chapter 5 energy. introduction energy is one of the most fundamental and far- reaching concepts in...
TRANSCRIPT
Chapter 5
Energy
Introduction Energy is one of the
most fundamental and far-reaching concepts in the physics world view. It took nearly 300 years to fully develop the ideas of energy and energy conservation—the idea that the total energy of an isolated system does not change.
Windmills transform the energy in the wind to electric energy.
Forms of Energy
Mechanical Position and Motion focus for now
NonMechanical chemical Electromagnetic Nuclear- Fusion, Fission Thermal
Using Energy Considerations
Energy can be transformed from one form to another Essential to the study of
physics, chemistry, biology, geology, astronomy
Can be used in place of Newton’s laws to solve certain problems more simply
A First Equation
Energy of Motion The most obvious form of energy is the
one an object has because of its motion. We call this quantity of motion the kinetic
energy of the object. kinetic energy depends on the mass
and the motion of the object. But the kinetic energy K of an object has its own equation.
K=1/2 mv2
Units of Energy
Energy of Motion The units for kinetic energy, and
therefore for all types of energy is a joule (J).
Kinetic energy is not a vector quantity.
An object has the same kinetic energy regardless of its direction as long as its speed does not change.
A typical textbook dropped from a height of 10 centimeters (about 4 inches) hits the floor with a kinetic energy of about 1 J.
Details of the Equation
Energy of Motion The factor of ½ makes the
kinetic energy compatible with other forms of energy, which we will study later.
Notice that the kinetic energy of an object increases with the square of its speed. This means that if an object
has twice the speed, it has 4× the kinetic energy; if it has 3× the speed, it has 9× the kinetic energy; and so on.
Doing the Math
Energy of Motion The kinetic energy of a
70-kilogram (154-pound) person running at a speed of 8 meters per second is:
KE = ½mv 2
= ½(70 kg)(8 m/s)2
= (35 kg)(64 m2/s2) = 2240 J
Arlo
Changing Kinetic Energy
A cart rolling along a frictionless, horizontal surface has a certain kinetic energy because it is moving.
A net force on the cart can change its speed and thus its kinetic energy.
If you push in the direction the cart is moving, you increase the cart’s kinetic energy.
Pushing in the opposite
direction slows the cart and
decreases its kinetic energy.
Changing Kinetic Energy. The displacement through which the force acts
determines how much the kinetic energy changes. The product of the net force F in the direction of motion
and the displacement moved d is known as the work W:
From the definition of work, we conclude that the units of work are newton-meters.
If we substitute our previous expression for a newton (kg · m/s2), we find that a newton-meter (N · m) equals a kg · m2/s2, which is the same as the units for energy—that is, a joule. Therefore, the units of work are the same as those of energy.
More About Work Scalar quantity The work done by a force is zero
when the force is perpendicular to the displacement cos 90° = 0
If there are multiple forces acting on an object, the total work done is the algebraic sum of the amount of work done by each force
Changing Kinetic Energy If the net force is in the
same direction as the velocity, the work is positive, and the kinetic energy increases.
If the net force and velocity are oppositely directed, the work is negative, and the kinetic energy decreases.
Practice1. In the figures below, identical boxes of mass 10 kg are moving at
the same initial velocity to the right on a flat surface. The same magnitude force, F, is applied to each box for the distance indicated in the figure.
Rank these situations in order of the work done on the box by F while the box moves the indicated distance to the right.
F FF
F
F
F
d=5m d=5m
d=5md=10m d=5m
d=10m
Newton’s 2nd law and our expressions for acceleration and distance traveled can be combined with the definition of work to show that the work done on an object is equal to the change in its kinetic energy:
Forces That Do No Work The meaning of work in physics is different
from the common usage of the word. Commonly, people talk about “playing” when they
throw a ball and “working” when they study physics. The physics definition of work is quite precise
—work occurs when the product of the force and the distance is nonzero.
When you throw a sphere, you are actually doing work on the ball; its kinetic energy is increased because you apply a force through a distance.
Although you may move pages and pencils as you study physics, the amount of work is quite small.
Holding Something Up
Forces That Do No Work Similarly, if you hold a suitcase above your head
for 30 minutes, you would probably claim it was hard work. According to the physics definition, however, you did not do any work on the suitcase;
the 30 minutes of straining and groaning did not change the suitcase’s kinetic energy.
It takes no work to hold a cheerleader in the air.
Forces That Do No Work
There are other situations in which a net force does not change an object’s kinetic energy.
If the force is applied in a direction perpendicular to its motion, the velocity of the object changes, but its speed doesn’t.
Therefore, the kinetic energy does not change. The definition of work takes this into account
by stating that it is only the force that acts along the direction of motion that can do work.
Pushing At An Angle
Forces That Do No Work Often, a force is neither parallel nor
perpendicular to the displacement of an object. Because force is a vector, we can think of it as having two components, one that is parallel and one that is perpendicular to the motion as illustrated. The parallel component does work, but the perpendicular one does not do any work.
Any force can be replaced by two perpendicular component forces.
Only the component along the direction of motion does work on the
box.
Circular Motion
Forces That Do No Work Consider an air-hockey puck moving in a circle
on the end of a string attached to the center of the table shown in Figure 7-5.
Because the speed is constant, the kinetic energy is also constant.
When the force is perpendicular to the velocity, the force does no work.
The force of gravity is balanced by the upward force of the table. These vertical forces cancel and do no work.
The tension that the string exerts on the puck is not canceled but does no work because it always acts perpendicular to the direction of motion.
Perfect Orbits
Forces That Do No Work If Earth’s orbit were a circle with the Sun at the center, the
gravitational force the Sun exerts on Earth would do no work.
However, because the orbit is an ellipse, the force is not always perpendicular to the direction of motion.
During one-half of each orbit, a small component of the force acts in the direction of motion, increasing Earth’s kinetic energy and speed.
During the other half of each orbit, the component is opposite the direction of motion, and Earth’s kinetic energy and speed decrease.
The gravitational force of the Sun does work on Earth whenever the force is not perpendicular to Earth’s velocity. The elliptical nature of the orbit has been exaggerated to show the parallel component.
Gravitational Potential Energy
When a sphere is thrown vertically upward, it has a certain amount of kinetic energy that
disappears as it rises. At the top of its flight, it has no kinetic
energy, but as it falls, the kinetic energy reappears.
If we believe that energy is an invariant, we must be missing one or more forms of energy.
Gravitational Potential Energy
The loss and subsequent reappearance of the ball’s kinetic energy can be understood by examining the work done on the ball.
As the ball rises, the force of gravity performs negative work on the ball, reducing its kinetic energy until it reaches zero.
On the way back down, the force of gravity increases the ball’s kinetic energy by the same amount it lost on the way up.
Rather than simply saying that the kinetic energy temporarily disappears, we can retain the idea of the conservation of energy by defining a new form of energy.
Kinetic energy is then transformed into this new form and later transformed back.
This new energy is called gravitational potential energy.
Gravitational Potential Energy
Its change must also be given by the work done by the force of gravity, and it must increase when the kinetic energy decreases, and vice versa.
Therefore, we define the gravitational potential energy of an object at a height h above some zero level as equal to the work done by the force of gravity on the object as it falls to height zero.
The gravitational potential energy of an object near Earth’s
surface is then given by mghThe units are the same as the units for
work.
gravitational potential energyforce of gravity x height =
Reference Levels for Gravitational Potential Energy
A location where the gravitational potential energy is zero must be chosen for each problem The choice is arbitrary since the change in the
potential energy is the important quantity Choose a convenient location for the zero reference
height often the Earth’s surface may be some other point suggested by the problemThe only thing that has any physical significance is the change in
gravitational potential energy. If a ball gains 20 joules of kinetic energy as it falls, it must lose 20
joules of gravitational potential energy. It does not matter how much gravitational potential energy it had
at the beginning; it is only the amount lost that has any meaning in physics.
Gravitational Potential Energy
As an example, we can calculate the gravitational potential energy of a 6-kg ball located 0.5 meter above the level that we choose to call zero:
GPE = mgh= (6 kg)(10 m/s2)(0.5 m)= 30 J
Notice that only the vertical height is important. Moving an object 100 meters sideways does not change the gravitational potential energy because the force of gravity is perpendicular to the motion and therefore does no work on the object.
Conceptual Question
Gravitational Potential Energy
Question: What is the change in gravitational potential energy of a 50-kg person who climbs a flight of stairs with a height of 3 meters and a horizontal extent of 5 meters?Answer: The change in gravitational potential energy is
GPE = mgh = (50 kg)(10 m/s2)(3 m) = 1500 J
The horizontal extent has no effect on the answer.
Flawed Reasoning
Gravitational Potential Energy
Bill and Will are calculating the gravitational potential energy of a 5-newton ball held 2 meters above the floor of their classroom.
Bill: “This is easy. Gravitational potential energy is mgh, where mg is the weight of the ball. We just multiply the 5 newtons by the 2 meters to get the gravitational potential energy of 10 joules.”
Will: “You are forgetting that our classroom is on the second floor. We are going to have to find out how high the ball is relative to the ground.”
Do you agree with either of these students?
Flawed Reasoning
Gravitational Potential Energy
Answer: It is only the difference in gravitational potential
energy that matters. We can either say that: the ball fell from a height of 2 meters to a height of
zero, or we can say that it fell from a height of 5 meters
(relative to the ground) to a height of 3 meters. Either way, we get the same decrease in
gravitational potential energy and the same increase in kinetic energy. Both students would get correct answers.
In general, it is usually easiest to take the lowest point in each problem to be the zero for height.
Conservation of Mechanical Energy
The sum of the gravitational potential and kinetic energies is conserved in some situations. This sum is called the mechanical energy of the system:
Total energy at an instant= Total energy at another instant
½ mv12+mgh1= ½ mv2
2+mgh2
Conservation of Mechanical Energy
When frictional forces can be ignored and the other nongravitational forces do not perform any work, the mechanical energy of the system does not change.
The simplest example of this circumstance is free fall (Figure 7-7). Any decrease in the gravitational potential energy shows up as an increase in the kinetic energy, and vice versa.
Notes About Conservation of Energy
We can neither create nor destroy energy Another way of saying energy is
conserved If the total energy of the system does
not remain constant, the energy must have crossed the boundary by some mechanism
Applies to areas other than physics
Galileo’s RampsConservation of Mechanical Energy
Galileo released a ball that rolled down a ramp, across a horizontal track, and up another ramp, as shown in. He remarked that the ball always returned to its original height.
This result is independent of the slopes of the ramps. During the horizontal portion of the ball’s trip, the
gravitational potential energy remained constant. Hence, the kinetic energy did not change and the speed remained constant, in agreement with Newton’s 1st law.
The Pendulum BobConservation of Mechanical Energy
The pendulum bob shown in Figure 7-9 cyclically gains and loses kinetic and gravitational potential energy.
Note that the tension exerted by the string does no work. Therefore, if we ignore frictional forces, the total mechanical energy is conserved.
Suppose that at the beginning the bob has zero speed at point A and a gravitational potential energy of 10 joules.
(We have chosen the zero for gravitational potential energy to be at the lowest point of the bob’s path.)
Because the kinetic energy is zero, the total mechanical energy is the same as the gravitational potential energy—that is, 10 joules.
The Pendulum BobConservation of Mechanical Energy
Even when the bob is someplace between the highest and lowest points of the swing, the total mechanical energy is still 10 joules.
If at this point we determine from the height of the bob that it has 6 joules of gravitational potential energy, we can immediately declare that it has 4 joules of kinetic energy.
Because we know the expression for the kinetic energy, we can calculate the speed of the bob at this point.
The Pendulum BobConservation of Mechanical Energy
Question: Suppose the bob is released at twice the height. What is the maximum kinetic energy?
Answer: The initial gravitational potential energy is now twice as big, so the maximum kinetic energy will also be twice as big—that is, 20 joules.
Roller CoastersConservation
Imagine trying to determine the speed of a roller coaster inside a thrilling loop-the-loop as it traverses the track.
Determining the speed at any spot using Newton’s 2nd law is difficult because the forces are continually changing magnitude and direction.
We can, however, use conservation of mechanical energy to determine the speed of an object without knowing the details of the net forces acting on it, providing we can ignore the frictional forces.
If we know the mass, speed, and height of the roller coaster at some spot, we can calculate its mechanical energy.
Now determining its speed at any other spot is greatly simplified. The height gives us the gravitational potential energy. Subtracting this from the total mechanical energy yields the kinetic energy from which we can obtain the roller coaster’s speed.
Notice that we don’t need to worry about all the energy transformations that occurred earlier in the ride.
Roller CoastersConservation of Mechanical Energy
Suppose the roller-coaster ride was designed like the one in, and upon reaching the top of the lower hill, your car almost comes to rest.
Assuming that there are no frictional forces to worry about (not true in real situations), is there any possibility that you can get over the higher hill?
The answer is no. Your gravitational potential energy at the top of the hill is nearly equal to the mechanical energy.
This energy is not enough to get you over the higher hill. You will gain speed and thus kinetic energy
as you coast down the hill, but as you start up the other hill, you will find that you cannot exceed the height of the original hill.
Roller CoastersConservation of Mechanical Energy
Is there any way that the roller-coaster car can make it over the second hill when starting on top of the first hill? If your car has some kinetic energy at the
top of the first hill, it might be possible to make it over the second hill.
You would need enough kinetic energy to equal or exceed the extra gravitational potential energy required to climb the next hill.
Flawed ReasoningConservation of Mechanical Energy
The following question appears on the final exam: “Three bears are throwing identical rocks from a bridge to the river below. Papa Bear throws his rock upward at an angle of 30° above the horizontal. Mama Bear throws hers horizontally. Baby Bear throws the rock at an angle of 30° below the horizontal. Assuming that all three bears throw with the same speed, which rock will be traveling fastest when it hits the water?” Three students meet after the exam and discuss their answers.
Emma: “Baby Bear’s rock will be going the fastest because it starts with a downward component of velocity.”
Hector: “But Papa Bear’s rock will stay in the air the longest, so it will have more time to speed up. I think his rock will be traveling the fastest.”
M’Lynn: “Papa Bear’s rock does stay in the air longer, but part of that time it is moving upward and slowing down. I think Mama Bear’s rock will be traveling fastest when it hits the water because it is in the air longer than Baby Bear’s and it is speeding up all of the time.”With which student (if any) do you agree?
Flawed ReasoningConservation of Mechanical Energy
Answer: All three students are wrong. They are making an
easy problem much too difficult by ignoring the power of the energy approach to problem solving.
Because each of the three rocks started with the same kinetic energy (same speed) and the same gravitational potential energy (same height), they must all end up with the same final kinetic energy before hitting the water. All three rocks must therefore hit the water with the same speed.
Note that the three rocks will not hit the water at the same time, with the same velocity, or at the same distance from the bridge. However, the energy method will not give us this information.
Friction
The blue path is shorter than the red path
The work required is less on the blue path than on the red path
Friction depends on the path and so is a nonconservative force
Energy changes to thermal… not mechanical= nonconservative
WNC=fd So energy at an instant of time
after motion might be thermal because friction did work
Bernoulli’s Effect If the fluid is not compressible, the fluid must
be moving faster in the narrow region. This is because the same amount of fluid must pass
by every point in the pipe, or it would pile up. Therefore, the fluid must flow faster in the narrow regions.
This might lead one to conclude incorrectly that the pressure would be higher in this region.
Swiss mathematician and physicist Daniel Bernoulli stated the correct result as a principle.The pressure in a fluid decreases as its
velocity increases.
Bernoulli’s Effect We can understand Bernoulli’s principle by
“watching” a small cube of fluid flow through the pipe.
The cube must gain kinetic energy as it speeds up entering the narrow region.
Because there is no change in its gravitational potential energy, there must be a net force on the cube that does work on it.
Therefore, the force on the front of the cube must be less than on the back. That is, the pressure must decrease as the cube moves into the narrow region.
As the cube of fluid exits the narrow region, it slows down. Therefore, the pressure must increase again.
Everyday Examples
Bernoulli’s Effect There are many examples of Bernoulli’s effect in
our everyday activities. Smoke goes up a chimney partly because hot air
rises but also because of the Bernoulli effect. The wind blowing across the top of the chimney
reduces the pressure and allows the smoke to be pushed up.
This effect is also responsible for houses losing roofs during tornadoes (or attacks by big bad wolves).
When a tornado reduces the pressure on the top of the roof, the air inside the house lifts the roof off.
Everyday Examples
Bernoulli’s Effect A fluid moving past an object is equivalent to the
object moving in the fluid, so the Bernoulli effect should occur in these situations.
A tarpaulin over the back of a truck lifts up as the truck travels down the road due to the reduced pressure on the outside surface of the tarpaulin produced by the truck moving through the air.
This same effect causes your car to be sucked toward a truck as it passes you going in the opposite direction.
The upper surfaces of airplane wings are curved so that the air has to travel a farther distance to get to the back edge of the wing. Therefore, the air on top of the wing must travel faster than that on the underside and the pressure on the top of the wing is less, providing lift to keep the airplane in the air.
Work Done by Varying Forces
The work done by a variable force acting on an object that undergoes a displacement is equal to the area under the graph of F versus x
Other Forms of Energy Roller coasters and pendulums do it; but we can
identify other places where kinetic energy is temporarily stored. For example, a moving ball can compress a spring and lose its kinetic energy (next slide, Figure 7-12).
While the spring is compressed, it stores energy much like a pendulum bob at the top of its arc. If we latch the spring while it is in the compressed state, we can store its energy indefinitely as elastic potential energy.
Releasing it at some future date will transform the spring’s elastic potential energy back into the ball’s kinetic energy.
Figure 7-12
Other Forms of Energy
Potential Energy Stored in a Spring Involves the spring constant (or
force constant), k Hooke’s Law gives the force
F = - k x F is the restoring force F is in the opposite direction of x k depends on how the spring was
formed, the material it is made from, thickness of the wire, etc.
Potential Energy in a Spring
Elastic Potential Energy related to the work required to
compress a spring from its equilibrium position to some final, arbitrary, position x
U= ½ kx2
Conceptual Question
Other Forms of Energy Question: When a ball is hung from a
vertical spring, it stretches the spring. As it drops, it loses gravitational potential energy, but this does not all show up as kinetic energy. What happens to the gravitational potential energy?
Answer: The gravitational potential energy is converted to kinetic energy and to elastic potential energy of the spring. At the bottom, it is all elastic potential energy.
Bungee Jumping
Other Forms of Energy Bungee jumping is a thrilling example of energy
storage and release. A nylon rope is securely fastened to the ankles
of the jumper, who then dives off a high platform. As the jumper falls, gravitational potential
energy is converted to kinetic energy. As the rope stretches, both the kinetic energy and
some additional gravitational potential energy are converted to elastic potential energy.
When the rope reaches its maximum stretch, the jumper bounces back up into the air because much of the elastic potential energy is converted back into kinetic and gravitational potential energy.
After several bounces the bungee jumper is lowered to the ground.
Notice that if there were no loss of mechanical energy, the bungee jumper would bounce forever!
Power In previous chapters we discussed how various
quantities change with time. For example, speed is the change of position with time, and acceleration is the change of velocity with time.
The change of energy with time is called power. Power P is equal to the amount of energy converted
from one form to another (W) divided by the time (Δt ) during which this conversion takes place:
Power is measured in units of joules per second, a metric unit known as a watt (W). One watt of power would raise a 1-kg mass (with a weight of 10 newtons) a height of 0.1 meter each second.
vFt
WP
2
2
s
mkg
s
JW
Some Different Units of Measurement
Power
The English unit for electric power is the watt, but a different English unit is used for mechanical power. One horsepower is equal to 746 watts. Can define units of work or energy in
terms of units of power: kilowatt hours (kWh) are often used in
electric bills 1kWh=3.6 x 106 J
Human Power A human can generate 1500 watts
(2 horsepower) for very short periods of time, such as in weightlifting.
The maximum average human power for an 8-hour day is more like 75 watts (0.1 horsepower).
Each person in a room generates thermal energy equivalent to that of a 75-watt lightbulb. That’s one of the reasons why crowded rooms warm up!
Working It Out
Power A compact car traveling at 27 m/s (60 mph) on a
level highway experiences a frictional force of about 300 N due to the air resistance and the friction of the tires with the road.
Therefore, the car must obtain enough energy by burning gasoline to compensate for the work done by the frictional forces each second:
This means that the power needed is 8100 W, or 8.1 kW.
This is equivalent to a little less than 11 horsepower.
Conceptual Question
Power
Question: How much energy is required to leave a 75-watt yard light on for 8 hours?
Answer: ΔE = PΔt
= (75 watts)(8 hours) = 600 watt-hours = 0.6 kWh.