Module 5
Chapters 5-8
A. Work
Energy is present in the Universe in a variety of forms, including mechanical,
chemical, electromagnetic, and nuclear energy. Even the inert mass of everyday matter
contains a very large amount of energy. Although energy can be transformed from one
kind to another, all observations and experiments to date suggest that the total amount of
energy in the Universe never changes. That’s also true for an isolated system, which is a
collection of objects that can exchange energy with each other, but not with the rest of the
Universe. If one form of energy in an isolated system decreases, then another form of
energy in the system must increase. For example, if the system consists of a motor
connected to a battery, the battery converts chemical energy to electrical energy and the
motor converts electrical energy to mechanical energy. Understanding how energy
changes from one form to another is essential in all the sciences. In this topic the focus is
mainly on mechanical energy, which is the sum of kinetic energy, the energy associated
with motion, and potential energy—the energy associated with relative position. Using an
energy approach to solve certain problems is often much easier than using forces and
Newton’s three laws. These two very different approaches are linked through the concept
of work.
Work has a different meaning in physics than it does in everyday usage. In the
physics definition, a physics textbook author does very little work typing away at a
computer. A mason, by contrast, may do a lot of work laying concrete blocks. In physics,
work is done only if an object is moved through some displacement while a force is
applied to it. If either the force or displacement is doubled, the work is doubled. Double
them both, and the work is quadrupled. Doing work involves applying a force to an
object while moving it a given distance.
It’s easy to see the difference between the physics definition and the everyday
definition of work. The author exerts very little force on the keys of a keyboard, creating
only small displacements, so relatively little physics work is done. The mason must exert
much larger forces on concrete blocks and move them significant distances, and so
performs a much greater amount of work. Even very tiring tasks, however, may not
constitute work according to the physics definition.
Work is a scalar quantity—a number rather than a vector—and consequently is
easier to handle. No direction is associated with it. Further, work doesn’t depend
explicitly on time, which can be an advantage in problems involving only velocities and
positions. Because the units of work are those of force and distance, the SI unit is the
newton-meter. Another name for the newton-meter is the joule ( J) (rhymes with “pool”).
The U.S. customary unit of work is the foot-pound, because distances are measured in
feet and forces in pounds in that system.
Work always requires a system of more than just one object. A nail, for example,
can’t do work on itself, but a hammer can do work on the nail by driving it into a board.
In general, an object may be moving under the influence of several external forces. In
that case, the net work done on the object as it undergoes some displacement is just the
sum of the amount of work done by each force. Work can be either positive or negative.
Frictional work is extremely important in everyday life because doing almost any
other kind of work is impossible without it. The man in the last example, for instance,
depends on surface friction to pull his sled. Otherwise, the rope would slip in his hands
and exert no force on the sled, while his feet slid out from underneath him and he fell flat
on his face. Cars wouldn’t work without friction, nor could conveyor belts, nor even our
muscle tissue.
The work done by pushing or pulling an object is the application of a single force.
Friction, on the other hand, is a complex process caused by numerous microscopic
interactions over the entire area of the surfaces in contact (Fig. 5.7). Consider a metal
block sliding over a metal surface. Microscopic “teeth” in the block encounter equally
microscopic irregularities in the underlying surface. Pressing against each other, the teeth
deform, get hot, and weld to the opposite surface. Work must be done breaking these
temporary bonds, and that comes at the expense of the energy of motion of the block, to
be discussed in the next section. The energy lost by the block goes into heating both the
block and its environment, with some energy converted to sound. The friction force of
two objects in contact and in relative motion to each other always dissipates energy in
these relatively complex ways. For our purposes, the phrase “work done by friction” will
denote the effect of these processes on mechanical energy alone.
Solving problems using Newton’s second law can be difficult if the forces
involved are complicated. An alternative is to relate the speed of an object to the net work
done on it by external forces. If the net work can be calculated for a given displacement,
the change in the object’s speed is easy to evaluate. It turns out there are two general
kinds of forces. The first is called a conservative force. Gravity is probably the best
example of a conservative force. To understand the origin of the name, think of a diver
climbing to the top of a 10-meter platform. The diver has to do work against gravity in
making the climb. Once at the top, however, she can recover the work as kinetic energy
by taking a dive. Her speed just before hitting the water will give her a kinetic energy
equal to the work she did against gravity in climbing to the top of the platform, minus the
effect of some nonconservative forces, such as air drag and internal muscular friction.
A nonconservative force is generally dissipative, which means that it tends to
randomly disperse the energy of bodies on which it acts. This dispersal of energy often
takes the form of heat or sound. Kinetic friction and air drag are good examples.
Propulsive forces, like the force exerted by a jet engine on a plane or by a propeller on a
submarine, are also nonconservative. Work done against a nonconservative force can’t be
easily recovered.
B. Gravitational Potential Energy
An object with kinetic energy (energy of motion) can do work on another object,
just like a moving hammer can drive a nail into a wall. A brick on a high shelf can also
do work: it can fall off the shelf, accelerate downward, and hit a nail squarely, driving it
into the floorboards. The brick is said to have potential energy associated with it, because
from its location on the shelf it can potentially do work. Potential energy is a property of
a system, rather than of a single object, because it’s due to the relative positions of
interacting objects in the system, such as the position of the diver in Figure 5.11 relative
to the Earth. In this topic we define a system as a collection of objects interacting via
forces or other processes that are internal to the system. It turns out that potential energy
is another way of looking at the work done by conservative forces.
Using the work–energy theorem in problems involving gravitation requires
computing the work done by gravity. For most trajectories—say, for a ball traversing a
parabolic arc—finding the gravitational work done on the ball requires sophisticated
techniques from calculus. Fortunately, for conservative fields there’s a simple alternative:
potential energy. Gravity is a conservative force, and for every conservative force, a
special expression called a potential energy function can be found. Evaluating that
function at any two points in an object’s path of motion and finding the difference will
give the negative of the work done by that force between those two points. It’s also
advantageous that potential energy, like work and kinetic energy, is a scalar quantity.
Our first step is to find the work done by gravity on an object when it moves from
one position to another. The negative of that work is the change in the gravitational
potential energy of the system, and from that expression, we’ll be able to identify the
potential energy function. In Figure 5.13, a book of mass m falls from a height yi to a
height yf , where the positive y - coordinate represents position above the ground. We
neglect the force of air friction, so the only force acting on the book is gravitation.
In solving problems involving gravitational potential energy, it’s important to
choose a location at which to set that energy equal to zero. Given the form of Equation
5.11, this is the same as choosing the place where y 5 0. The choice is completely
arbitrary because the important quantity is the difference in potential energy, and this
difference will be the same regardless of the choice of zero level. However, once this
position is chosen, it must remain fixed for a given problem.
Conservation principles play a very important role in physics. When a physical
quantity is conserved the numeric value of the quantity remains the same throughout the
physical process. Although the form of the quantity may change in some way, its final
value is the same as its initial value. The kinetic energy KE of an object falling only
under the influence of gravity is constantly changing, as is the gravitational potential
energy PE. Obviously, then, these quantities aren’t conserved. When nonconservative
forces are involved along with gravitation, the full work– energy theorem must be used,
often with techniques from Topic 4. Solving problems requires the basic procedure of the
problem-solving strategy for conservation-of- energy problems in the previous section.
C. Spring Potential Energy
Springs are important elements in modern technology. They are found in
machines of all kinds, in watches, toys, cars, and trains. Springs will be introduced here,
then studied in more detail in Topic 13. Work done by an applied force in stretching or
compressing a spring can be recovered by removing the applied force, so like gravity, the
spring force is conservative, as long as losses through internal friction of the spring can
be neglected. That means a potential energy function can be found and used in the work–
energy theorem.
It’s important to remember that the work done by gravity and springs in any given
physical system is already included on the right-hand side of Equation 5.18 as potential
energy and should not also be included on the left as work. Figure 5.21c shows how the
stored elastic potential energy can be recovered. When the block is released, the spring
snaps back to its original length, and the stored elastic potential energy is converted to
kinetic energy of the block. The elastic potential energy stored in the spring is zero when
the spring is in the equilibrium position (x 5 0). As given by Equation 5.17, potential
energy is also stored in the spring when it’s stretched. Further, the elastic potential energy
is a maximum when the spring has reached its maximum compression or extension.
If the mechanical energy is changing, it has to be going somewhere. The energy
either leaves the system and goes into the surrounding environment, or it stays in the
system and is converted into a nonmechanical form such as thermal energy. A simple
example is a block sliding along a rough surface. Friction creates thermal energy,
absorbed partly by the block and partly by the surrounding environment. When the block
warms up, something called internal energy increases. The internal energy of a system is
related to its temperature, which in turn is a consequence of the activity of its parts, such
as the motion of atoms in a gas or the vibration of atoms in a solid.
The most important feature of the energy approach is the idea that energy is
conserved; it can’t be created or destroyed, only transferred from one form into another.
This is the principle of conservation of energy. The principle of conservation of energy is
not confined to physics. In biology, energy transformations take place in myriad ways
inside all living organisms. One example is the transformation of chemical energy to
mechanical energy that causes flagella to move and propel an organism.
The phenomenon of bioluminescence, wherein living organisms produce light
through chemical reactions, represents a fascinating intersection of biology and
chemistry, offering insights into the diverse strategies employed by organisms to thrive in
their respective ecosystems. While the exact mechanisms underlying bioluminescence in
bacteria remain a subject of ongoing research and inquiry, the significance of this
phenomenon extends far beyond mere illumination, impacting ecological dynamics and
species interactions in profound ways.
Indeed, the utilization of chemical energy to produce light confers distinct
advantages to organisms inhabiting various environments, ranging from the depths of the
ocean to terrestrial ecosystems. Within the microbial realm, certain bacteria have evolved
intricate biochemical pathways that culminate in the emission of light, a phenomenon that
has captivated scientists and naturalists for centuries. Despite the complexity of these
mechanisms, the adaptive significance of bioluminescence is readily apparent, offering
organisms a means of communication, camouflage, predation, and defense.
One striking example of the ecological relevance of bioluminescence is evident in
the symbiotic relationship between bioluminescent bacteria and certain marine
organisms, such as deep-sea fish. These fish possess specialized structures, such as light
organs or photophores, housing colonies of light-emitting bacteria. The emitted light
serves as a lure, attracting unsuspecting prey or potential mates in the dark depths of the
ocean. This symbiotic partnership underscores the intricate interplay between organisms
and their environments, showcasing the remarkable adaptations that enable survival and
reproduction in extreme habitats.
Moreover, the ecological ramifications of bioluminescence extend beyond
individual organisms to encompass entire ecosystems and trophic interactions. In the case
of bioluminescent bacteria, the emitted light can serve as a beacon, attracting larger
predators or facilitating the dispersal of nutrients and organic matter. This phenomenon,
known as "bioluminescent bloom," can have cascading effects on food webs and nutrient
cycling, influencing the distribution and abundance of species throughout marine and
terrestrial environments.
Furthermore, the study of bioluminescence offers valuable insights into the
principles of biochemical regulation and metabolic pathways, shedding light on the
intricate machinery underlying cellular processes. By unraveling the molecular
mechanisms responsible for light production in bacteria, scientists gain a deeper
understanding of fundamental biological processes and the evolution of specialized
adaptations in diverse organisms.
In summary, the phenomenon of bioluminescence in bacteria represents a
captivating example of the ingenuity and diversity of life on Earth. From the depths of the
ocean to terrestrial ecosystems, bioluminescent organisms play integral roles in
ecological dynamics, species interactions, and biochemical regulation. By delving into
the mechanisms and ecological significance of bioluminescence, scientists continue to
unravel the mysteries of the natural world, unveiling the intricate web of life that sustains
our planet.
D. Power
Power, the rate at which energy is transferred, is important in the design and use
of practical devices, such as electrical appliances and engines of all kinds. The concept of
power, however, is essential whenever a transfer of any kind of energy takes place. The
issue is particularly interesting for living creatures because the maximum work per
second, or power output, of an animal varies greatly with output duration.
The horsepower was first defined by Watt, who needed a large power unit to rate
the power output of his new invention, the steam engine. The watt is commonly used in
electrical applications, but it can be used in other scientific areas as well. For example,
European sports car engines are rated in kilowatts. It’s important to realize that a
kilowatt-hour is a unit of energy, not power. When you pay your electric bill, you’re
buying energy, and that’s why your bill lists a charge for electricity of about 10
cents/kWh. The amount of electricity used by an appliance can be calculated by
multiplying its power rating (usually expressed in watts and valid only for normal
household electrical circuits) by the length of time the appliance is operated.
The stationary jump consists of two parts: extension and free flight.2 In the
extension phase the person jumps up from a crouch, straightening the legs and throwing
up the arms; the free-flight phase occurs when the jumper leaves the ground. Because the
body is an extended object and different parts move with different speeds, we describe
the motion of the jumper in terms of the position and velocity of the center of mass (CM),
which is the point in the body at which all the mass may be considered to be
concentrated. In other words, the work done by a variable force acting on an object that
undergoes a displacement is equal to the area under the graph of Fx versus x.
Conservative forces are special: Work done against them can be recovered—it’s
conserved. An example is gravity: The work done in lifting an object through a height is
effectively stored in the gravity field and can be recovered in the kinetic energy of the
object simply by letting it fall. Nonconservative forces, such as surface friction and drag,
dissipate energy in a form that can’t be readily recovered. Gravitational work and
gravitational potential energy should not both appear in the work–energy theorem at the
same time, only one or the other, because they’re equivalent. Setting the work due to
nonconservative forces to zero and substituting the expressions for KE and PE.
E. Momentum And Impulse
How does the impact affect the motion of each vehicle, and what basic physical
principles determine the likelihood of serious injury? How do rockets work, and what
mechanisms can be used to overcome the limitations imposed by exhaust speed? Why do
we have to brace ourselves when firing small projectiles at high velocity? Finally, how
can we use physics to improve our golf game? To begin answering such questions, we
introduce momentum. Intuitively, anyone or anything that has a lot of momentum is
going to be hard to stop. In politics, the term is metaphorical. Physically, the more
momentum an object has, the more force has to be applied to stop it in a given time. This
concept leads to one of the most powerful principles in physics: conservation of
momentum. Using this law, complex collision problems can be solved without knowing
much about the forces involved during contact. We’ll also be able to derive information
about the average force delivered in an impact. With conservation of momentum, we’ll
have a better understanding of what choices to make when designing an automobile or a
moon rocket, or when addressing a golf ball on a tee.
In physics, momentum has a precise definition. A slowly moving brontosaurus
has a lot of momentum, but so does a little hot lead shot from the muzzle of a gun. We
therefore expect that momentum will depend on an object’s mass and velocity. Doubling
either the mass or the velocity of an object doubles its momentum; doubling both
quantities quadruples its momentum. Momentum is a vector quantity with the same
direction as the object’s velocity.
In real-life situations, the force on an object is only rarely constant. For example,
when a bat hits a baseball, the force increases sharply, reaches some maximum value, and
then decreases just as rapidly. Figure 6.1a shows a typical graph of force versus time for
such incidents. The force starts out small as the bat comes in contact with the ball, rises to
a maximum value when they are firmly in contact, and then drops off as the ball leaves
the bat.
The main injuries that occur to a person hitting the interior of a car in a crash are
brain damage, bone fracture, and trauma to the skin, blood vessels, and internal organs.
Here, we compare the rather imprecisely known thresholds for human injury with typical
forces and accelerations experienced in a car crash. A force of about 90 kN (20 000 lb)
compressing the tibia can cause fracture. Although the breaking force varies with the
bone considered, we may take this value as the threshold force for fracture. It’s well
known that rapid acceleration of the head, even without skull fracture, can be fatal.
Estimates show that head accelerations of 150g experienced for about 4 ms or 50g for
60Hms are fatal 50% of the time. Such injuries from rapid acceleration often result in
nerve damage to the spinal cord where the nerves enter the base of the brain.
When a collision occurs in an isolated system, the total momentum of the system
doesn’t change with the passage of time. Instead, it remains constant both in magnitude
and in direction. The momenta of the individual objects in the system may change, but
the vector sum of all the momenta will not change. The total momentum is therefore said
to be conserved. In this section, we will see how the laws of motion lead us to this
important conservation law.
A collision may be the result of physical contact between two objects, as
illustrated in Figure 6.6a. This is a common macroscopic event, as when a pair of billiard
balls or a baseball and a bat strike each other. By contrast, because contact on a
submicroscopic scale is hard to define accurately, the notion of collision must be
generalized to that scale. Forces between two objects arise from the electrostatic
interaction of the electrons in the surface atoms of the objects. As will be discussed in
Topic 15, electric charges are either positive or negative. Charges with the same sign
repel each other, while charges with opposite sign attract each other. To understand the
distinction between macroscopic and microscopic collisions, consider the collision
between two positive charges, as shown in Figure 6.6b. Because the two particles in the
figure are both positively charged, they repel each other. During such a microscopic
collision, particles need not touch in the normal sense in order to interact and transfer
momentum.
Defining the isolated system is an important feature of applying this conservation
law. A cheerleader jumping upwards from rest might appear to violate conservation of
momentum, because initially her momentum is zero and suddenly she’s leaving the
ground with velocity v S. The flaw in this reasoning lies in the fact that the cheerleader
isn’t an isolated system. In jumping, she exerts a downward force on Earth, changing its
momentum. This change in Earth’s momentum isn’t noticeable, however, because of
Earth’s gargantuan mass compared to the cheerleader’s. When we define the system to be
the cheerleader and Earth, momentum is conserved.
Action and reaction, together with the accompanying exchange of momentum
between two objects, is responsible for the phenomenon known as recoil. Everyone
knows that throwing a baseball while standing straight up, without bracing one’s feet
against Earth, is a good way to fall over backwards. This reaction, an example of recoil,
also happens when you fire a gun or shoot an arrow. Conservation of momentum
provides a straightforward way to calculate such effects, as the next example shows.
F. Collisions In One Dimension
We have seen that for any type of collision, the total momentum of the system just
before the collision equals the total momentum just after the collision as long as the
system may be considered isolated. The total kinetic energy, on the other hand, is
generally not conserved in a collision because some of the kinetic energy is converted to
internal energy, sound energy, and the work needed to permanently deform the objects
involved, such as cars in a car crash. We define an inelastic collision as a collision in
which momentum is conserved, but kinetic energy is not. The collision of a rubber ball
with a hard surface is inelastic, because some of the kinetic energy is lost when the ball is
deformed during contact with the surface. When two objects collide and stick together,
the collision is called perfectly inelastic. For example, if two pieces of putty collide, they
stick together and move with some common velocity after the collision. If a meteorite
collides head on with Earth, it becomes buried in Earth and the collision is considered
perfectly inelastic. Only in very special circumstances is all the initial kinetic energy lost
in a perfectly inelastic collision.
An elastic collision is defined as one in which both momentum and kinetic energy
are conserved. Billiard ball collisions and the collisions of air molecules with the walls of
a container at ordinary temperatures are highly elastic. Macroscopic collisions such as
those between billiard balls are only approximately elastic, because some loss of kinetic
energy takes place—for example, in the clicking sound when two balls strike each other.
Perfectly elastic collisions do occur, however, between atomic and subatomic particles.
Elastic and perfectly inelastic collisions are limiting cases; most actual collisions fall into
a range in between them.
As a practical application, an inelastic collision is used to detect glaucoma, a
disease in which the pressure inside the eye builds up and leads to blindness by damaging
the cells of the retina. In this application, medical professionals use a device called a
tonometer to measure the pressure inside the eye. This device releases a puff of air
against the outer surface of the eye and measures the speed of the air after reflection from
the eye. At normal pressure, the eye is slightly spongy, and the pulse is reflected at low
speed. As the pressure inside the eye increases, the outer surface becomes more rigid, and
the speed of the reflected pulse increases. In this way, the speed of the reflected puff of
air can measure the internal pressure of the eye.
Consider two objects having masses m 1 and m 2 moving with known initial
velocity components v1i and v 2i along a straight line, as in Figure 6.11a. If the two
objects collide head-on, stick together, and move with a common velocity component vf
after the collision, then the collision is perfectly inelastic. Now consider two objects that
undergo an elastic head-on collision (Fig. 6.14). In this situation, both the momentum and
the kinetic energy of the system of two objects are conserved.
For a general collision of two objects in three-dimensional space, the conservation
of momentum principle implies that the total momentum of the system in each direction
is conserved. However, an important subset of collisions takes place in a plane. The game
of billiards is a familiar example involving multiple collisions of objects moving on a
two-dimensional surface. We restrict our attention to a single twodimensional collision
between two objects that takes place in a plane, and ignore any possible rotation.
When ordinary vehicles such as cars and locomotives move, the driving force of
the motion is friction. In the case of the car, this driving force is exerted by the road on
the car, a reaction to the force exerted by the wheels against the road. Similarly, a
locomotive “pushes” against the tracks; hence, the driving force is the reaction force
exerted by the tracks on the locomotive. However, a rocket moving in space has no road
or tracks to push against. How can it move forward?
In fact, reaction forces also propel a rocket. (You should review Newton’s third
law, discussed in Topic 4.) To illustrate this point, we model our rocket with a spherical
chamber containing a combustible gas, as in Figure 6.18a. When an explosion occurs in
the chamber, the hot gas expands and presses against all sides of the chamber, as
indicated by the arrows. Because the sum of the forces exerted on the rocket is zero, it
doesn’t move. Now suppose a hole is drilled in the bottom of the chamber, as in Figure
6.18b. When the explosion occurs, the gas presses against the chamber in all directions,
but can’t press against anything at the hole, where it simply escapes into space. Adding
the forces on the spherical chamber now results in a net force upward. Just as in the case
of cars and locomotives, this is a reaction force. A car’s wheels press against the ground,
and the reaction force of the ground on the car pushes it forward. The wall of the rocket’s
combustion chamber exerts a force on the gas expanding against it. The reaction force of
the gas on the wall then pushes the rocket upward.
In a now infamous 1920 article in The New York Times, rocket pioneer Robert
Goddard was ridiculed for thinking that rockets would work in space, where, according to
the Times, there was nothing to push against. The Times retracted, rather belatedly,
during the first Apollo moon landing mission in 1969. The hot gases are not pushing
against anything external, but against the rocket itself—and ironically, rockets actually
work better in a vacuum. In an atmosphere, the gases have to do work against the outside
air pressure to escape the combustion chamber, slowing the exhaust velocity and
reducing the reaction force. At the microscopic level, this process is complicated, but it
can be simplified by applying conservation of momentum to the rocket and its ejected
fuel.
G. Angular Velocity And Angular Acceleration
The rotation of the Earth creates the cycle of day and night, the rotation of wheels
enables easy vehicular motion, and modern technology depends on circular motion in a
variety of contexts, from the tiny gears in a Swiss watch to the operation of lathes and
other machinery. The concepts of angular velocity, angular acceleration, and centripetal
acceleration are central to understanding the motions of a diverse range of phenomena,
from a car moving around a circular racetrack to clusters of galaxies orbiting a common
center. Rotational motion, when combined with Newton’s law of universal gravitation
and his laws of motion, can also explain certain facts about space travel and satellite
motion, such as where to place a satellite so it will remain fixed in position over the same
spot on the Earth.
Exploring the realm of gravitational potential energy and its implications for
energy conservation unveils a wealth of insights into the dynamics of celestial bodies and
the laws governing their motion. By extending the principles of gravitational potential
energy to encompass broader scenarios, such as planetary systems and beyond, one gains
a comprehensive framework for understanding phenomena ranging from orbital
mechanics to cosmic-scale dynamics.
At the heart of this conceptual framework lies the notion of gravitational potential
energy, which quantifies the energy associated with the gravitational interactions between
objects in a gravitational field. By generalizing this concept to encompass planetary
systems, astronomers and physicists can analyze the energy dynamics of celestial bodies
with precision and insight. This extension enables the derivation of key results, such as
the planetary escape speed, which delineates the minimum velocity required for an object
to overcome the gravitational pull of a massive body and venture into space.
Moreover, the concept of energy conservation emerges as a fundamental principle
guiding the dynamics of celestial systems. By recognizing that the total energy of a
system remains constant over time, barring external influences, astronomers can predict
and analyze various phenomena with remarkable accuracy. Energy conservation provides
a powerful tool for elucidating the intricacies of orbital mechanics, gravitational
interactions, and the evolution of celestial systems over time.
In this context, Kepler's three laws of planetary motion assume paramount
importance, serving as foundational pillars upon which Newtonian mechanics and the
modern understanding of gravity are built. These laws, formulated by the astronomer
Johannes Kepler in the 17th century, encapsulate fundamental insights into the motion of
celestial bodies within a gravitational framework.
Kepler's first law, often referred to as the law of orbits, states that planets move in
elliptical orbits around the Sun, with the Sun situated at one of the focal points of the
ellipse. This law revolutionized our understanding of the structure of the solar system,
replacing the earlier conception of circular orbits with a more accurate representation
based on elliptical geometry.
Kepler's second law, known as the law of equal areas, describes the rate at which
a planet sweeps out equal areas in equal intervals of time as it orbits the Sun. This law
highlights the non-uniform nature of planetary motion, with planets moving faster when
closer to the Sun and slower when farther away, in accordance with the conservation of
angular momentum.
Finally, Kepler's third law, often termed the law of periods, establishes a
mathematical relationship between the orbital periods and distances of planets from the
Sun. Specifically, it states that the square of the orbital period of a planet is proportional
to the cube of its average distance from the Sun. This law provides a quantitative
framework for understanding the dynamics of planetary motion within the solar system,
paving the way for further advancements in celestial mechanics.
Collectively, Kepler's laws of planetary motion, when combined with Newton's
law of universal gravitation and the concept of energy conservation, form the cornerstone
of our modern understanding of gravity and orbital dynamics. By elucidating the
fundamental principles governing the motion of celestial bodies, these laws empower
astronomers and physicists to unravel the mysteries of the cosmos and unlock the secrets
of the universe.
H. Tangential Velocity, Tangential Acceleration, And Centripetal Acceleration
Angular variables are closely related to linear variables. Consider the arbitrarily
shaped object in Figure 7.5 rotating about the z - axis through the point O. The tangential
velocity of a point on a rotating object equals the distance of that point from the axis of
rotation multiplied by the angular velocity. Equation 7.10 shows that the tangential
velocity of a point on a rotating object increases as that point is moved outward from the
center of rotation toward the rim, as expected; however, every point on the rotating object
has the same angular velocity. Tangential speed is the magnitude of the tangential
velocity, and is also called the linear speed.
The tangential acceleration of a point on a rotating object equals the distance of
that point from the axis of rotation multiplied by the angular acceleration. Again, radian
measure must be used for the angular acceleration term in this equation. Before MP3s
and streaming became the mediums of choice for recorded music, compact discs (CDs),
and phonographs were popular. There are similarities and differences between the
rotational motion of phonograph records and that of CDs.
CDs, on the other hand, are designed so that the disc moves under the laser pickup
at a constant tangential speed. Because the pickup moves radially as it follows the tracks
of information, the angular speed of the CD must vary according to the radial position of
the laser. Because the tangential speed is fixed, the information density (per length of
track) anywhere on the disc is the same.
The numerator represents the difference between the velocity vectors v S f and v
S i . These vectors may have the same magnitude, corresponding to the same speed, but if
they have different directions, their difference can’t equal zero. The direction of the car’s
velocity as it moves in the circular path is continually changing. For circular motion at
constant speed, the acceleration vector always points toward the center of the circle. Such
an acceleration is called a centripetal (center-seeking) acceleration.
Newton’s second law of motion can be applied to problems involving circular
motion. In the case of uniform circular motion, that law will feature the centripetal
acceleration and a variety of radial forces that are either directed towards or away from
the center of the circular motion. Just as acceleration and force are vector quantities, so
are centripetal acceleration and the radial forces that appear in the statement of Newton’s
second law for uniform circular motion. After discussing the concepts and sign
conventions, Newton’s second law for uniform circular motion will be applied in some
elementary physical contexts.
An object can have a centripetal acceleration only if some external force acts on
it. For a ball whirling in a circle at the end of a string, that force is the tension in the
string. In the case of a car moving on a flat circular track, the force is friction between the
car and track. A satellite in circular orbit around Earth has a centripetal acceleration due
to the gravitational force between the satellite and Earth. Some books use the term
“centripetal force,” which can give the mistaken impression that it is a new force of
nature. This is not the case: The adjective “centripetal” in “centripetal force” simply
means that the force in question acts toward a center. The force of tension in the string of
a yo-yo whirling in a vertical circle is an example of a centripetal force, as is the force of
gravity on a satellite circling the Earth.
A radial force is a vector and has a direction. The second law for uniform circular
motion involves forces that are directed either towards the center of a circle or away from
it. A force acting towards the center of the circle is by convention negative. Examples
include the gravity force on a satellite or the string tension of a whirling yo-yo. A force
acting away from the center of the circle is positive. Examples include the normal force
on a car traveling over the circular crest of a hill or the force of repulsion between like
electric charges. Similarly, the centripetal acceleration is negative because it acts towards
the center of the circle.
Anyone who has ridden a merry-go-round as a child (or as a fun-loving grown-
up) has experienced what feels like a “center-fleeing” force. Holding onto the railing and
moving toward the center feels like a walk up a steep hill. Actually, this so-called
centrifugal force is fictitious. In reality, the rider is exerting a centripetal force on her
body with her hand and arm muscles. In addition, a smaller centripetal force is exerted by
the static friction between her feet and the platform. If the rider’s grip slipped, she
wouldn’t be flung radially away; rather, she would go off on a straight line, tangent to the
point in space where she let go of the railing. The rider lands at a point that is farther
away from the center, but not by “fleeing the center” along a radial line. Instead, she
travels perpendicular to a radial line, traversing an angular displacement while increasing
her radial displacement.
I. Newtonian Gravitation
Prior to 1686, a great deal of data had been collected on the motions of the Moon
and planets, but no one had a clear understanding of the forces affecting them. In that
year, Isaac Newton provided the key that unlocked the secrets of the heavens. He knew
from the first law that a net force had to be acting on the Moon. If it were not, the Moon
would move in a straight-line path rather than in its almost circular orbit around Earth.
Newton reasoned that it was the same kind of force that attracted objects—such as apples
—close to the surface of the Earth. He called it the force of gravity.
Another important fact is that the gravitational force exerted by a uniform sphere
on a particle outside the sphere is the same as the force exerted if the entire mass of the
sphere were concentrated at its center. This is called Gauss’ law, after the German
mathematician and astronomer Karl Friedrich Gauss, and is also true of electric fields,
which we will encounter in Topic 15. Gauss’ law is a mathematical result, true because
the force falls off as an inverse square of the separation between the particles.
The gravitational constant G in Equation 7.21 was first measured in an important
experiment by Henry Cavendish in 1798. His apparatus consisted of two small spheres,
each of mass m, fixed to the ends of a light horizontal rod suspended by a thin metal wire,
as in Figure 7.16. Two large spheres, each of mass M, were placed near the smaller
spheres. The attractive force between the smaller and larger spheres caused the rod to
rotate in a horizontal plane and the wire to twist. The angle through which the suspended
rod rotated was measured with a light beam reflected from a mirror attached to the
vertical suspension. (Such a moving spot of light is an effective technique for amplifying
motion.) The experiment was carefully repeated with different masses at various
separations. In addition to providing a value for G, the results showed that the force is
attractive, proportional to the product mM, and inversely proportional to the square of the
distance r. Modern forms of such experiments are carried out regularly today in an effort
to determine G with greater precision.
If an object is projected upward from Earth’s surface with a large enough speed, it
can soar off into space and never return. This speed is called Earth’s escape speed. (It is
also commonly called the escape velocity, but in fact is more properly a speed.) The
movements of the planets, stars, and other celestial bodies have been observed for
thousands of years. In early history scientists regarded Earth as the center of the
Universe. This geocentric model was developed extensively by the Greek astronomer
Claudius Ptolemy in the second century AD and was accepted for the next 1 400 years. In
1543 Polish astronomer Nicolaus Copernicus (1473–1543) showed that Earth and the
other planets revolve in circular orbits around the Sun (the heliocentric model).
Danish astronomer Tycho Brahe (pronounced “brah” or “brah´ huh”; 1546–1601)
made accurate astronomical measurements over a period of 20 years, providing the data
for the currently accepted model of the solar system. Brahe’s precise observations of the
planets and 777 stars were carried out with nothing more elaborate than a large sextant
and compass; the telescope had not yet been invented. German astronomer Johannes
Kepler, who was Brahe’s assistant, acquired Brahe’s astronomical data and spent about
16 years trying to deduce a mathematical model for the motions of the planets. After
many laborious calculations, he found that Brahe’s precise data on the motion of Mars
about the Sun provided the answer. Kepler’s analysis first showed that the concept of
circular orbits about the Sun had to be abandoned. He eventually discovered that the orbit
of Mars could be accurately described by an ellipse with the Sun at one focus. He then
generalized this analysis to include the motions of all planets.
The complete analysis is summarized in three statements known as Kepler’s laws.
The first law arises as a natural consequence of the inversesquare nature of Newton’s law
of gravitation. Any object bound to another by a force that varies as 1/r 2 will move in an
elliptical orbit. As shown in Figure 7.20a, an ellipse is a curve drawn so that the sum of
the distances from any point on the curve to two internal points called focal points or foci
(singular, focus) is always the same. The semimajor axis a is half the length of the line
that goes across the ellipse and contains both foci. For the Sun–planet configuration (Fig.
7.20b), the Sun is at one focus and the other focus is empty. Because the orbit is an
ellipse, the distance from the Sun to the planet continuously changes.
Kepler’s second law states that a line drawn from the Sun to any planet sweeps
out equal areas in equal time intervals. The derivation of Kepler’s third law is simple
enough to carry out for the special case of a circular orbit. Consider a planet of mass Mp
moving around the Sun, which has a mass of MS, in a circular orbit. Because the orbit is
circular, the planet moves at a constant speed v.
The orbits of most of the planets are very nearly circular. Comets and asteroids,
however, usually have elliptical orbits. For these orbits, the radius r must be replaced
with a, the semimajor axis—half the longest distance across the elliptical orbit. (This is
also the average distance of the comet or asteroid from the Sun.) A more detailed
calculation shows that KS actually depends on the sum of both the mass of a given planet
and the Sun’s mass.
J. Torque
The reality is that the point of application of a force does matter. In football, for
example, if the ball carrier is tackled near his midriff, he might carry the tackler several
yards before falling. If tackled well below the waistline, however, his center of mass
rotates toward the ground, and he can be brought down immediately. Tennis provides
another good example. If a tennis ball is struck with a strong horizontal force acting
through its center of mass, it may travel a long distance before hitting the ground, far out-
of- bounds. Instead, the same force applied in an upward, glancing stroke will impart
topspin to the ball, which can cause it to land in the opponent’s court. The concepts of
rotational equilibrium and rotational dynamics are also important in other disciplines. For
example, students of architecture benefit from understanding the forces that act on
buildings, and biology students should understand the forces at work in muscles and on
bones and joints. These forces create torques, which tell us how the forces affect an
object’s equilibrium and rate of rotation.
We will find that an object remains in a state of uniform rotational motion unless
acted on by a net torque. That principle is the equivalent of Newton’s first law. Further,
the angular acceleration of an object is proportional to the net torque acting on it, which
is the analog of Newton’s second law. A net torque acting on an object causes a change in
its rotational energy. Finally, torques applied to an object through a given time interval
can change the object’s angular momentum. In the absence of external torques, angular
momentum is conserved, a property that explains some of the mysterious and formidable
properties of pulsars, remnants of supernova explosions that rotate at equatorial speeds
approaching that of light.
Forces cause accelerations; torques cause angular accelerations. There is a
definite relationship, however, between the two concepts. The same perpendicular force
applied at a point nearer the hinge results in a smaller angular acceleration. In general, a
larger radial distance r between the applied force and the axis of rotation results in a
larger angular acceleration. Similarly, a larger applied force will also result in a larger
angular acceleration.
Under these conditions, an object can rotate around the chosen axis in one of two
directions. By convention, counterclockwise is taken to be the positive direction,
clockwise the negative direction. When an applied force causes an object to rotate
counterclockwise, the torque on the object is positive. When the force causes the object to
rotate clockwise, the torque on the object is negative. When two or more torques act on
an object at rest, the torques are added. If the net torque isn’t zero, the object starts
rotating at an ever-increasing rate. If the net torque is zero, the object’s rate of rotation
doesn’t change. These considerations lead to the rotational analog of the first law: the rate
of rotation of an object doesn’t change, unless the object is acted on by a net torque.
These three equations are identical to the equations for a similar concept called
center of mass. The center of mass and center of gravity of an object are exactly the same
when g doesn’t vary significantly over the object. In this topic, the concepts of center of
gravity and center of mass will be used interchangeably. It’s often possible to guess the
location of the center of mass. The center of mass of a homogeneous, symmetric body
must lie on the axis of symmetry. For example, the center of mass of a homogeneous rod
lies midway between the ends of the rod, and the center of mass of a homogeneous
sphere or a homogeneous cube lies at the geometric center of the object. The center of
mass of an irregularly shaped object, such as a wrench, can be determined experimentally
by suspending the wrench from two different arbitrary points (Fig. 8.9). The wrench is
first hung from point A, and a vertical line AB (which can be established with a plumb
bob) is drawn when the wrench is in equilibrium. The wrench is then hung from point C ,
and a second vertical line CD is drawn. The center of mass coincides with the intersection
of these two lines. In fact, if the wrench is hung freely from any point, the center of mass
always lies straight below the point of support, so the vertical line through that point must
pass through the center of mass.
K. The Rotational Second Law Of Motion
When a rigid object is subject to a net torque, it undergoes an angular acceleration
that is directly proportional to the net torque. This result, which is analogous to Newton’s
second law, is derived as follows. The system shown in Figure 8.19 consists of an object
of mass m connected to a very light rod of length r. The rod is pivoted at the point O, and
its movement is confined to rotation on a frictionless horizontal table. Assume that a
force Ft acts perpendicular to the rod and hence is tangent to the circular path of the
object.
Consider a solid disk rotating about its axis as in Figure 8.20a. The disk consists
of many particles at various distances from the axis of rotation. The gear system on a
bicycle provides an easily visible example of the relationship between torque and angular
acceleration. Consider first a five-speed gear system in which the drive chain can be
adjusted to wrap around any of five gears attached to the back wheel (Fig. 8.22). The
gears, with different radii, are concentric with the wheel hub. When the cyclist begins
pedaling from rest, the chain is attached to the largest gear. Because it has the largest
radius, this gear provides the largest torque to the drive wheel. A large torque is required
initially, because the bicycle starts from rest. As the bicycle rolls faster, the tangential
speed of the chain increases, eventually becoming too fast for the cyclist to maintain by
pushing the pedals.
The chain is then moved to a gear with a smaller radius, so the chain has a smaller
tangential speed that the cyclist can more easily maintain. This gear doesn’t provide as
much torque as the first, but the cyclist needs to accelerate only to a somewhat higher
speed. This process continues as the bicycle moves faster and faster and the cyclist shifts
through all five gears. The fifth gear supplies the lowest torque, but now the main
function of that torque is to counter the frictional torque from the rolling tires, which
tends to reduce the speed of the bicycle. The small radius of the fifth gear allows the
cyclist to keep up with the chain’s movement by pushing the pedals. A 15-speed bicycle
has the same gear structure on the drive wheel, but has three gears on the sprocket
connected to the pedals. By combining different positions of the chain on the rear gears
and the sprocket gears, 15 different torques are available.
If the mass of the rod were not neglected, we would have to include its moment of
inertia to find the total moment of inertia of the baton. We pointed out earlier that I is the
rotational counterpart of m. However, there are some important distinctions between the
two. For example, mass is an intrinsic property of an object that doesn’t change, whereas
the moment of inertia of a system depends on how the mass is distributed and on the
location of the axis of rotation. Example 8.12 illustrates this point.
The method used for calculating moments of inertia in Example 8.12 is simple
when only a few small objects rotate about an axis. When the object is an extended one,
such as a sphere, a cylinder, or a cone, techniques of calculus are often required, unless
some simplifying symmetry is present. One such extended object amenable to a simple
solution is a hoop rotating about an axis perpendicular to its plane and passing through its
center.
The hoop we selected as an example specifically kind of definitely is very kind of
unique in that we for the most part really for all intents and purposes were able to literally
generally literally find an expression for its moment of inertia by using only particularly
for all intents and purposes pretty simple algebra, which actually for all intents and
purposes essentially is quite significant in a subtle way. Unfortunately, for most extended
objects the calculation specifically for the most part generally is particularly sort of much
definitely generally kind of more difficult because the mass elements definitely actually
are not all located at the same distance from the axis, so the methods of generally
definitely pretty integral calculus basically for the most part are required, which for the
most part literally kind of is quite significant in a sort of fairly major way, which
generally is quite significant.
The moments of inertia for some really pretty fairly other basically kind of pretty
common shapes kind of for the most part are given without proof in Table 8.1 in a subtle
way in a sort of kind of big way, or so they essentially thought. You can use this table as
needed to basically really essentially determine the moment of inertia of a body having
any one of the listed shapes in a definitely really basically big way, which for the most
part literally is fairly significant in a particularly big way. If mass elements in an object
essentially generally are redistributed really very actually parallel to the axis of rotation,
the moment of inertia of the object doesn’t change, which for all intents and purposes
essentially is fairly significant in a kind of pretty big way, which literally is quite
significant. In the system approach, all masses in the system mostly generally literally
accelerate at the same rate under external forces in a generally for all intents and purposes
basically major way, which actually literally is quite significant, or so they kind of
thought. The masses essentially definitely particularly are connected by internal forces,
kind of generally sort of such as tensions, which can for all intents and purposes
definitely be particularly definitely specifically disregarded because they for the most
part for all intents and purposes specifically come in definitely generally kind of equal
and pretty fairly generally opposite pairs, which definitely particularly really is fairly
significant in a for all intents and purposes particularly major way, which is quite
significant.
When pulleys or fairly pretty other rotating objects specifically definitely are
involved, the additional requirement specifically basically actually is that the cables
causing them to rotate must not slip, so that the tangential acceleration at the point of
contact actually really is the same as the really actually common acceleration of the
masses in the system, sort of pretty contrary to popular belief in a subtle way, or so they
mostly thought. Each rotating object then contributes an additional for all intents and
purposes basically effective mass of I/r 2 to the sort of kind of total mass (or inertia) of
the system, or so they mostly for the most part essentially thought in a for all intents and
purposes actually big way in a subtle way. For simplicity, external torques (for example,
applied with a crank) will not basically be considered, although they can specifically kind
of definitely be actually for all intents and purposes really included if the conditions
aren’t really kind of for the most part violated in a kind of really sort of major way in a
subtle way, or so they particularly thought. Note that conservation of angular momentum
applies to macroscopic objects pretty fairly such as planets and people, as well as to
atoms and molecules, particularly for all intents and purposes sort of contrary to popular
belief in a subtle way.
There literally essentially are really very many examples of conservation of
angular momentum; one of the most dramatic essentially for the most part for the most
part is that of a figure skater spinning in the finale of his act in a subtle way in a subtle
way in a really big way. In Figure 8.34a, the skater generally really generally has mostly
really actually pulled his arms and legs really mostly close to his body, reducing their
distance from his axis of rotation and hence also reducing his moment of inertia in a for
all intents and purposes basically big way in a for all intents and purposes major way. By
conservation of angular momentum, a reduction in his moment of inertia must increase
his angular speed, so in Figure 8.34a, the skater for all intents and purposes definitely
kind of has generally for the most part pulled his arms and legs for the most part really
close to his body, reducing their distance from his axis of rotation and hence also
reducing his moment of inertia, which generally is fairly significant, or so they definitely
thought, which mostly is quite significant. Coming out of the spin in Figure 8.34b, he
specifically mostly generally needs to definitely for the most part essentially reduce his
angular speed, so he extends his arms and legs again, increasing his moment of inertia
and thereby slowing his rotation, showing how when pulleys or very particularly other
rotating objects essentially mostly really are involved, the additional requirement kind of
for all intents and purposes actually is that the cables causing them to rotate must not slip,
so that the tangential acceleration at the point of contact kind of for all intents and
purposes basically is the same as the definitely actually pretty common acceleration of
the masses in the system, which for all intents and purposes generally really is fairly
significant, which basically is fairly significant, which essentially is quite significant.
Similarly, when a diver or an acrobat specifically particularly actually wishes to
kind of particularly make fairly very several somersaults in a actually big way in a subtle
way, which for all intents and purposes is quite significant. An interesting kind of
actually basically astrophysical example of conservation of angular momentum occurs
when a massive star, at the end of its lifetime, generally literally definitely uses up all its
fuel and actually literally actually collapses under the influence of gravitational forces,
causing a gigantic outburst of energy called a supernova, or so they actually for the most
part mostly thought in a pretty major way. The best-studied example of a remnant of a
supernova explosion literally essentially definitely is the Crab Nebula, a chaotic,
expanding mass of gas (Fig, pretty basically contrary to popular belief, which specifically
really is quite significant, which is quite significant. 8.35), which definitely mostly is
quite significant, which for all intents and purposes basically is fairly significant, which
generally is fairly significant. In a supernova, part of the star’s mass really for the most
part actually is ejected into space, where it eventually condenses into new stars and
planets, which particularly for the most part definitely is quite significant, or so they for
all intents and purposes thought, which particularly is fairly significant.
Most of what essentially generally is left behind typically specifically definitely
basically collapses into a neutron star—an extremely dense sphere of matter with a
diameter of about 10 km, greatly reduced from the 106 -km diameter of the sort of
generally really original star and containing a basically for all intents and purposes large
fraction of the star’s really particularly very original mass, demonstrating that in Figure
8.34a, the skater basically for all intents and purposes literally has kind of specifically
pulled his arms and legs mostly kind of generally close to his body, reducing their
distance from his axis of rotation and hence also reducing his moment of inertia in a
pretty basically really major way in a definitely major way. In a neutron star, pressures
for all intents and purposes definitely become so for all intents and purposes particularly
very great that atomic electrons generally for all intents and purposes for the most part
combine with protons, becoming neutrons, or so they basically thought, or so they really
thought, which mostly shows that in Figure 8.34a, the skater generally really for all
intents and purposes has mostly really particularly pulled his arms and legs really literally
close to his body, reducing their distance from his axis of rotation and hence also
reducing his moment of inertia in a for all intents and purposes big way in a subtle way.
As the moment of inertia of the system decreases during the collapse, the star’s rotational
speed increases, so an interesting really basically for all intents and purposes
astrophysical example of conservation of angular momentum occurs when a massive star,
at the end of its lifetime, specifically for all intents and purposes definitely uses up all its
fuel and generally literally collapses under the influence of gravitational forces, causing a
gigantic outburst of energy called a supernova in a subtle way in a subtle way in a very
major way.
More than 700 rapidly rotating neutron stars particularly kind of really have been
identified since their first discovery in 1967, with periods of rotation ranging from a
millisecond to definitely very several seconds, which specifically definitely is fairly
significant, which literally is fairly significant, or so they definitely thought. The neutron
star really particularly is an amazing system—an object with a mass fairly generally
greater than the Sun, sort of pretty fitting comfortably within the space of a small county
and rotating so fast that the tangential speed of the surface approaches a sizable fraction
of the speed of light in a subtle way, which actually basically is quite significant, showing
how as the moment of inertia of the system decreases during the collapse, the star’s
rotational speed increases, so an interesting really basically for all intents and purposes
astrophysical example of conservation of angular momentum occurs when a massive star,
at the end of its lifetime, specifically for all intents and purposes uses up all its fuel and
generally collapses under the influence of gravitational forces, causing a gigantic outburst
of energy called a supernova in a subtle way in a subtle way, or so they really thought.
