WORK AND ENERGY
_104. Work._--"Whenever a force moves a body upon which it acts, it is
said to do work upon that body." For example, if a man pushes a
wheelbarrow along a path, he is doing work on it as long as the
wheelbarrow moves, but if the wheelbarrow strikes a stone and the man
continues to push and no motion results, from a scientific point of view
he is then doing no work on it.
"Work signifies the overcoming of resistance," and unless the resistance
is overcome no work is done. Lifting a weight is doing work on it,
supporting a weight is not, although the latter may be nearly as
tiresome as the former. Work as used in science is a technical term. Do
not attach to it meanings which it has in every-day speech.
=105. Measurement of Work.=--Work is measured by the product of the
force by the displacement caused in the direction of the force, that is
_W_ = _fs_. Therefore if a unit of force acts through a unit of space, a
unit of work will be done. There are naturally several units of work
depending upon the units of force and space employed.
_English Work Unit._--If the force of one _pound_ acts through the
distance of one _foot_, a _foot-pound_ of work is done. A foot-pound is
defined as the work done when 1 lb. is lifted 1 ft. against the force of
gravity.
_Metric Work Unit._--If the force is one _kilogram_ and the distance one
_meter_, _one kilogram-meter_ of work is done.
_Absolute Work Unit._--If the force of one _dyne_ acts through the
distance of one _centimeter_ a _dyne-centimeter_ of work is done. This
usually is called an i. Other work units are sometimes used depending
upon the force and distance units employed. One, the i, is equal to
10,000,000 ergs or 10⁷ ergs.
=Problem.=--If a load is drawn 2 miles by a team exerting 500 lbs.
force, how much work is done?
=Solution.=--Since the force employed is 500 lbs., and the distance
is 2 × 5280 ft., the work done is 500 × 2 × 5280 or 5,280,000
ft.-lbs.
=106. Energy.=--In the various cases suggested in the paragraphs upon
work, an agent, a man, an animal or a machine, was mentioned as putting
forth an effort in order to do the work. It is also true that in order
to perform work an agent must employ _energy, or the energy of a body is
its capacity for doing work_. Where an agent does work upon a body, as
in winding up a spring or in lifting a weight, the body upon which the
work has been done may acquire energy by having work done upon it. That
is, it may become able to do work itself upon some other body. For
instance, a lifted weight in falling back to its first position may turn
wheels, or drive a post into the ground against resistance; a coiled
spring may run clock work, strike a blow, or close a door. Hence the
energy, or the capacity for doing work, is often acquired by a body
because work has first been done upon that body.
=107. Potential Energy.=--The wound up spring may do work because work
has first been done upon it. The lifted weight may also do work because
work has first been done in raising it to its elevated position since in
falling it may grind an object to powder, lift another weight or do some
other kind of work. _The energy that a body possesses on account of its
position or shape and a stress to which it is subjected is called
potential energy._ The potential energy of a body is measured by the
work done in lifting it, changing its shape, or by bringing about the
conditions by which it can do work. Thus if a block of iron weighing
2000 lbs. is lifted 20 ft., it possesses 40,000 ft.-lbs. of potential
energy. It is therefore able to do 40,000 ft.-lbs. of work in falling
back to its first position. If the block just mentioned should fall from
its elevated position upon a post, it could drive the post into the
ground because its motion at the instant of striking enables it to do
work. To compute potential energy you compute the work done upon the
body. That is, _P.E._ = _w_ × _h_ or _f_ × _s_.
=108. Kinetic Energy.=--_The energy due to the motion of a body is
called kinetic energy_. The amount of kinetic energy in a body may be
measured by the amount of work done to put it in motion. It is usually
computed, however, by using its mass and velocity on striking. To
illustrate, a 100-lb. ball is lifted 16 ft. The work done upon it, and
hence its potential energy, is 1600 ft.-lbs. On falling to the ground
again, this will be changed into kinetic energy, or there will be 1600
ft.-lbs. of kinetic energy on striking. It will be noted that since
energy is measured by the work it can do, work units are always used in
measuring energy. To compute the kinetic energy of a falling body by
simply using its mass and velocity one proceeds as follows, in solving
the above problem:
First, find the velocity of the falling body which has fallen 16
ft. A body falls 16 ft. in _one_ second. In this time it gains a
velocity of 32 ft. per second. Now using the formula for kinetic
energy _K.E._ = _wv_²/(2_g_), we have _K.E._ = 100 × 32 × 32/(2 ×
32) = 1600 ft.-lbs. as before. The formula, _K.E._ = _wv_²/(2_g_),
may be derived in the following manner:
The kinetic energy of a falling body equals the work done in giving
it its motion, that is, _K.E._ = _w_ × _S_, in which, _w_ = the
weight of the body and _S_ = the distance the body must fall freely
in order to acquire its velocity. The distance fallen by a freely
falling body, _S_, = 1/2_gt_² = _g_²_t_²/(2_g_) (Art. 98, p. 111).
Now, _v_ = _gt_ and _v_² = _g_²_t_².
Substituting for _g_²_t_², its equal _v_², we have _S_ =
_v_²/(2_g_). Substituting this value of S in the equation _K.E._ =
_w_ × _S_, we have _K.E._ = _wv_²/(2_g_).
Since the kinetic energy of a moving body depends upon its mass and
velocity and not upon the _direction_ of motion, this formula may
be used to find the kinetic energy of any moving body. Mass and
weight in such problems may be considered numerically equal.
=Important Topics=
1. Work defined.
2. Work units, foot-pound, kilogram-meter, erg.
3. Energy defined.
4. Kinds of energy, potential and kinetic.
=Problems=
1. How much work will a 120-lb. boy do climbing a mountain 3000 ft.
high? Should the vertical or slant height be used? Why?
2. In a mine 4000 kg. of coal are lifted 223 meters: how much work
is done upon the coal? What is the kind and amount of energy
possessed by the coal?
3. A pile driver weighs 450 lbs. It is lifted 16 ft. How much work
has been done upon it? What kind and amount of energy will it have
after falling 16 ft. to the pile?
4. A train weighing 400 tons is moving 30 miles per hour. Compute
its kinetic energy. (Change its weight to pounds and velocity to
feet per second.)
5. What would be the kinetic energy of the train in problem 4 if it
were going 60 miles per hour? If it were going 90 miles per hour?
How does doubling or trebling the speed of an object affect its
kinetic energy? How does it affect its momentum?
6. What is the kinetic energy of a 1600-lb. cannon ball moving 2000
ft. per second?
7. Mention as many kinds of mechanical work as you can and show how
each satisfies the definition of work.
8. A pile driver weighing 3000 lbs. is lifted 10 ft. How much work
is done upon it?
9. If the pile driver in problem 8 is dropped upon the head of a
pile which meets an average resistance of 30,000 lbs., how far will
one blow drive it?
10. A 40 kg. stone is placed upon the top of a chimney 50 meters
high. Compute the work done in kilogram-meters and foot-pounds.
(2) POWER AND ENERGY
=109. Horse-power.=--In computing work, no account is taken of the time
required to accomplish it. But since the time needed to perform an
undertaking is of much importance, the rate of work, or the _power or
activity_ of an agent is an important factor. Thus if one machine can do
a piece of work in one-fifth the time required by another machine, it is
said to have five times the power of the other. Therefore the power of a
machine is _the rate at which it can do work_. James Watt (1736-1819),
the inventor of the steam-engine, in _expressing_ the power of his
engine, used as a unit a _horse-power_. He considered that a horse could
do 33,000 ft.-lbs. of work a minute. This is equal to 550 ft.-lbs. per
second or 76.05 kg.-m. per second. This is too high a value but it has
been used ever since his time. Steam engines usually have their power
rated in horse-power. That is, locomotives produce from 500 to 1500
horse-power. Some stationary and marine engines develop as high as
25,000 horse-power. The power of an average horse is about 3/4
horse-power and of a man about 1/7 horse-power when working continuously
for several hours.
=110. The Watt.=--In the metric system, the erg as a unit of work would
give as a unit of power 1 erg per second. This amount is so small,
however, that a larger unit is usually employed, the practical unit
being 10,000,000 ergs a second, that is, one joule per second. (See Art.
105.) This practical unit is called a _Watt_ after James Watt. The
power of dynamos is usually expressed in kilowatts, a kilowatt
representing 1000 watts. Steam-engines in modern practice are often
rated in kilowatts instead of horse-power. A horse-power is equivalent
to 746 watts, or is nearly 3/4 of a kilowatt.
=111. Energy. Its Transference and Transformation.= We have considered
energy as the capacity for doing work, and noted the two kinds,
potential and kinetic, and the facility with which one may change into
another. In fact, the transference of energy from one body to another,
and its transformation from one form to another is one of the most
common processes in nature. Take a pendulum in motion, at the _end_ of a
swing, its energy being entirely due to its elevated position is all
_potential_; at the _lowest_ point in its path its energy being entirely
due to its motion is all _kinetic_. The change goes on automatically as
long as the pendulum swings. A motor attached by a belt to a washing
machine is started running. The energy of the motor is transferred by
the belt to the washer where it is used in rubbing and moving the
clothes.
The heat used in warming a house is usually obtained by burning coal or
wood. Coal is believed to be formed from the remains of plants that grew
in former geologic times. These plants grew through the help of the
radiant energy of the sun. The following are transformations of energy
that have occurred: The radiant energy of sunlight was transformed into
the _chemical_ energy of the plants. This remained as chemical energy
while the plants were being converted into coal, was mined, brought to
the stove or furnace and burned. The burning transformed the chemical
energy into heat energy in which form we use it for warming rooms. Take
the energy used in running a street car whose electrical energy comes
from a waterfall. The energy of the car itself is mechanical. Its
motor, however, receives electrical energy and transforms it into
mechanical. This electrical energy comes along a wire from a dynamo at
the waterfall, where water-wheels and generators transform into
electrical energy the mechanical energy of the falling water. The water
obtained its energy of position by being evaporated by the heat of the
radiant energy of the sun. The vapor rising into the air is condensed
into clouds and rain, and falling on the mountain side, has, from its
elevated position, potential energy. The order of transformation,
therefore, is in this case, radiant, heat, mechanical, electrical, and
mechanical. Can you trace the energy from the sun step by step to the
energy you are using in reading this page?
=112. Forms of Energy.=--A steam-engine attached to a train of cars
employs its energy in setting the cars in motion, _i.e._, in giving them
kinetic energy and in overcoming resistance to motion. But what is the
source of the energy of the engine? It is found in the coal which it
carries in its tender. But of what kind? Surely not kinetic, as no
motion is seen. It is therefore potential. What is the source of the
energy of the coal? This question leads us back to the time of the
formation of coal beds, when plants grew in the sunlight and stored up
the energy of the sun's heat and light as _chemical_ energy. The sun's
light brings to the earth the energy of the sun, that central storehouse
of energy, which has supplied nearly all the available energy upon the
earth. Five _forms_ of energy are known, viz., mechanical, heat,
electrical, radiant, and chemical.
=113. Energy Recognized by its Effects.=--Like force, energy is
invisible and we are aware of the forms only by the effects produced by
it.
We recognize _heat_ by _warming_, by expansion, by pressure.
We recognize _light_ by _warming_, by its affecting vision.
We recognize _electrical_ energy by its heat, light, motion, or magnetic
effect. We recognize _mechanical_ energy by the _motion_ that it
produces. We recognize _chemical_ energy by knowing that the source of
energy does not belong to any of the foregoing.
A boy or girl is able to do considerable work. They therefore possess
energy. In what form does the energy of the body mainly occur? One can
determine this for himself by applying questions to each form of energy
in turn as in Art. 114.
_114. Source of the Energy of the Human Body._--Is the energy of the
human body mostly heat? No, since we are not very warm. Is it light or
electrical? Evidently not since we are neither luminous nor electrical.
Is it mechanical? No, since we have our energy even when at rest. Is it
chemical? It must be since it is none of the others. Chemical energy is
contained within the molecule.
It is a form of potential energy and it is believed to be due to the
position of the atoms within the molecule. As a tightly coiled watch
spring may have much energy within it, which is set free on allowing the
spring to uncoil, so the chemical energy is released on starting the
chemical _reaction_. Gunpowder and dynamite are examples of substances
containing chemical energy. On exploding these, heat, light, and motion
are produced. Gasoline, kerosene, and illuminating gas are purchased
because of the potential energy they contain. This energy is set free by
burning or exploding them.
The source of the energy of our bodies is of course the food we eat. The
energy contained in the food is also chemical. Vegetables obtain their
energy from the sunlight (radiant energy). This is why plants will not
grow in the dark. The available energy is mostly contained in the form
of starch, sugar and oil. Digestion is employed principally to dissolve
these substances so that the blood may absorb them and carry them to the
tissues of the body where they are needed. The energy is set free by
oxidation (burning), the oxygen needed for this being supplied by
breathing. Breathing also removes the carbon dioxide, which results from
the combustion. It is for its energy that our food is mostly required.
=115. Conservation of Energy.=--In the study of matter we learned that
it is indestructible. Energy is also believed to be indestructible. This
principle stated concisely teaches that _despite the innumerable changes
which energy undergoes the amount in the universe is unchangeable_, and
while energy may leave the earth and be lost as far as we are concerned,
that it exists somewhere in some form. The principle which teaches this
is called the "Conservation of Energy." The form into which energy is
finally transformed is believed to be heat.
Important Topics
1. Power defined. Units. Horse-power. Watt.
2. Transference and transformations of energy.
3. Forms of energy; heat, electrical, mechanical, radiant, chemical.
4. Effects of the several forms of energy.
5. Energy of the human body.
6. Conservation of energy.
Exercises
1. A boy weighing 110 lbs. ran up a stairs 10 ft. high, in 4 seconds.
How much work was done? What was his _rate_ of work (foot-pounds per
second)? Express also in horse-power.
2. A locomotive drawing a train exerts a draw bar pull of 11,000 lbs.
How much work does it do in moving 3 miles? What is its _rate_ of work
if it moves 3 miles in 5 minutes? Express in horse-power.[G]
[G] The following formula is of assistance in computing
_horse-power_ in problems: H. p. = (lbs. × ft.)/(550 × sec.).
3. If 400 kg. are lifted 35 meters in 5 seconds what work is done? What
is the rate of work? Express in horse-power, watts and kilowatts.
4. Trace the energy of a moving railway train back to its source in the
sun.
5. Why does turning the propeller of a motor boat cause the boat to
move?
6. Does it require more power to go up a flight of stairs in 5 seconds
than in 10 seconds? Explain. Is more work done in one case than in the
other? Why?
7. Can 1 man carrying bricks up to a certain elevation for 120 days do
as much work as 120 men carrying up bricks for 1 day?
8. If the 1 man and 120 men of problem 7 do the same amount of work have
they the same power? Explain.
9. If 160 cu. ft. of water flow each second over a dam 15ft. high what
is the available power?
10. What power must an engine have to fill a tank 11 × 8 × 5 ft. with
water 120 ft. above the supply, in 5 minutes?
11. A hod carrier weighing 150 lbs. carries a load of bricks weighing
100 lbs. up a ladder 30 ft. high. How much work does he do?
12. How much work can a 4-horse-power engine do in 5 minutes?
13. Find the horse-power of a windmill that pumps 6 tons of water from a
well 90 ft. deep in 30 minutes.
14. How many horse-power are there in a waterfall 20 ft. high over which
500 cu. ft. of water pass in a minute?
15. The Chicago drainage canal has a flow of about 6000 cu. ft. a
second. If at the controlling works there is an available fall of 34 ft.
how many horse-power can be developed?
16. How long will it take a 10-horse-power pump to fill a tank of 4000
gallons capacity, standing 300 ft. above the pump?
17. A boy weighing 162 lbs. climbs a stairway a vertical height of 14
ft. in 14.6 seconds. How much power does he exert?
18. The same boy does the same work a second time in 4.2 seconds. How
much power does he exert this time? What causes the difference?
19. What is a horse-power-hour? a kilowatt-hour?
(3) SIMPLE MACHINES AND THE LEVER
=116. Machines and Their Uses.=--A man, while standing on the ground,
can draw a flag to the top of a pole, by using a rope passing over a
pulley.
A boy can unscrew a tightly fitting nut that he cannot move with his
fingers, by using a wrench.
A woman can sew a long seam by using a sewing machine in much less time
than by hand.
A girl can button her shoes much quicker and easier with a button-hook
than with her fingers.
These illustrations show some of the reasons why machines are used. In
fact it is almost impossible to do any kind of work efficiently without
using one or more machines.
=117. Advantages of Machines.=--(a) Many machines make possible an
_increased speed_ as in a sewing machine or a bicycle.
(b) Other machines exert an _increased force_. A rope and a set of
pulleys may enable a man to lift a heavy object such as a safe or a
piano. By the use of a bar a man can more easily move a large rock. (See
Fig. 83.)
[Illustration: FIG. 83.--The rock is easily moved.]
(c) The _direction_ of a force may be changed thus enabling work to be
done that could not be readily accomplished otherwise. As, e.g., the use
of a pulley in raising a flag to the top of a flag pole, or in raising a
bucket of ore from a mine by using a horse attached to a rope passing
over two or more pulleys. (See Fig. 84.)
(d) _Other agents_ than man or animals _can be used_ such as
electricity, water power, the wind, steam, etc. Fig. 85 represents a
windmill often used in pumping water.
_A machine is a device for transferring or transforming energy._ It is
usually therefore an instrument for doing work. An electric motor is a
machine since it _transforms_ the energy of the electric current into
motion or mechanical energy, and _transfers_ the energy from the wire to
the driving pulley.
[Illustration: FIG. 84.--The horse lifts the bucket of ore.]
=118. A Machine Cannot Create Energy.=--Whatever does work upon a
machine (a man, moving water, wind, etc.) loses energy which is employed
in doing the work of the machine. A pair of shears is a machine since it
transfers energy from the hand to the edges that do the cutting. Our own
bodies are often considered as machines since they both transfer and
transform energy.
We must keep in mind that _a machine cannot create energy_. The
principle of "Conservation of Energy" is just as explicit on one side as
the other. Just as energy, cannot be destroyed, so energy cannot be
created. A machine can give out no more energy than is given to it. It
acts simply as an agent in transferring energy from one body to
another. Many efforts have been made to construct machines that when
once started will run themselves, giving out more energy than they
receive. Such efforts, called seeking for _perpetual motion_, have never
succeeded. This fact is strong evidence in favor of the principle of the
conservation of energy.
[Illustration: FIG. 85.--A windmill.]
=119. Law of Machines.=--When a body receives energy, work is done upon
it. Therefore work is done upon a machine when it receives energy and
the machine does work upon the body to which it gives the energy. In the
operation of a machine, therefore, two quantities of work are to be
considered and by the principle of the conservation of energy, these two
must be equal. _The work done by a machine equals the work done upon it,
or the energy given out by a machine equals the energy received by it._
These two quantities of work must each be composed of a _force_ factor
and a _space_ factor. Therefore two forces and two spaces are to be
considered in the operation of a machine. The force factor of the work
done on the machine is called the _force_ or _effort_. It is the force
applied to the machine. The force factor of the work done by a machine
is called the _weight or resistance_. It is the force exerted by the
machine in overcoming the resistance and equals the resistance
overcome.
If _f_ represents the force or effort, and _D_{f}_ the space it acts
through, and _w_ represents the weight or resistance, and _D_{w}_ the
space it acts through, then the law of machines may be expressed by an
equation, _f × D_{f} = w × D_{w}_. That is, _the effort times the
distance the effort acts equals the resistance times the distance the
resistance is moved or overcome_. When the product of two numbers equals
the product of two other numbers either pair may be made the means and
the other the extremes of a proportion. The equation given above may
therefore be expressed _w: f = D_{f}: D_{w}_. Or the resistance is to
the effort as the effort distance is to the resistance distance. The law
of machines may therefore be expressed in several ways. One should keep
in mind, however, that the _same_ law of machines is expressed even
though the form be different. What two ways of expressing the law are
given?
=120. The Simple Machines.=--There are but six _simple machines_. All
the varieties of machines known are simply modifications and
combinations of the six simple machines. The six simple machines are
more easily remembered if we separate them into two groups of three
each. The first or _lever_ group consists of those machines in which a
part revolves about a fixed axis. It contains the _lever_, _pulley_ and
_wheel and axle_. The second or _inclined plane group_ includes those
having a sloping surface. It contains the _inclined plane_, the _wedge_,
and the _screw_.
=121. The Lever.=--The _lever_ is one of the simple machines most
frequently used, being seen in scissors, broom, coal shovel, whip,
wheelbarrow, tongs, etc. _The lever consists of a rigid bar capable of
turning about a fixed axis called the fulcrum._ In studying a lever, one
wishes to know what weight or resistance it can overcome when a certain
force is applied to it. Diagrams of levers, therefore, contain the
letters _w_ and _f_. In addition to these, _O_ stands for the fulcrum
on which it turns. By referring to Fig. 86, _a_, _b_, _c_, one may
notice that each of these may occupy the middle position between the
other two. The two forces (other than the one exerted by the fulcrum)
acting on a lever always oppose each other in the matter of changing
rotation. They may be considered as a pair of parallel forces acting on
a body, each tending to produce rotation.
[Illustration: FIG. 86.--The three classes of levers.]
=122. Moment of Force.=--The _effectiveness_ of each force may therefore
be determined by computing its _moment_ about the fixed axis (see Art.
84), that is, by multiplying each force by its distance to the fulcrum
or axis of rotation. Let a meter stick have a small hole bored through
it at the 50 cm. mark near one edge, and let it be mounted on a nail
driven into a vertical support and balanced by sliding a bent wire along
it. Suspend by a fine wire or thread a 100 g. weight, 15 cm. from the
nail and a 50 g. weight 30 cm. from the nail, on the other side of the
support. These two weights will be found to balance. When viewed from
this side _A_ (Fig. 87) tends to turn the lever in a clockwise direction
(down at right), _B_ in the counter-clockwise direction (down at left).
Since the lever balances, the forces have equal and opposite effects in
changing its rotation as may also be computed by determining the moment
of each force by multiplying each by its distance from the fulcrum.
Therefore the _effectiveness_ of a force in changing rotation depends
upon the distance from it to the axis as well as upon the magnitude of
the force.
[Illustration: FIG. 87.--The two moments are equal about _C_. 100 × 15 =
50 × 30.]
From the experiment just described, the moment of the acting force
equals the moment of the weight or _f × D_{f} = w × D_{w}_, or the
effort times the effort arm equals the weight times the weight arm. This
equation is called the law of the lever. It corresponds to the general
law of machines and may also be written _w: f = D_{f}: D_{w}_.
=123. Mechanical Advantage.=--A lever often gives an advantage because
by its use one may lift a stone or weight which the unaided strength of
man could not move. If the lever is used in lifting a stone weighing 500
lbs., the force available being only 100 lbs., then its _mechanical
advantage_ would be 5, the ratio of _w:f_. In a similar way, the
mechanical advantage of any machine is found by finding the ratio of the
resistance or weight to the effort. What must be the relative lengths of
the effort arm and resistance or weight arm in the example just
mentioned? Since the effort times the effort arm equals the weight times
the weight arm, if _f × D_{f} = w × D_{w}_, then _D_{f}_ is five times
_D_{w}_. Hence the mechanical advantage of a lever is easily found by
finding the ratio of the effort arm to the weight arm.
Important Topics
1. Advantage of machines.
2. Machines cannot create energy.
3. Law of machines.
4. Six simple machines.
5. Lever and principle of moments.
6. Mechanical advantage of a machine.
Exercises
1. Give six examples of levers you use.
2. Fig. 88_a_ represents a pair of paper shears, 88_b_ a pair of
tinner's shears. Which has the greater mechanical advantage? Why?
Explain why each has the most effective shape for its particular work.
[Illustration: FIG. 88.--(_a_) Paper shears. (_b_) Tinner's shears.]
3. Find examples of levers in a sewing machine.
4. What would result if, in Art. 122, the 100 g. weight were put 25 cm.
from O and the 50 g. weight 45 cm. from O? Why? Explain using principle
of moments.
5. How is the lever principle applied in rowing a boat?
6. When you cut cardboard with shears, why do you open them wide and cut
near the pivot?
7. In carrying a load on a stick over the shoulder should the pack be
carried near the shoulder or out on the stick? Why?
8. How can two boys on a see-saw start it without touching the ground?
9. In lifting a shovel full of sand do you lift up with one hand as hard
as you push down with the other? Why?
[Illustration: FIG. 89.--The hammer is a bent lever. What is its
mechanical advantage?]
10. Why must the hinges of a gate 3 ft. high and 16 ft. wide be stronger
than the hinges of a gate 16 ft. high and 3 ft. wide?
11. When one sweeps with a broom do the hands do equal amounts of work?
Explain.
12. A bar 6 ft. long is used as a lever to lift a weight of 500 lbs. If
the fulcrum is placed 6 in. from the weight, what will be the effort
required? Note: two arrangements of weight, fulcrum and effort are
possible.
13. The handle of a hammer is 12 in. long and the claw that is used in
drawing a nail is 2.5 in. long. (See Fig. 89.) A force of 25 lbs. is
required to draw the nail. What is the resistance of the nail?
14. The effective length of the head of a hammer is 2 in. The handle is
15 in. long and the nail holds in the wood with a force of 500 lbs. Only
60 lbs. of force is available at the end of the handle. What will be the
result?
15. If an effort of 50 lbs. acting on a machine moves 10 ft., how far
can it lift a weight of 1000 lbs.?
16. A bar 10 ft. long is to be used as a lever. The weight is kept 2 ft.
from the fulcrum. What different levers can it represent?
17. The effort arm of a lever is 6 ft., the weight arm 6 in. How long
will the lever be? Give all possible answers.
18. Two boys carry a weight of 100 lbs. on a pole 5 ft. long between
them. Where should the weight be placed in order that one boy may carry
one and one-fourth times as much as the other?
(4) THE WHEEL AND AXLE AND THE PULLEY
=124. The Wheel and Axle.=--1. One of the simple machines most commonly
applied in compound machines is the _wheel_ and _axle_. It consists of a
wheel _H_ mounted on a cylinder _Y_ so fastened together that both turn
on the same axis. In Fig. 90, ropes are shown attached to the
circumferences of the wheel and axle. Sometimes a hand wheel is used as
on the brake of a freight or street car, or simply a crank and handle is
used, as in Fig. 91. The _capstan_ is used in moving buildings.
Sometimes two or three wheels and axles are geared together as on a
derrick or crane as in Fig. 92.
[Illustration: FIG. 90.--The wheel and axle.]
[Illustration: FIG. 91.--Windlass used in drawing water from a well.]
[Illustration: FIG. 92.--A portable crane.]
[Illustration: FIG. 93.--The wheel and axle considered as a lever.]
[Illustration: FIG. 94.--View of transmission gears in an automobile. 1,
Drive gear; 2, High and intermediate gear; 3, Low and reverse gear; 4,
8, Reverse idler gears; 5, 6, 7, Countershaft gears. (_Courtesy of the
Automobile Journal_.)]
[Illustration: FIG. 95.--Reducing gear of a steam turbine.]
Fig. 93 is a diagram showing that the wheel and axle acts like a lever.
The axis _D_ is the fulcrum, the effort is applied at _F_, at the
extremity of a radius of the wheel and the resisting weight _W_ at the
extremity of a radius of the axle. Hence, if _D_{f}_, the effort
distance, is three times _D_{w}_, the weight distance, the weight that
can be supported is three times the effort. Here as in the lever, _f ×
D_{f} = w × D_{w}_, or _w:f = D_{f}:D_{w}_, or _the ratio of the weight
to the effort equals the ratio of the radius of the wheel to the radius
of the axle_. This is therefore the mechanical advantage of the wheel
and axle. Since the diameters or circumferences are in the same ratio as
the radii these can be used instead of the radii. Sometimes, when
_increased speed_ instead of increased force is desired, the radius of
the wheel or part to which power is applied is less than that of the
axle. This is seen in the bicycle, buzzsaw, and blower. Sometimes geared
wheels using the principle of the wheel and axle are used to reduce
speed, as in the _transmission_ of an automobile (see Fig. 94), or the
reducing gear of a steam turbine. (See Figs. 95 and 293.)
A _bevel gear_ is frequently used to change the direction of the force.
(See Fig. 94.)
[Illustration: FIG. 96.--A single movable pulley.]
[Illustration: FIG. 97.--Block and tackle.]
[Illustration: FIG. 98.--The fixed pulley considered as a lever.]
[Illustration: FIG. 99.--The movable pulley considered as a lever.]
=125. The Pulley.=--The _pulley_ consists of a wheel turning on an axis
in a frame. The wheel is called a sheave and the frame a block. The rim
may be smooth or grooved. The grooved rim is used to hold a cord or
rope. One use of the pulley is to change the _direction_ of the acting
force as in Fig. 84, where pulley _B_ changes a horizontal pull at _H_
to a downward force and pulley _A_ changes this into an upward force
lifting the weight _W_. These pulleys are fixed and simply change the
direction. Without considering the loss by friction, the pull at _W_
will equal that at _F_. Sometimes, a pulley is attached to the weight
and is lifted with it. It is then called a _movable pulley_. In Fig. 96
the _movable pulley_ is at _P_, a fixed pulley is at _F_. When _fixed
pulleys_ are used, a single cord runs through from the weight to the
effort, so that if a force of 100 lbs. is applied by the effort the same
force is received at the weight. But with movable pulleys several
sections of cord may extend upward from the weight each with the force
of the effort upon it. By this arrangement, a weight several times
larger than the effort can be lifted. Fig. 97 represents what is called
a _block and tackle_. If a force of 50 lbs. is exerted at _F_, each
section of the rope will have the same tension and hence the six
sections of the rope will support 300 lbs. weight. The _mechanical
advantage of the pulley_ or the _ratio of the weight_ to the effort,
therefore, _equals the number of sections of cord supporting the
weight_. The fixed pulley represents a lever, see Fig. 98, where the
effort and weight are equal. In the movable pulley, the fulcrum (see
Fig. 99) is at _D_; the weight, _W_, is applied at the center of the
pulley and the effort at _F_. The weight distance, _D_{w}_, is the
radius, and the effort distance, _D_{f}_, is the diameter of the
pulley. Since _W/F = D_{f} / D_{w} = 2_ in a movable pulley, the weight
is twice the effort, or its mechanical advantage is 2.
Important Topics
1. Wheel and Axle, Law of Wheel and Axle.
2. Pulley, Fixed and Movable, Block and Tackle, Law of Pulley.
Exercises
1. Why do door knobs make it easier to unlatch doors? What simple
machine do they represent? Explain.
2. What combination of pulleys will enable a 160-lb. man to raise a
900-lb. piano?
3. When you pull a nail with an ordinary claw hammer, what is the effort
arm? the resistance arm?
4. How much work is done by the machine in problem 2 in lifting the
piano 20 ft.? How much work must be done upon the machine to do this
work?
5. The pilot wheel of a boat has a diameter of 60 in.; the diameter of
the axle is 6 in. If the resistance is 175 lbs., what force must be
applied to the wheel?
6. Four men raise an anchor weighing {1 1/2} tons, with a capstan (see
Fig. 110) having a barrel 9 in. in diameter. The circle described by the
hand-spikes is {13 1/2} ft. in diameter. How much force must each man
exert?
[Illustration: FIG. 100.--The Capstan.]
7. A bicycle has a 28-in. wheel. The rear sprocket is 3 in. in
diameter,[H] the radius of the pedal crank is 7 in.; 24 lbs. applied to
the pedal gives what force on the rim of the wheel? What will be the
speed of the rim when the pedal makes one revolution a second?
[H] Consider the diameter of the front sprocket as 6 inches.
8. Measure the diameters of the large and small pulleys on the
sewing-machine at your home. What mechanical advantage in number of
revolutions does it give? Verify your computation by turning the wheel
and counting the revolutions.
9. What force is required with a single fixed pulley to raise a weight
of 200 lbs.? How far will the effort move in raising the weight 10 ft.?
What is the mechanical advantage?
10. In the above problem substitute a single movable pulley for the
fixed pulley and answer the same questions.
11. What is the smallest number of pulleys required to lift a weight of
600 lbs. with a force of 120 lbs.? How should they be arranged?
12. A derrick in lifting a safe weighing 2 tons uses a system of pulleys
employing 3 sections of rope. What is the force required?
13. Name three instances where pulleys are used to do work that
otherwise would be difficult to do.
14. Draw a diagram for a set of pulleys by means of which 100 lbs. can
lift 400 lbs.
(5) THE INCLINED PLANE. EFFICIENCY
=126. Efficiency.=--The general law of machines which states that the
work done by a machine equals the work put into it requires a
modification, when we apply the law in a practical way, for the reason
that in using any machine there is developed more or less friction due
to parts of the machine rubbing on each other and to the resistance of
the air as the parts move through it. Hence the statement of the law
that accords with actual working conditions runs somewhat as follows:
_The work put into a machine equals the useful work done by the machine
plus the wasted work done by it._ The _efficiency_ of a machine is the
ratio of the _useful_ work done by it to the _total_ work done on the
machine. If there were no friction or wasted work, the efficiency would
be perfect, or, as it is usually expressed, would be 100 per cent.
Consider a system of pulleys into which are put 600 ft.-lbs. of work.
With 450 ft.-lbs. of useful work resulting, the efficiency would be 450
÷ 600 = {3/4}, or 75 per cent. In this case 25 per cent. of the work
done on the machine is wasted. In a simple lever the friction is slight
so that nearly 100 per cent. efficiency is often secured.
Some forms of the wheel and axle have high efficiencies as in bicycles
with gear wheels. Other forms in which ropes are employed have more
friction. Pulleys have sometimes efficiencies as low as 40 per cent.
when heavy ropes are used.
=127. Inclined Plane.=--We now come to a type of _simple machine of
lower efficiency_ than those previously mentioned. These belong to the
inclined plane group, which includes the inclined plane (see Fig. 101),
the wedge and the screw. They are extensively used, however,
notwithstanding their low efficiency, on account of often giving a high
mechanical advantage. The _relation between these machines may be easily
shown_, as the _wedge_ is obviously _a double inclined plane_. In Art.
82 it is shown that the effort required to hold a weight upon an
inclined plane is to the _weight_ supported as the _height_ of the plane
is to its _length_.
[Illustration: FIG. 101.--An inclined plane.]
Or while the weight is being lifted the vertical height _BC_, the effort
has to move the length of the plane _AC_. Since by the law of machines
the effort times its distance equals the weight times its distance, or
the weight is to the effort as the effort distance is to the weight
distance, therefore the mechanical advantage of the inclined plane is
the ratio of the length to the height of the inclined plane.
_Inclined planes_ are used to raise heavy objects short distances, as
barrels into a wagon, and iron safes into a building. Stairways are
inclined planes with steps cut into them.
=128. The Wedge.=--Wedges are used to separate objects, as in splitting
wood (see Fig. 102), cutting wood, and where great force is to be
exerted for short distances. An axe is a wedge, so is a knife. A fork
consists of several round wedges set in a handle. The edge of any
cutting tool is either an inclined plane or a wedge. Our front teeth are
wedges. Numerous examples of inclined planes may be seen about us.
No definite statement as to the mechanical advantage of the wedge can be
given as the work done depends largely on friction. The force used is
generally applied by blows on the thick end. In general, the longer the
wedge for a given thickness the greater the mechanical advantage.
[Illustration: FIG. 102.--One use of the wedge.]
=129. The Screw.=--The screw is a cylinder around whose circumference
winds a spiral groove. (See Fig. 103.) The raised part between the two
adjacent grooves is the =thread= of the screw. The screw turns in a
block called a =nut=, within which is a spiral groove and thread exactly
corresponding to those of the screw. The distance between two
consecutive threads measured parallel to the axis is called the =pitch=
of the screw. (See Fig. 104.) If the thread winds around the cylinder
ten times in the space of 1 in., the screw is said to have ten threads
to the inch, the pitch being {1/10} in. The screw usually is turned by
a lever or wheel with the effort applied at the end of the lever, or at
the circumference of the wheel. While the effort moves once about the
circumference of the wheel the weight is pushed forward a distance equal
to the distance between two threads (the pitch of the screw). The work
done by the effort therefore equals _F × 2πr_, _r_ being the radius of
the wheel, and the work done on the weight equals _W × s_, _s_ being the
pitch of the screw. By the law of machines _F × 2πr = W × s_ or _W / F =
(2πr) / s_. Therefore the mechanical advantage of the screw equals
_(2πr) / s_. Since the distance the weight moves is small compared to
that the power travels, there is a great gain in force. The screw is
usually employed where _great force_ is to be exerted through small
distances as in the vise (Fig. 105) the jack screw (Fig. 106), screw
clamps, to accurately measure small distances as in the micrometer (Fig.
107) and spherometer, and to lessen the motion in speed-reducing
devices. The worm gear (Fig. 108) is a modification of the screw that is
sometimes used where a considerable amount of speed reduction is
required.
[Illustration: FIG. 103.--The screw is a spiral inclined plane.]
[Illustration: FIG. 104.--The pitch is _S_.]
[Illustration: FIG. 105.--A vise.]
[Illustration: FIG. 106.--A jack screw.]
[Illustration: FIG. 107.--A micrometer screw.]
[Illustration: FIG. 108.--This large worm-wheel is a part of the
hoisting mechanism employed for the lock gates of the Sault Ste. Marie
Canal.]
Important Topics
1. Efficiency of machines.
2. The inclined plane, wedge and screw. Applications.
Exercises
1. A plank 12 ft. long is used to roll a barrel weighing 200 lbs. into a
wagon 3 ft. high. Find the force required parallel to the incline.
2. How long a plank will be needed to roll an iron safe weighing 1-1/2
tons into a wagon 3 ft. high using a pull of 600 lbs. parallel to the
incline.
3. An effort of 50 lbs. acting parallel to the plane prevents a 200-lb.
barrel from rolling down an inclined plane. What is the ratio of the
length to the height of the plane?
4. A man can push with a force of 150 lbs. and wishes to raise a box
weighing 1200 lbs. into a cart 3 ft. high. How long a plank must he use?
5. The radius of the wheel of a letter press is 6 in., the pitch of its
screw is 1/4 in. What pressure is produced by a force of 40 lbs.?
6. The pitch of a screw of a vice is 1/4 in., the handle is 1 ft. long.
what pressure can be expected if the force used is 100 lbs.?
7. A jackscrew is used to raise a weight of 2 tons. The bar of the
jackscrew extends 2 ft. from the center of the screw. There are two
threads to the inch. Find the force required.
(6) FRICTION, ITS USES AND LAWS
=130. Friction.=--Although often inconvenient and expensive, requiring
persistent and elaborate efforts to reduce it to a minimum, friction has
its uses, and advantages. Were it not for friction between our shoes and
the floor or sidewalk, we could not keep our footing. _Friction is the
resistance that must be overcome when one body moves over another._ It
is of two kinds, _sliding_ and _rolling_. If one draws a block and then
a car of equal weight along a board, the force employed in each case
being measured by a spring balance, a large difference in the force
required will be noticed, showing how much less rolling friction is than
sliding friction.
=131. Ways of Reducing Friction.=--(a) Friction is often caused by the
minute projections of one surface sinking into the depressions of the
other surface as one moves over the other. It follows, therefore, that
if these projections could be made as small as possible that friction
would be lessened. Consequently _polishing_ is one of the best means for
reducing friction. In machines all moving surfaces are made as smooth as
possible. In different kinds of materials these little ridges and
depressions are differently arranged. (b) In Fig. 109 the friction
between _R_ and _S_ would be greater than between _R_ and _T_. In _R_
and _S_ the surfaces will fit closer together than in _R_ and _T_. The
_use of different materials will reduce friction_. The iron axles of car
wheels revolve in bearings of brass. Jewels are used in watches for the
same reason. (c) Another very common method of reducing friction is by
the use of _lubricants_. The oil or grease used fills up the
irregularities of the bearing surfaces and separates them. _Rolling
friction_ is frequently substituted for sliding friction by the use of
ball and roller bearings. These are used in many machines as in
bicycles, automobiles, sewing machines, etc. (See Fig. 110.)
[Illustration: FIG. 109.--The friction between _R_ and _S_ is greater
than between _R_ and _T_.]
=132. Value of Friction.=--_Friction always hinders motion_ and whenever
one body moves over or through another the energy used in overcoming the
friction is transformed into heat which is taken up by surrounding
bodies and usually lost. Friction is therefore the great obstacle to
perfect efficiency in machines. Friction, however, like most afflictions
_has its uses_. We would find it hard to get along without it. Without
friction we could neither walk nor run; no machines could be run by
belts; railroad trains, street cars, in fact all ordinary means of
travel would be impossible, since these depend upon friction between the
moving power and the road for propulsion.
[Illustration: FIG. 110.--Timken roller bearings. As used in the front
wheel of an automobile.]
=133. Coefficient of Friction.=--The ratio between the friction when
motion is just starting and the force pushing the surfaces together is
called the _coefficient of friction_.
If the block in Fig. 111 is drawn along the board with uniform motion,
the reading of the spring balances indicates the amount of friction.
Suppose the friction is found to be 500 g., and the weight of the block
to be 2000 g. Then the coefficient of friction for these two substances
will be {500/2000} = {1/4}, or 25 per cent.
=134. Laws of Friction, Law I.=--_The friction when motion is occurring
between two surfaces is proportional to the force holding them
together._ Thus if one measures the friction when a brick is drawn along
a board, he will find that it is doubled if a second brick is placed on
the first. On brakes greater pressure causes greater friction. If a rope
is drawn through the hands more pressure makes more friction.
[Illustration: FIG. 111.--A method for testing the friction between
surfaces.]
=Law II.=--_Friction is independent of the extent of surface in
contact._ Thus a brick has the same friction drawn on its side as on its
edge, since, although the surface is increased, the weight is unchanged.
=Law III.=--Friction is greatest at starting, but after starting is
practically the same for all speeds.
=135. Fluid Friction.=--When a solid moves through a fluid, as when a
ship moves through the water or railroad trains through the air, the
resistance encountered is not the same as with solids but increases with
the square of the velocity for slow speeds and for high speeds at a
higher rate. This is the reason why it costs so much to increase the
speed of a fast train, since the resistance of the air becomes the
prominent factor at high speeds. The resistance to the motion of a ship
at high speed is usually considered to increase as the cube of the
velocity so that to double the speed of a boat its driving force must be
eight times as great.
Important Topics
1. Friction: two kinds; sliding and rolling.
2. Four ways of reducing friction.
3. Uses of friction.
4. Coefficient of friction. Three laws of friction.
5. Fluid friction.
Exercises
1. How long must an inclined plane be which is 10 meters high to enable
a car weighing 2000 kg. to be pushed up its length by a force of 100 kg.
parallel to the incline?
2. State how and where friction is of use in the operation of the
inclined plane, the wedge, the screw, the wheel and axle.
3. A wheelbarrow has handles 6 ft. long. If a load of 300 lbs. is placed
18 in. from the axis of the wheel, what force placed at the end of the
handles will be required to lift it?
4. A jackscrew has 3 threads to the inch, and the lever used to turn it
is 4 ft. long. If the efficiency of the screw is 60 per cent., what
force must be applied to raise a load of 5 tons?
5. In problem 4 how far must the force move in raising the weight 3 in.
Compute the work done upon the weight, the work done by the power and
the efficiency of the machine from these two amounts of work.
6. What simple machines are represented in a jackknife, a
sewing-machine, a screw-driver, a plane, a saw, a table fork?
7. A laborer carries 1500 lbs. of brick to a platform 40 ft. high. How
much useful work does he do?
8. If he weighs 150 lbs. and his hod weighs 10 lbs., how much useless
work does he do in taking 30 trips to carry up the bricks of problem 7?
What is his efficiency?
9. If the laborer hoists the brick of problem 7 in a bucket weighing 50
lbs., using a fixed pulley and rope, what is the useless work done if it
takes 12 trips to carry up the brick? What is the efficiency of the
device?
10. The efficiency of a set of pulleys is 70 per cent. How much force
should be applied if acting through 100 ft. it is to raise a load of 400
lbs. 20 ft.?
11. The spokes of the pilot wheel of a motor-boat are 1 ft. long, the
axle around which the rudder ropes are wound is 3 in. in diameter. What
effort must be applied if the tension in the ropes is 50 lbs.?
12. Why are the elevated railway stations frequently placed at the top
of an incline, the tracks sloping gently away in both directions?
13. The screw of a press has 4 threads to the inch and is worked by a
lever of such length that an effort of 25 lbs. produces a force of 2
tons. What is the length of the lever?
14. It takes a horizontal force of 10 lbs. to draw a sled weighing 50
lbs. along a horizontal surface. What is the coefficient of friction?
15. The coefficient of rolling friction of a railroad train on a track
is 0.009. What pull would an engine have to exert to haul a train
weighing 1000 tons along a level track?
16. How heavy a cake of ice can be dragged over a floor by a horizontal
force of 20 lbs., if the coefficient of friction is 0.06?
17. The coefficient of friction of iron on iron is 0.2. What force can a
switch engine weighing 20 tons exert before slipping?
18. Using a system of pulleys with a double movable block a man weighing
200 lbs. is just able to lift 600 lbs. What is the efficiency of the
system?
19. What is the horse-power of a pump that can pump out a cellar full of
water 40 ft. × 20 ft. by 10 ft. deep, in 30 minutes?
20. How many tons of coal can a 5 horse-power hoisting engine raise in
30 minutes from a barge to the coal pockets, a height of 50 ft.?
(7) WATER POWER
=136. Energy of Falling Water.=--The energy of falling and running water
has been used from the earliest times for developing power and running
machinery. The energy is derived from the action of the moving water in
striking and turning some form of _water-wheel_, several varieties of
which are described below.
=The Overshot Wheel.=--The overshot wheel (Fig. 112) is turned by the
weight of the water in the buckets. It was formerly much used in the
hilly and mountainous sections of this country for running sawmills and
grist mills as it is very easily made and requires only a small amount
of water. Its efficiency is high, being from 80 to 90 per cent., the
loss being due to friction and spilling of water from the buckets. To
secure this high efficiency the overshot wheel must have a diameter
equal to the height of the fall which may be as much as 80 or 90 ft.
[Illustration: FIG. 112.--Overshot water wheel.]
[Illustration: FIG. 113.--Undershot water wheel.]
[Illustration: FIG. 114.--Diagram illustrating the principle of the
Pelton wheel.]
=The Undershot Wheel.=--The old style undershot wheel (Fig. 113) is used
in level countries, where there is little fall, often to raise water for
irrigation. Its efficiency is very low, seldom rising more than 25 per
cent. The principle of the undershot wheel, however, is extensively used
in the water motor and the Pelton wheel (Fig. 114). In these the water
is delivered from a nozzle in a jet against the lower buckets of the
wheel. They have an efficiency of about 80 per cent. and are much used
in cities for running small machines, washing machines, pipe organ
blowers, etc., and in mountainous districts where the head is great.
[Illustration: FIG. 115.--Diagram of a hydro-electric power house
showing a vertical turbine _A_ with penstock _B_ and tail race _C_.]
[Illustration: FIG. 116.--The outer case of a turbine showing the
mechanism for controlling the gates.]
[Illustration: FIG. 117.--Inner case of a turbine showing the gates and
the lower end of the runner within.]
[Illustration: FIG. 118.--The runner of a turbine.]
[Illustration: FIG. 119.--Turbine and generator of the Tacoma
hydro-electric power plant.]
=137. The Turbine.=--The turbine is now used more than any other form of
water-wheel. It was invented in 1827 by De Fourneyron in France. It can
be used with a small or large amount of water, the power depending on
the head (the height of the water, in the reservoir above the wheel). It
is the most efficient type of water-wheel, efficiencies of 90 per cent.
often being obtained. The wheel is entirely under water (Fig. 115). It
is enclosed in an outer case (Fig. 116) which is connected with the
reservoir by a penstock or pipe and is always kept full of water. The
wheel itself is made in two parts, a rotating part called the runner
(see Fig. 118) and an inner case (Fig. 117) with gates that regulate
the amount of water entering the wheel. This case has blades curved so
that the water can strike the curved blades of the rotating part (Fig.
118) at the angle that is best adapted to use the energy of the water.
The water then drops through the central opening into the tail race
below (see Fig. 115). The energy available is the product of the weight
of the water and the head. The turbine is extensively used to furnish
power for generating electricity at places where there is a sufficient
fall of water. The electrical energy thus developed is transmitted from
50 to 200 miles to cities where it is used in running street cars,
electric lighting, etc. Turbines can be made to revolve about either
vertical or horizontal axes. Fig. 119 represents a _horizontal_ water
turbine connected to a dynamo. Compare this with the _vertical_ turbine
in Fig. 115.
Exercises
1. Does a person do more work when he goes up a flight of stairs in 5
seconds than when he goes up in 15 seconds? Explain.
2. A motorcycle has a 4 horse-power motor and can go at a rate of 50
miles per hour. Why cannot 4 horses draw it as fast?
3. What is the efficiency of a motor that is running fast but doing no
useful work?
4. What horse-power can be had from a waterfall, 12 ft. high, if 20 cu.
ft. of water pass over it each second?
5. What is the horse-power of a fire engine if it can throw 600 gallons
of water a minute to a height of 100 ft.?
6. Why are undershot wheels less efficient than the overshot wheel or
turbine?
7. A revolving electric fan is placed on the stern of a boat. Does the
boat move? Why? Place the fan under water. Does the boat now move? Why?
8. Why does an electric fan produce a breeze?
9. Explain the action of the bellows in an organ.
10. At Niagara Falls the turbines are 136 ft. below the surface of the
river. Their average horse-power is 5000 each. 430 cu. ft. of water each
second pass through each turbine. Find the efficiency.
11. At Laxey on the Isle of Man is the largest overshot wheel now in
use. It has a horse-power of 150, a diameter of 72.5 ft., a width of 10
ft., and an efficiency of 85 per cent. How many cubic feet of water pass
over it each second?
12. The power plant at the Pikes Peak Hydro-electric Company utilizes a
head of 2150 ft., which is equal to a pressure of 935 lbs. per square
inch, to run a Pelton wheel. If the area of the nozzle is 1 sq. in. and
the jet has a velocity of 22,300 ft. per minute, what is the horse-power
developed if the efficiency is 80 per cent.?
13. A test made in 1909 of the turbines at the Centerville power house
of the California Gas and Electric Corporation showed a maximum
horse-power of 9700, speed 400 r.p.m. under a head of 550 ft. The
efficiency was 86.25 per cent. How many cubic feet of water passed
through the turbines each second?
14. The turbine in the City of Tacoma Power Plant (see Fig. 120) uses a
head of 415 ft. 145 cu. ft. a second pass through the turbine. Calculate
the horse-power.
15. In problem 14, what is the water pressure per square inch at the
turbine?
16. The power plant mentioned in problem 13 develops 6000 kw. What is
the efficiency?
Review Outline: Work and Energy
Work; how measured, units, foot-pound, kilogram meter, erg.
Energy; how measured, units, potential, _P.E._ = _w × h_, or _f × s_.
Kinetic = _(wv²)/(2g)_.
Power; how measured, units, horse power, watt, 5 forms of energy,
conservation. H.p. = (lbs. × ft.)/(550 × sec.).
Machines; 6 simple forms, 2 groups, advantages, uses, Law: _W × D_{w}_ =
_F × D_{f}_.
Lever; moments, mechanical advantage, uses and applications.
Wheel and Axle and Pulley; common applications, mechanical advantage.
Inclined Plane, Wedge, and Screw; mechanical advantage and efficiency.
Friction; uses, how reduced, coefficient of, laws (3).
Water Wheels; types, efficiency, uses.