CHAPTER
 1
INTRODUCTION
The mass moment of inertia is an
important concept in rotational motion. The mass moment of inertia also called
as the rotational inertia of a body is a measure of how hard it is to get it
rotating about some axis. Rotational inertia is one indicator of the ability of
rotating body to store kinetic energy. It is also an indicator of the amount of
torque that will be needed to rotationally accelerate the body. Just as the
mass is a measure of resistance to linear acceleration, moment of inertia is a
measure of resistance to angular acceleration. In the design of devices that
are intended for the storage and transfer of large amount of energy, maximum
moment of inertia of the rotating parts is desirable. Flywheels used in
automobiles are a good example for this. Flywheels are designed to have very
large moment of inertia. A large moment of inertia will require a large driving
torque. The flywheel is set in rotation when the vehicle is started. It stores energy given to it as energy of
rotation. Because of its large moment of
inertia, it prevents the engine from stopping when the pistons are not
supplying energy. The flywheel prevents
the engine from accelerating too much when energy is supplied to it. In this way, it keeps the engine turning
through its cycles even when the piston is not supplying power (pistons supply
power for only a fraction of the total cycle time). The function of the flywheel is to make the
motion of the vehicle smooth and jerkfree. Flywheels are not only used in
automobiles but also in generators, electric motors etc.
Since the moment of inertia of flywheels
is of high importance, its measurement is very relevant. There are theoretical
formulas and expressions for the calculation of moment inertia. But most of
these empirical formulas are restricted to simple geometries. In practical
applications the shape and geometry of flywheels employed will be complex and
hence such formulas will not serve the purpose. Here arises the need of some
method for the determination of rotary inertia of flywheels.
In this project we deal with an
experimental method for the determination of moment of inertia of a flywheel. A
particular experimental setup is fabricated such that the moment of inertia of
the flywheel can be calculated using the measurable or known parameters such as
number of rotations of the flywheel, radius of the axle, the weight applied,
time for rotation etc.
CHAPTER  2
METHODOLOGY
·
Detailed study
of the experiment.
·
Study of the
requirements.
·
Experimental
setup is designed.
·
Components are
fabricated and assembled.
·
Testing of the
setup is done, for different loads.
CHAPTER
 3
LITERATURE REVIEW
v Moment of
Inertia
In
classical mechanics, moment of inertia, also called mass moment of inertia,
rotational inertia, polar moment of inertia of mass, or the angular mass, (SI
units kg·m²) is a measure of an object's resistance to changes to its rotation.
It is the inertia of a rotating body with respect to its rotation. The moment
of inertia plays much the same role in rotational dynamics as mass does in
linear dynamics, describing the relationship between angular momentum and
angular velocity, torque and angular acceleration, and several other
quantities. The symbol I and sometimes J are usually used to refer to the
moment of inertia or polar moment of inertia.
While
a simple scalar treatment of the moment of inertia suffices for many
situations, a more advanced tensor treatment allows the analysis of such
complicated systems as spinning tops and gyroscopic motion.
The
concept was introduced by Leonhard Euler in his book ‘Theoria motus corporum
solidorum seu rigidorum’ in 1765. In this book, he discussed the moment of
inertia and many related concepts, such as the principal axis of inertia.
The
moment of inertia of an object about a given axis describes how difficult it is
to change its angular motion about that axis. Therefore, it encompasses not
just how much mass the object has overall, but how far each bit of mass is from
the axis. The farther out the object's mass is, the more rotational inertia the
object has, and the more force is required to change its rotation rate. For
example, consider two hoops, A and B, made of the same material and of equal
mass. Hoop A is larger in diameter but thinner than B. It requires more effort to
accelerate hoop A (change its angular velocity) because its mass is distributed
farther from its axis of rotation: mass that is farther out from that axis
must, for a given angular velocity, move more quickly than mass closer in. So
in this case, hoop A has a larger moment of inertia than hoop B.
The
moment of inertia of an object can change if its shape changes. Skaters who
begin a spin with arms outstretched provide a striking example. By pulling in
their arms, they reduce
their moment of inertia, causing them to spin faster (by the conservation of
angular momentum).
The moment of inertia has two forms, a
scalar form, I, (used when the axis of rotation is specified) and a more
general tensor form that does not require the axis of rotation to be specified.
The scalar moment of inertia, I, (often called simply the "moment of
inertia") allows a succinct analysis of many simple problems in rotational
dynamics, such as objects rolling down inclines and the behavior of pulleys.
For instance, while a block of any shape will slide down a frictionless decline
at the same rate, rolling objects may descend at different rates, depending on
their moments of inertia. A hoop will descend more slowly than a solid disk of
equal mass and radius because more of its mass is located far from the axis of
rotation. However, for (more complicated) problems in which the axis of
rotation can change, the scalar treatment is inadequate, and the tensor
treatment must be used (although shortcuts are possible in special situations).
Examples requiring such a treatment include gyroscopes, tops, and even
satellites, all objects whose alignment can change.
The moment of inertia is also called the
mass moment of inertia (especially by mechanical engineers) to avoid confusion
with the second moment of area, which is sometimes called the area moment of
inertia (especially by structural engineers). The easiest way to differentiate
these quantities is through their units (kg·m² as opposed to m^{4}). In
addition, moment of inertia should not be confused with polar moment of inertia
(more specifically, polar moment of inertia of area), which is a measure of an
object's ability to resist torsion (twisting) only, although, mathematically,
they are similar if the solid for which the moment of inertia is being
calculated has uniform thickness in the direction of the rotating axis, and
also has uniform mass density, the difference between the two types of moments
of inertia is a factor of mass.
CHAPTER  4
DESCRIPTION OF EQUIPMENT
The main components of the
experimental setup are
1)
Flywheel
2)
Axle on which flywheel is mounted
3)
End bearings to support the axle
4)
The base and stands on which bearings are fixed
5)
Thread to hang the
slotted masses to the flywheel
6)
A set of slotted
masses with a hanger
v Flywheel
Fig. 1

v Axle
Fig. 2

v Bearing
A
bearing is a device to allow
constrained relative motion between two or more parts, typically rotation or
linear movement. Bearings may be classified broadly according to the motions
they allow and according to their principle of operation as well as by the directions
of applied loads they can handle. Plain bearings use surfaces in rubbing
contact, often with a lubricant such as oil or graphite. A plain bearing may or
may not be a discrete device. It may be nothing more than the bearing surface
of a hole with a shaft passing through it, or of a planar surface that bears
another (in these cases, not a discrete device); or it may be a layer of
bearing metal either fused to the substrate (semidiscrete) or in the form of a
separable sleeve (discrete). With suitable lubrication, plain bearings often
give entirely acceptable accuracy, life, and friction at minimal cost.
Therefore, they are very widely used.
However, there are many applications
where a more suitable bearing can improve efficiency, accuracy, service intervals,
reliability, and speed of operation, size, weight, and costs of purchasing and
operating machinery.
Thus, there are many types of
bearings, with varying shape, material, lubrication, principle of operation,
and so on. For example, rollingelement bearings use spheres or drums rolling
between the parts to reduce friction; reduced friction allows tighter
tolerances and thus higher precision than a plain bearing and reduced wear
extends the time over which the machine stays accurate. Plain bearings are commonly
made of varying types of metal or plastic depending on the load, how corrosive
or dirty the environment is, and so on. In addition, bearing friction and life
may be altered dramatically by the type and application of lubricants. For
example, a lubricant may improve bearing friction and life, but for food
processing a bearing may be lubricated by an inferior foodsafe lubricant to
avoid food contamination; in other situations a bearing may be run without
lubricant because continuous lubrication is not feasible, and lubricants
attract dirt that damages the bearings.
v Pulley
A pulley, also called a
sheave or a drum, is a mechanism composed of a wheel on an axle or shaft that
may have a groove between two flanges around its circumference. A rope, cable, belt,
or chain usually runs over the wheel and inside the groove, if present. Pulleys
are used to change the direction of an applied force, transmit rotational
motion, or realize a mechanical advantage in either a linear or rotational
system of motion. It is one of the six simple machines. Two or more pulleys
together are called a block and tackle.
v Cord
A rope is a length of fibers, twisted or braided together to improve
strength for pulling and connecting. It has tensile strength but is too
flexible to provide compressive strength (i.e. it can be used for pulling, but
not pushing). Rope is thicker and stronger than similarly constructed cord,
line, string, and twine. Common materials for rope include natural fibers such
as Manila hemp, hemp, linen, cotton, coir, jute, and sisal.
Fig. 4

v The base
and stands on which bearings are fixed
The base is made up of plywood. The stands on which bearings
are fixed is made up of teak wood. The two stands are joined to the base and
they are fastened using screws.
Fig. 5

v Hanging
slotted weight
The slotted weight consists of four 50gram weights which can
be added or removed from the default 50gram weight provided with the hook.
Fig. 6

CHAPTER
 5
PRINCIPLE
The flywheel is mounted in its bearings with its
axle horizontal and at a suitable height from the ground, and a string carrying
a suitable mass m at its one end is wrapped completely and evenly round
the axle. When the mass m is released, the string unwinds itself, thus
setting the flywheel in rotation. As the mass m descends further and
further the rotation of the flywheel goes on increasing till it becomes maximum
when the mass touches the floor.
Let h be the distance fallen through by the
mass and let v and w be the linear velocity of the mass and
angular velocity of the flywheel respectively at the instant the mass drops
off. Then, as the mass descends a distance h, it loses potential energy (mgh) which is used up: (i) partly
in providing kinetic energy of translation 1/2mv^{2}^{ }to the
falling mass itself, (ii) partly in giving kinetic energy of rotation 1/2Iω^{2} to the flywheel (where
I is the moment of inertia of the flywheel about the axis of rotation)
and (iii) partly in doing work against friction. If the work done
against friction is steady and F per turn, and, if the number of
rotations made by the flywheel till the mass detaches is equal to n_{1},
the work done against friction is equal to n_{1}F. Hence
by the principle of conservation of energy, we have
After
the mass reaches the floor, the flywheel continues to rotate for a considerable
time t before it is brought to rest by friction. If it makes n_{2
}rotations in this time, the work done against friction is equal to
n_{2}F and evidently it is equal to the kinetic energy of
the flywheel at the instant the mass touches the floor. Thus
Fig. 7

After
the mass has detached, its angular velocity decreases on account of friction
and after some time t, the
flywheel finally comes to rest. At the time of detachment of the mass the angular
velocity of the wheel is ω and when it
comes to rest its angular velocity is zero. Hence, if the force of friction is
steady, the motion of the flywheel is uniformly retarded and the average
angular velocity during this interval is equal to ω/2,
Thus
...(4)

Thus
observing the time‘t’ and counting the rotations n_{1} and n_{2}
made by the flywheel its moment of inertia can be calculated from equation (3)
and (4).
For
a flywheel with large moment of inertia (I),
1/2mv^{2} may be
neglected, the equation (3a) becomes
Substituting
the value of ω from Eq. (4) in above
equation, we get
This
formula is used to calculate the moment of inertia of a flywheel.
v ALTERNATIVE METHOD
Fig. 8

Moment of inertia is calculated by plotting the graph and
obtaining its slope.
CHAPTER – 6
DESIGN AND DRAWING
Fig. 9

Fig. 10

CHAPTER  7
PROCEDURE
1. Attach
a mass m to one end of a thin thread.
2. The
thread is wrapped evenly round the axle of the wheel.
3. Allow
the mass to descend slowly and count the number of revolutions n_{1}
during descent.
4. When
the thread has unwound itself and the mass reaches the floor after n1
turns, start the stopwatch.
5. Count
the number of revolutions n_{2} before the flywheel comes to rest
and stop the stop watch. Thus n2 and t are known.
6. Repeat
the experiment with three different masses.
7. Calculate
the value of I using the equation derived earlier.
v Alternative
method
1. Attach
a mass m to one end of a thin thread.
2.
The thread is wrapped evenly round the
axle of the wheel.
3.
The mass is released from
fixed height h unwinding the thread around the axle.
4.
Record the time t for
the load to reach the ground.
5.
Vary m, and determine
the corresponding value of t.
6.
Plot a graph of α against T.
7.
Calculate the moment of
inertia of the flywheel.
CHAPTER 8
OBSERVATION AND CALCULATIONS
Total load applied
m
( kg)

Height of fall of the mass
h
(m)

No. of revolutions of flywheel before
the mass touches the floor
n_{1}

No. of revolutions of flywheel after
the mass touching the floor
n_{2}

Time for n_{2 }revolutions
t (seconds)

Moment of inertia
I
(kgm^{2})

0.15

0.5

6

7

4.75

2.31×10^{3}

0.2

0.5

7

8

5.46

3.086×10^{3}

0.25

0.5

8

12

6.76

2.957×10^{3}

Mean I = 2.78×10^{3 }kgm^{2}
Total load applied
m
(in kg)

Height of fall of the mass
h
(m)

Time taken by the mass to touch
the floor
t
(s)

Linear acceleration
(m/s^{2})

Tension
(N)

Angular acceleration
(rad/ s^{2})

0.15

0.5

5.38

0.0345

1.466

3.622

0.2

0.5

4.15

0.0580

1.9504

6.089

0.25

0.5

3.20

0.097

2.428

10.1837

SAMPLE CALCULATION
We have the expression for moment of inertia,
Sl. No. 1
m = 0.15 kg g = 9.81 m/s^{2} h = 0.5 m
t = 4.75
s n_{1}
= 6 n_{2 }=
7
I = (0.15×9.81×0.5)
_{/ }(8×3.14×3.14×7×7_{/}4.75×4.75)
× (1+6/7)_{ }
= 2.3127×10^{3 }Kgm^{2}
Alternative method
The
graph plotted with T in Xaxis and α
in Yaxis
Moment of inertia, I = R/s where R = radius of axle
and s =slope from the graph
We have,
R = 9.525×10^{3 }
s = 3.5 from the graph
I = 9.525×10^{3 }/ 3.5
= 2.72×10^{3}
Kgm^{2}
Angular acceleration (α)
vs. Tension (T) Graph

T

CHAPETR  9
SOURCES OF ERRORS AND PRECAUTIONS
1.
The loop should be sufficient so that loop
remains even after the mass after reach the floor and there may be no tendency
for it to rewind in the opposite direction and stop the free rotation of
flywheel.
2.
The
string should be evenly wound on the axle, i.e., there should be no
overlapping of, or a gap left between, the various coils of the string.
3.
The
string used should be of very small diameter compared with the diameter of the
axle. If the string is of appreciable thickness half of its thickness should be
added to the radius of the axle to get the effective value of r.
4.
The
friction at the bearings should not be great and the mass tied to the end of
the string should be sufficient to be able to overcome the bearingfriction and
so to start falling of its own accord.
5.
Take
extra care to start the stopwatch immediately the string starts to move.
CHAPTER  10
CRITICISM OF THE METHOD
In this method the exact instant at which the mass drops off
cannot be correctly found out and hence the values of n_{1}, n_{2}
and t cannot be determined very accurately. The angular velocity ω of the
flywheel at the instant the mass drops off has been calculated from the formula
ω = (4Πn_{2})/t on the assumption that the force of friction remains
constant while the angular velocity of the flywheel decreases from w to zero.
But as the friction is less at greater velocities, we have no justification for
this assumption. Hence for more accurate result, ω should be measured by a
method in which no such assumption is made e.g., with a tuning fork.
CHAPTER  11
CONCLUSION
After the detail study about moment of inertia and related
concepts the experimental method for determination of moment of inertia of a
flywheel was designed. The experimental setup for this purpose was fabricated
successfully. The result obtained upon conducting the experiment was very
satisfactory and nearly accurate. The project gave us more confidence that we
will be able to put into practice whatever theoretical knowledge we gained
during our course of study till now.
I really like the information which you have provided in blog about this concept of Physics.Rotation motion is different than Newton's basic laws of motion.
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