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Title: Moment Of Inertia Of A Rotor
Aim: See your Lab Manual
Apparatus: See your laboratory Manual
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THEORY:
**General Knowledge**
The law of inertia states that it is the
tendency of an object to resist a change in motion. Copernicus and then Galileo
were the first to dispute Aristotle's thought on movement and in doing so they
developed the first thoughts on inertia. Galileo Galilei was the first state “A
body moving on a level surface will continue in the same direction at a
constant speed unless disturbed”. Johannes Kepler was the first to look
specifically at Inertia, he even gave the name which come from the latin for
“laziness”. But it was Newton who echoed Gallilei with added precision and
quantification buy stating that an object will remain in motion or at rest.
This resistive force that objects contain is described and “Inertia”, and
should be considered the single term that describes Newton’s First Law.
The
rotational inertia of an object is dependent on the mass the the arrangement of
the mass within the object. A simple rule of thumb is- the more compact an
object's mass, the less rotational inertia an object will have. We studied to
shapes and their inertia. A ring and a disk. The rotational inertia of a ring
with consistent density is dependent of its mass and the inner and outer
radius. The relationship between the mass and the radii is described as Iring=12M(R12+R22).
A disk is nothing but a ring with no inner radius so its inertia is simply a
function of its mass and its outer radius, specifically Idisk=12MR2.
Experimentally and inertia can be found by applying a known torque to the
object and dividing that torque by the resulting acceleration. Iobject=.
We gathered this data by suspending a rotary motion sensor with its rotating
axis perpendicular to the earth, thus its pulley is parallel. A second
pulley was mounted with it’s axis parallel and its face perpendicular to the
earth. In order run a cord downward, thus a weight could be hung from the cord
to induce a force on the rotary sensor. Careful attention was given to the
alignment of the two pulleys. we wanted the face of the second pulley to run
tangential to the circumference of the rotary sensors pulley. If this was not
achieved the force applied to the rotary sensor would actually be the Fcosθ
where theta is the angle off from ideal.
THEORY
1:
In this experiment, we are going to estimate the moment of inertia of a
rotating rigid body using dynamical relations (specifically energy
conservation). Since moment of inertia, relative to rotational motion,
corresponds to mass in linear motion, this is an indicator of the more complex
nature of rotational motion, as we cannot, in any straightforward, easy way,
estimate the mass of a body by a free fall experiment. This experiment will be
an analogous experiment to such a free fall experiment, and will supply a
simple procedure for estimating moment of inertia. The reason for this is that
we will use the known mass of the falling body in the rotation experiment.
Recall that the potential energy
of a particle of mass m at a height h above the terminal position is mgh ,
where g is the acceleration due to gravity, while the final kinetic energy is
1/2mv2 , where v is the speed of the mass in the terminal
position. If we equate these two quantities, using conservation of energy
(ignoring such losses as those due to air resistance), the mass m simply
cancels out of the equation, i.e. simple free fall cannot determine the
particle mass. On the other hand, if we release the mass m , connected by a
string around a pulley (see Figure 1), from a height h above a terminal
position, we find that we can estimate the moment of inertia, I , of the
pulley, assuming m to be known. The initial potential energy is mgh as before,
but the final kinetic energy must include the kinetic energy of the pulley
(1/2)Iω2
where ω is
the angular speed(of the pulley),
and the final kinetic energy of
the string holding the mass (which we shall ignore).
Ignoring losses due to friction,
energy conservation now yields:
mgh = ½ Iω2 + ½mv2 (1)
We can easily solve for I in this
equation, provided that we have some adequate theory for ω and v.
As will be seen, in the
discussion of the theory, by making some simple assumptions, this is indeed the
case, and we are able to estimate the moment of inertia I , at least in
principle. We will be able to do this directly from the dynamical relations
involving torque in this case.
THEORY 2:
The purpose of this experiment is to determine the
experimental moment of inertia of a disk and of a ring by using the principle
of conservation of energy. The experimental moment of inertia will be compared
with the theoretical moment of inertia
for each.
The theoretical moments of inertia of the bodies
used as unknowns in this experiment may be calculated from the following
equations:
I
= ½ MR2 for
the disk when horizontal.
I
= ¼ MR2 for
disc when vertical.
I=
½ M(R12 + R22) for
the ring.
A vertical axle supports various bodies whose
moment of inertia are to be determined. A small drum is rigidly attached to the
axle. A string is wound around the drum and passes over a pulley. A mass, attached
to the end of the string, falls to the floor, unwinding the string and
accelerating the axle and the experimental body rotationally. We will refer to
this as the driving mass.
As the driving mass falls to the floor, the
gravitational force exerted by the Earth does positive work on the system. This
work is equal to the weight of the driving mass times the distance the mass
falls. Friction also does work in this experiment.
The work
done by friction will be computed by determining what driving mass will fall at
a constant speed once set into downward motion. This particular driving mass
will be called the friction mass. When the driving mass is equal to the
friction mass, the kinetic energy of the system does not change. If the kinetic
energy does not change, the total work done on the system is zero.
We will
assume that the work done by friction is independent of the driving mass,
negative, and equal in magnitude to the weight of the friction mass times the
height the driving mass falls.
By the work energy theorem, the total work done on
the system will equal the change in kinetic energy of the system. The kinetic
energy of our system will be the sum of the translational kinetic energy of the
driving mass and the rotational kinetic energy of the experimental body.
For Procedures, See your Mechanical Laboratory Manual.
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Observations :-
1.
Applications:-
1.
PRECAUTIONS:
For
Precautions, See General Laboratory Precautions
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