, in that it described gravitation as a curvature of space and
time.
In Einstein's theory of general relativity, he proposed that space and time may
be united into a single, four-dimensional geometry consisting of 3 space
dimensions and 1 time dimension. In this geometry, called spacetime, the
motions of particles from point to point as time progresses are represented by
curves called world lines. If there is no gravity acting, the most natural lines in
this geometry are straight lines, and they represent particles that are movingalways in the same direction with the same speedthat is, particles that have
no force acting on them. If a particle is acted on by a force, then its world line
will not be straight. Einstein also proposed that the effect of gravitation should
not be represented as the deviation of a world line from straightness, as it
would be for an electrical force. If gravitation is present, it should not be
considered a force. Rather, gravitation changes the most natural world lines
and thereby curves the geometry of spacetime. In a curved geometry, such as
the two-dimensional surface of the earth, there are no straight lines. Instead,
there are special curves called geodesics, an example of which are great
circles around the earth. These special curves are at each point as straight as
possible, and they are the most natural lines in a curved geometry. The effect
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of gravitation would be to influence the geodesics in spacetime. Near sources
of gravitation the space is strongly curved and the geodesics behave less and
less like those in flat, uncurved spacetime. In the solar system, for example, the
effect of the sun and the earth is to cause the moon to move on a geodesic that
winds around the geodesic of the earth 12 times a year.
Testing Einstein's Theory
Einstein's theory required verification, but the level of precision in
measurement needed to distinguish between Einstein's theory and Newton's
theory was difficult to achieve in the early 20th century and remains so today.
There were two predictions, however, that could be tested. One involved
deviations in the orbit of Mercury. Astronomers had observed that the ellipse
of Mercury's orbit itself rotatedthat is, the point nearest the sun, called the
perihelion, slowly advanced around the sun. The rate of advancement
predicted by Newton's theory differed slightly from what astronomers had
measured, but Einstein's theory predicted the correct rate.
The second test involved measuring the bending of light as it passed around
the sun. Both Newton's and Einstein's theories predicted that light would be
deflected by gravitation. But the amount of deflection predicted by the two
theories differed. The light to be measured in such a test originates in distant
stars. However, under ordinary conditions the sun's brightness prevents
scientists from observing the light from these stars. This problem disappears
during an eclipse, when the moon blocks the sun's light. In 1919 a special
British expedition took photographs during an eclipse. Scientists measured the
deflection of starlight as it passed by the sun and arrived at numbers that
agreed with Einstein's prediction. Subsequent eclipse observations also have
confirmed Einstein's theory.
Other physicists have proposed relativistic theories of gravitation to compete
with Einstein's. In these competing theories, almost all of which are
geometrical like Einstein's, gravitational effects move at the speed c. They
differ mostly in the mathematical details. Even using the technology of the late
20th century, scientists still find it very difficult to test these theories with
experiments and observations. But Einstein's theory has passed all tests thathave been made so far.
Applications of Einstein's Theory
Einstein's general relativity theory predicts special gravitational conditions.
The Big Bang theory, which describes the origin and early expansion of the
universe, is one conclusion based on Einstein's theory that has been verified in
several independent ways.
Another conclusion suggested by general relativity, as well as other relativistic
theories of gravitation, is that gravitational effects move in waves.Astronomers have observed a loss of energy in a pair of neutron stars (stars
composed of densely packed neutrons) that are orbiting each other. The
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astronomers theorize that energy-carrying gravitational waves are radiating
from the pair, depleting the stars of their energy. Very violent astrophysical
events, such as the explosion of stars or the collision of neutron stars, can
produce gravitational waves strong enough that they may eventually be
directly detectable with extremely precise instruments. Astrophysicists are
designing such instruments with the hope that they will be able to detectgravitational waves by the beginning of the 21st century.
Another gravitational effect predicted by general relativity is the existence of
black holes. The idea of a star with a gravitational force so strong that light
cannot escape from its surface can be traced to Newtonian theory. Einstein
modified this idea in his theory of general relativity. Because light cannot
escape from a black hole, for any objecta particle, spacecraft, or wavetoescape, it would have to move past light. But light moves outward at the speed
c. According to relativity, c is the highest attainable speed, so nothing can pass
it. The black holes that Einstein envisioned, then, allow no escape whatsoever.An extension of this argument shows that when gravitation is this strong,
nothing can even stay in the same place, but must move inward. Even the
surface of a star must move inward, and must continue the collapse that
created the strong gravitational force. What remains then is not a star, but a
region of space from which emerges a tremendous gravitational force.
Other Modern Theories
Einstein's theory of gravitation revolutionized 20th-century physics. Another
important revolution that took place was quantum theory. Quantum theory
states that physical interactions, or the exchange of energy, cannot be made
arbitrarily small. There is a minimal interaction that comes in a packet called
the quantum of an interaction. For electromagnetism the quantum is called the
photon. Like the other interactions, gravitation also must be quantized.
Physicists call a quantum of gravitational energy a graviton. In principle,
gravitational waves arriving at the earth would consist of gravitons. In
practice, gravitational waves would consist of apparently continuous streams
of gravitons, and individual gravitons could not be detected.
Einstein's theory did not include quantum effects. For most of the 20th century,theoretical physicists have been unsuccessful in their attempts to formulate a
theory that resembles Einstein's theory but also includes gravitons. Despite the
lack of a complete quantum theory, it is possible to make some partial
predictions about quantized gravitation. In the 1970s, British physicist Stephen
Hawking showed that quantum mechanical processes in the strong
gravitational pull just outside of black holes would create particles and quanta
that move away from the black hole, thereby robbing it of energy.
Theory of Everything
An important trend in modern theoretical physics is to find a theory ofeverything (TOE), in which all four of the fundamental forces are seen to be
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really different aspects of the same single universal force. Physicists already
have unified electromagnetism and the weak nuclear force and have made
progress in joining these two forces with the strong nuclear force (see Grand
Unification Theories). However, relativistic gravitation, with its geometric and
mathematically complex character, poses the most difficult challenge. Einstein
spent most of his later years searching for an all-encompassing theory bytrying to make electromagnetism geometrical like gravitation. The modern
approach is to try to make gravitation fit the pattern of the other fundamental
forces. Much of this work involves looking for mathematical patterns. A TOE
is difficult to test using experiments because the effects of a TOE would be
important only in the rarest circumstances.2
Gravitational ForcesBecause the moon has significantly less mass than the earth, the weight of an object on
its surface is only one-sixth the objects weight on the earths surface. This graph shows
how the weight of an object with weight w on earth varies with respect to its position
between the earth and moon. Since the earth and moon pull in opposite directions, there
2"Gravitation,"Microsoft Encarta 97 Encyclopedia. 1993-1996 Microsoft Corporation.
All rights reserved.
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is a point, 346,000 km (215,000 mi) from the earth, where the opposite gravitational
forces cancel, and the weight is zero. Microsoft Illustration3
Gravitation
From Issac Newton,
F = G m m
d
1 2
2(1)
F= force of attraction between 2 bodies of masses m1& m2
d= distance between m1 & m2
G = universal gravitational constant
= (6.670 +/- 0.005) x 10-11
m3s2kg
Example :-
mass of Earth = 5.98 x 10
24
kgradius of Earth = 6.368 x 106
m
From (1), for a 1 kg mass on earth surface
F=
6 607 10 5 98 10 1
6 368 10
11 24
62
. .
.
x x x x
x
= 9.8516 N
If this force acts on a unit mass, the acceleration is 9.8516 m/s2
( often called the accelerationdue to gravity and is given symbol g )
Therefore Gravitational force = m x g
However, since the Earth is not a perfect sphere and it also rotates, the declared standard g
value is 9.80665 m/s2
or N/kg
Energy
The quantity calledenergy ties together all branches of physics. In the field of
mechanics, energy must be provided to do work; work is defined as the
product of force and the distance an object moves in the direction of the force.
When a force is exerted on an object but the force does not cause the object to
move, no work is done. Energy and work are both measured in the same
unitsergs, joules, or foot-pounds, for example.
If work is done lifting an object to a greater height, energy has been stored in
the form of gravitational potential energy. Many other forms of energy exist:
electric and magnetic potential energy; kinetic energy; energy stored in
3"Gravitational Forces,"Microsoft Encarta 97 Encyclopedia. 1993-1996 Microsoft
Corporation. All rights reserved.
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stretched springs, compressed gases, or molecular bonds; thermal energy; and
mass itself. In all transformations from one kind of energy to another, the total
energy is conserved. For instance, if work is done on a rubber ball to raise it,
its gravitational potential energy is increased. If the ball is then dropped, the
gravitational potential energy is transformed to kinetic energy. When the ball
hits the ground, it becomes distorted and thereby creates friction between themolecules of the ball material. This friction is transformed into heat, or
thermal energy.4
Work & Kinetic Energy
Since F = m a
= mdv
dt
therefore in its component form
Fx = mdv
dtm
dx
dt
dv
dxmv
dv
dx
x x
x
x
Similarly, Fy = mvydv
dy
y
Fz = mvz
dv
dz
z
therefore F dx mv dv mv C x x x x 1
2
2
Similarly F dy mv C y y 1
2
2
F dz mv C z z 1
2
2
Therefore F dx F dy F dz mv Cons t x y z 1
2
2tan
where v2 = vx2
+ vy2
+ vz2
therefore Fds = (Fx ii + Fy j + Fz k) (dx i + dy j + dz k)
= (Fx dx + Fy dy + Fz dz)
4"Mechanics,"Microsoft Encarta 97 Encyclopedia. 1993-1996 Microsoft Corporation.
All rights reserved.
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Therefore Fds =1
2
2mv constant
Fds - work done by force Fwhen acting on a particle moving along a given
path.
1
2
2mv - kinetic energy of the particle
dimension of K.E. is kgm2/s
2or (kg m/s
2)m or Nm or J
note :- both work done & energy are scalar.
Power - rate at which work is performed.
I.e. Power = d
dtwork
= ddt
Fds
=d
dtF
ds
dtdt
= Fds
dt= F v (1)
or Power = d
dtK E
d
dtmv. .
1
2
2
Combine with (1), therefore F v = mvdv
dt
= mva
Dimension :- Nm/s or J/s or Watts(W)