Chapter 6: Work, Power, Energy, Moment of Inertia Introduction
Water powered mills were the factories up until about 1865. This was before the widespread
use of steam power and much before the use of electricity. In parts of rural America water
power was the cheapest and many times the only source of energy for industry until the mid
to late 1930’s (Animal and human power was often used in the farming operation, but few
farmers ground their own grain.). One of the reasons that many major cities developed along
rivers, especially at falls in the river, was due to the power available from the falling water.
The potential energy of the water was used in the mills in several different ways and we will
discuss two of the more common in this chapter. In addition we will discuss power and how
items rotate. The goal of this chapter is to gain a historical perspective how physics affected
the development of the United States and specifically how the principles of energy, work,
power and rotational motion were employed.
Objectives
•
•
•
•
•
To
To
To
To
To
explain conservation of mechanical energy
understand the functional of both overshot and undershot mills
understand the concepts of potential and kinetic energy
explain rotational motion
understand the concepts of work and power
Gravitational Potential Energy
Gravitational Potential Energy is the
energy that is stored in an object
that is located a height above the
surface of the earth. For example if
a pond of water is located on a hill
20 feet above a house the water in
the pond has stored energy and thus
can be used to create water
pressure in the home. Energy is
defined as the ability to do work.
The cycle for water on the earth is the following: the sun evaporates water from the
surface of the earth, and it goes into the atmosphere and forms clouds. The cloud
eventually becomes saturated and the water falls back to the earth as rain. Once on the
ground the water at higher elevations begins to flow toward lower elevations and
eventually may reach the ocean. As the water falls from higher to lower elevations the
gravitational potential energy of the water is converted into kinetic energy, heat and
sound.
Overshot Mill - Part 1
The simplest and most common type mill has an overshot waterwheel. An overshot mill is a
potential energy device in which the water falls a vertical distance. This causes the
waterwheel to rotate which changes the potential energy of the water into work inside the
mill. The mill to the right is Mabry Mill just off the Blue Ridge Parkway in Southwestern
Virginia. You can see the waterwheel and the water flume which supplies the water to the top
of the waterwheel. The water is supplied by a small lake back in the trees. The water is
released on the wheel just slightly forward of the vertical center of the wheel. If the water was
supplied exactly at the center of the wheel it would be impossible for the wheel to rotate
because part of the water would flow forward and part backward. There would be no net
force, and the wheel would not rotate. The weight (a force which is a product of mass and the
acceleration of gravity) of the water does cause an unbalanced torque (a force applied at a
distance from a pivot point, i.e. the axle of the wheel) on the wheel. Because this torque is
not balanced by any other torques it causes the wheel to rotate. This rotation carries the
water from a higher level to a lower level and reduces the potential energy of the water. The
greater the vertical distance that the water remains on the wheel or the larger diameter of the
wheel, the more energy removed from the water. The potential energy removed from the
water will be converted into work inside the mill. Not all mills were used to grind grain, but we
will only discuss grain grinding mills (flour and grist). Other types of mills were carding mills,
saw mills, woolen mills and others.
Overshot Mill - Part 2
How is physics applied in the operation of an overshot water powered mill? Potential Energy is
defined as:
(6.1)
U represents the potential energy. m is the mass of the water. g is the acceleration of gravity
which is a constant of
waterwheel.
. h is the height that the water falls while in contact with the
Acceleration is the speeding up or slowing down of an item
(Acceleration will be discussed in more depth in the Chapter 7).
Any item that falls will accelerate at a constant rate, and this is
the acceleration of gravity. The water wheel has buckets that
will catch a finite amount of water before the rotation of the
wheel has reached a point that the next bucket is filled by the
water. Note the water cannot remain on the wheel for a full half
of rotation because it will pour out of the buckets before it
reaches the bottom of the path.
Work is defined as a force moved through a parallel distance to
the direction the force was applied. Energy gives you the ability
to do work. The energy of the water is transferred to doing
work inside the mill.
Figure 6.3 Overshot
Waterwheel
By Old Sturbridge Village
Work is defined as:
(6.2)
W represents the work. F is a force. d is a distance parallel to the direction of the force. The
force is the product of the mass of the water and the acceleration (of gravity). This
relationship is Newton’s Second Law of Motion and is defined as:
(6.3)
The acceleration, a will be the acceleration of gravity and is represented by g. The force is the
weight of the water and is represented with w.
(6.4)
When you drop a weight (force) vertically (height) you change the potential energy to work.
(6.5)
Overshot Mill - Part 3
Determining the mass of the water on a waterwheel is not a measurement that can be easily
made in the real world (and would have been very hard 200 years ago) so we must look at
what can be measured. We could determine the maximum water per bucket, but in reality the
buckets do not get completely filled with water.
Power is the amount of work done in a given amount of time. It is defined by the following
relationship:
(6.6)
P represents power. W is work. t is time.
First of all a time period must be substituted into our relationship to utilize the concept of
power. Using equation (6.5) and substituting it into equation (6.6) the following is obtained.
(6.7)
We have converted our relationship into a time dependent relationship. Finding the mass per
time, a rate, is still a measurement that is not easily determined. We need to convert the
mass of the water into the volume of the water. To accomplish this we will use the density of
water which is a constant for fresh water and is defined to be mass per volume (refer to
chapter 2):
(6.8)
(rho) is the density. m is the mass of the water. V is the volume of the water.
Overshot Mill - Part 4
To determine the volume of the water we need to understand how the water was delivered to
the waterwheel. This was usually done by a wooden flume (sometimes by a metal pipe). The
water might travel to the flume via of an earthen or stone mill race. The wooden flume is the
most common delivery device and we will only analysis this device for our purposes. The
wooden flume is a rectangular shaped device as can be observed in the image below. The
volume of the flume can be calculated by the following relationship.
(6.9)
The volume is defined for the flume as length (l) times the width (w) times the depth (d).
Substituting into equation (6.8) and solving for mass gives the following.
(6.10)
If we take equation (6.10) and substitute it into equation (6.7) we get the following.
(6.11)
Velocity for the water is defined as the distance the water travels in a given amount of time
(see chapter 7 for more detail). The relationship for velocity is defined as:
(6.12)
The l will represent the distance the water travels (length) and t is the time period in which
the distance was measured. We will discuss velocity in more detail in Chapter 7.
Velocity of the water is measured as it travels in the flume. To determine this velocity today
we have devices that we can be placed in the water. In the days of the water powered mill
they would observe a leaf floating in the water or determine how much water flowed down the
flume in a given amount of time and knowing the size of the flume they could determine the
velocity.
Overshot Mill - Part 5
If we substitute equation (6.12) into equation (6.11) we get the following relationship.
(6.13)
We will now substitute the values for the constants
and simplify the equation. The density of water is
1000 kg/m3. The acceleration of gravity is 9.8
m/s2. The constants in equation (6.13) will become
the weight density value of water which was
discussed Chapter 2.
(6.14)
The width times the depth is the cross sectional
area and will be represented with an A.
(6.15)
The height is the vertical distance the water falls while in contact with the wheel which is
usually between 70% and 90% of the height of the wheel. Some highly engineered wheels
this can reach nearly 100%
We can use the equation in the form of equation (6.15) or (6.14).
Example: Mill Springs, KY - Part 1
We will use the overshot mill at
Mill Springs, KY for our example
problem. The water for the mill
comes from 13 springs and the
water is retained by a dam. The
present mill building was built in
1877 and the present water
wheel was installed in 1906. The
first mill on this site was built
around 1817. A famous Civil War
battle was fought near this site in
1862.
The velocity of the water in
the flume (entering the
wheel) is about 2 m/s.
Water remains on the
wheel for approximately
the entire half revolution of
the wheel therefore we will
use the vertical height of
the wheel.
The wheel is approximately 13 meters in height and thus
the value of h is approximately 13 meters. The width of
the wheel is approximately 62 centimeter and the depth
of the water is about 10 centimeters.
If we substitute into equation (6.14) the following is obtained.
(6.16)
Thus the input power of the system is 15,798 watts, but the normal unit of power in the
United States for mechanical systems is the horsepower. One horsepower is equal to 746
watts. Therefore:
(6.17)
This mill, with one of the largest waterwheels in the country, has an input power of only 21.2
hp which is about the same amount of power that your car generates when it is traveling at
55 miles per hour and not accelerating on a highway.
Example: Mill Springs, KY - Part 2
Many of the old mills were shut down as their mill
dams broke or their water wheel decayed or
broke. Some of the mills remained in operation
longer by using a farm tractor or a Model T to
power the mill through a belt. These usually were
the smaller country mills. The larger commercial
mills (like Mill Springs) converted from water
power to steam power or possibly electricity.
Many mills were lost due to fire or floods. The mill at Mill Springs has a gasoline engine that
was obtained when the steel Fitz (manufacturer of the water wheel) waterwheel was installed
in 1906. This engine provided additional power for the mill. It was used to power some of the
other machinery within the mill, too.
Mill Springs operated until 1949 when the Army Corp of Engineers purchased the property as
part of the Lake Cumberland project.
The input power is always less than the output power and many mills were approximately
10% efficient. The mill at Mill Springs has a very high efficiency because of the balance of
the wheel and the speed at which the stones are rotated for demonstration purposes. The
water depth on the waterwheel was much greater and the velocity of the water was much
higher when the mill was operational and the dam was in good repair.
(6.18)
Mills Today
If you visit an old mill they usually grind grain for only a short period of time a few times a
day. This is generally for demonstration purposes. The machinery could grind more grain, but
if they produced large amounts of corn meal or flour they would not have a market for their
product. At many of the old mills the flour they produce often does not meet the health
standards of today. There still are water powered mills that are commercial. One of these is in
Kentucky near Midway.
When the mill is not grinding very little water is required to rotate the wheel. The power
output of the mill can be controlled by the amount of water which is put on the wheel. The
gate box which is visible in Figure 6.8 can control the amount of water released. These gates
are positioned at different places on different mills. When the miller is ready to grind grain he
will raise the gate and allow more water to travel onto the wheel. The wheel will begin to
rotate at a greater speed. The miller will then engage either a gear or belt so that the stones
or roller mills will begin to turn. The miller will continue to increase the flow of water until the
stones are turning at the proper speed. If other equipment is also being used, the gate is
raised even more. The process is reversed when the grinding is complete. If the large amount
of water was allowed to flow over the waterwheel as when the stones and other equipment
were engaged, the wheel would rotate at such a high velocity that the wheel would be
damaged.
The water mills were either large commercial business or small family farm related operations.
Many of the name brands of flour that we use today began with small water powered mills. At
these mills in their early days they would grind 24 hours per day when the crops were being
harvested. At other times of the year they would grind only occasionally. During the harvest
season the farmers might wait in long lines for days to get their corn and wheat ground. Of
course the mill needed to have water and many mills had mill ponds to assure that they had
an ample supply of water. Floods and droughts were serious problems for these early
factories. Remember also that water power was used for other purposes than just grinding
grain. It was used to make cloth and yarn, for power to run the assembly lines for the
manufacturing of the early automobiles, for cutting lumber and many other manufacturing
purposes. Many times sawmills would be built prior to the grist mill to produce the lumber for
the new community. The byproducts of the grinding operation would be used for animals feed
or as the grain for a distillery or just thrown into the mill pond to produce some extremely
large fish.
Kinetic Energy
Not all of the mills were overshot mills as discussed
above. In a few locations undershot water mills were
built however, very few remain in the United States.
Many of these mills were converted into water
powered steel turbines. One of the few remaining is
the mill pictured on the previous page, Graue Mill.
Graue Mill is located outside of Chicago, IL. The stories
that are told are that when Abraham Lincoln was a
young lawyer he would stop and visit the mill. They
would talk with the miller about how the mill worked.
It is known that Lincoln made several trips between
his home and Chicago during this period of time. This
mill was also a stop on the Underground Railroad.
Undershot Mill
Undershot Mills use the principles of kinetic energy. Kinetic
means to be in motion. Therefore this source of energy is the
energy of moving water. An undershot mill uses the velocity
of the water to turn the waterwheel. The water gains it kinetic
energy as part of a fast moving stream or being stored in a
mill pond behind a dam as discussed in Chapter 2. When the
water does not already have sufficient kinetic energy a dam is
placed across a river. Water is taken off the bottom of the
dam and flows through a millrace. The deeper the water held
by the dam the faster the water will flow down the millrace.
This occurs because there is a greater amount of pressure on
the water at the bottom of the dam due to the weight of the
water. You may have experienced this effect as you swim
deeper in a pool the pressure you feel on your body is
greater. This is because more water is above you pressing
down. Review our discussion of dams in Chapter 2.
Kinetic Energy is defined as:
(6.19)
K represents the kinetic energy of the system. m is the mass. v is the velocity.
The water slows down upon striking the waterwheel therefore the kinetic energy of the
water is also less. The kinetic energy lost by the water is converted to work inside the mill.
If the water comes nearly to rest then all of the kinetic energy of the water has been
converted into work. Combining equations (6.2) and (6.19):
(6.20)
Similar to the overshot wheel discussed earlier we will
convert the equations from energy to power equations
by making the equations time dependent. This is done
because of the problem defining the mass of the water
striking the waterwheel. We will start with equations
(6.6) and (6.20) and combine them together.
(6.21)
Using the same logic as done with the overshot mill we will substitute equation (6.10) and
the following is obtained:
(6.22)
As before length per time is velocity and the value for density of water is a constant. Width
times depth is the cross section of the flume as before. Therefore the equation simplifies to:
(6.23)
Graue Mill, IL
For our example we will use Graue Mill near
Chicago, IL. Note that the waterwheel is not like
the wheel discussed in the overshot mill.
Overshot wheels have buckets to hold the water
as the wheel turns. Also there was an elevated
flume. The wheel on Graue Mill and other
undershot mills is more like the paddle wheel of
a steamboat. This is because we are only using
the kinetic energy of the water. It is not falling;
instead it is pushing the waterwheel.
As you can see in Figure 6.16 this wheel is much wider than the wheel previously discussed,
but it is also much less massive and therefore has much less inertia. The wheel must be
extremely well balanced. The cross sectional area of the flume is 3m x .5m or 1.5 m2. The
velocity of water is approximately 3 m/s. If we substitute into equation (6.23):
(6.24)
Again the input power is much greater than the output power of the mill. Therefore the
efficient of the mill can be calculated as discussed previously.
Other Types of Mills
There are other types of waterwheels these including the
breastshot wheel which uses both kinetic and potential
energy. There are two types of breastshot wheel. With the
first type the water hits the wheel slightly below center
(also known as pitchback wheel).With the other type the
water hit slightly above the center (not very efficient).
Another type of wheel is the tub wheel which took many
different forms including ones that look like paddles to
others that look like an overshot wheel lying on its side.
In the 1840’s the first turbines were used which were
made of metal and had a much higher efficiency than did
the wooden waterwheels.
The newer overshot waterwheels were also made of metal
which lasted longer and did not have the typical problems
of wood wheels such as warping and deterioration due to
the water. Therefore as wooden wheels decayed they
were replaced with metal wheels or turbines. Waterwheels
were ordered and installed in Kentucky into the early
parts of the 20th century. There are still commercial mills
in Kentucky which operate on water power and use
turbines. There are other parts of the country that have
commercial mills that still use overshot waterwheels.
Energy Transfer
Many of the early water powered mills were built to
grind grain (corn and wheat). To transfer the input
power to the turning of the stones took massive
gearing both to change the direction of the power
and also to increase the velocity of the stones.
Similarly when you ride most bicycles you will
peddle at one speed but the rear wheel will turn at a
different speed due to the gears.
The power take off from the wheel at Mill
Springs is typical of many of the large metal
wheels. The wheel has a gear ring visible in
figure 6.8. The power is transferred to a smaller
gear. The large ring has 36 teeth in each of the
16 sections for a total of 576 gear teeth. The
smaller gear has 38 teeth on its circumference.
Therefore for every rotation of the large wheel
the small wheel turned 15.2 rotations.
The large wheel turns at 2 or 3 rotations per minute so the small wheel turns at about one
revolution every two seconds.
Figure 6.21 Energy Transfer by Old Sturbridge Village
Rotation Kinetic Energy
To grind grain solid stones with grooves
cut into them are used. These stones
called burrs, do not crush the grain but
instead it is sheared. The stones are
large and massive and must rotate at a
speed much faster then the water
wheel turns.
The equations and solutions for an
object that rotates is different from an
object moving in a straight line. For an
object that rotates we look at the
angular velocity instead of the linear
velocity and the Greek letter omega (ω)
is used to represent the angular
velocity. Measurements are made
usually in radians per second. A radian
is part of a circle. All circles have 2π
radians and π is equal to 3.14. A radian
does not depend on the radius of the
circle. Angular velocity is defined as the
angle the object turn through during a
defined period of time.
We are only going to use solid circular shaped objects but be aware that different shaped
objects with the same mass will rotate at different rates dependent on the moment of
inertia of the object. Rings, solid disks, bars and spheres will all have different equations
for the moment of inertia. For the millstone which is a solid disk the equation is:
(6.25)
I represents the moment of Inertia. m is the mass of the object, and r is the radius of the
object. For a typical stone in a large mill the stone is usually about 1.5 m in diameter and
has a mass of 1000 kg.
Therefore:
(6.26)
Note that .75 meter is used since this is the radius, i.e. radius is half of the diameter and
the height was converted into metric units instead of English.
Equation for Rotational Kinetic Energy
The equation for Rotational Kinetic Energy is represented by the following relationship.
(6.27)
The stones begin at rest and increase in speed until they reach operational speed. The
work done on the stones is equal to the rotational kinetic energy of the stones. A stone in
a typical mill will turn at 60 revolutions per minute. Revolutions per minute need to be
converted to radians per second. There are 60 seconds per minute and 2π radians per
revolution. We will substitute equation (6.27) into the power equation (6.6).
(6.28)
Therefore the power required to start a millstone turning to operational speed at the Mill
Spring grist mill, is:
P = 7.43 hp
Note this is much less than the input power of the mill. The energy which was lost was
mainly due to friction (assume that all the power was used only by the millstones). This
also assumes that it takes only one second to go from rest to operational speed and at
many mills this time period is much longer.
Efficiency
If the output power is the amount of power required to do the work for which the mill is
designed and the input power is the amount of power which is derived from the water
we can determine a simple efficiency of the mill using equation, (6.18). We will use the
data from equations (6.17) and (6.28) to calculate the result. These results will be for
Mill Springs with the constrains previously noted.
(6.29)
Old College Book: Waterwheels, Part 1 & 2
The concepts covered in this chapter were taught in colleges and universities in the past.
I have copied a few of the pages from an old textbook for you to review. These images
are used with permission of the publisher at the Society for the Preservation of Old Mills
(SPOOM, http://www.spoom.org).