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Moment Of Inertia
Where does moment of inertia originate?
1. Angular acceleration is proportional to torque.
2. This corresponds to Newton’s law for translational motion’
(where Force is replaced by torque and angular acceleration
takes the place of linear acceleration)
3. In a linear case the acceleration is proportional to the Force
and inversely proportional to the mass.
Newton's first law of motion says "A
body maintains the current state of
motion unless acted upon by an
external force." The measure of the
inertia in the linear motion is the
mass of the system and its angular
counterpart is the so-called moment
of inertia. The moment of inertia of a
body is not only related to its mass
but also the distribution of the mass
throughout the body. So two bodies
of the same mass may possess
different moments of inertia.
It appears in the relationships for the
dynamics of rotational motion. The
moment of inertia must be specified
with respect to a chosen axis of
rotation. For a point mass the moment of
inertia is just the mass times the square
of perpendicular distance to the
rotation axis, I = mr2. That point mass
relationship becomes the basis for all
other moments of inertia since any
object can be built up from a collection
of point masses.
Relationship of formulas
F = ma
and atan = ra so F = mra
t = Fr
t = mr2a
Here we have a direct relation between the angular
acceleration and the applied torque. The quantity mr2
represents the rotational inertia of the particles of a
certain mass rotating in a circle of radius “r” about a
fixed point.
Common Moments of Inertia
The moment of inertia of a system describes how the
mass is distributed around the rotating object.
Which object will reach the bottom of the incline first?
They have the same radius and are the same mass
By calculating a value for “m” and ”R” you
can see the hoop has a larger moment of
inertia and therefore requires mrre energy
to get it started.
Newton’s Second Law for rotation is seen below as it is
compared to the linear second law. Also you can see how
moment of inertia is used to compare linear and angular
values for momentum, kinetic energy, and work
Rotational Kinetic Energy
An object rotating around an axis is said to have
rotational kinetic energy.
The equation to the right
represents the kinetic
energy of a rigid rotating
An object that rotates while its center of mass undergoes
translational motion will have both translational and
rotational kinetic energies.
Therefore the total kinetic energy of an object that
rotates and moves in a linear direction is the sum of the
rotational and translational kinetic energies.
KE = ½ mv2 + ½ Iw2
If an object rolls down an incline the potential energy
(mgy) = PE is converted to both rotational and
translational kinetic energies But if an object slides down
an incline all the potential energy is converted to kinetic
So using the kinetic energy equation which object will
be moving the fastest at the bottom of the incline?
Would a sliding object
beat both these objects?
Work done by torque
Work done on an object rotating about a fixed axis, such
as a pulley can be written in angular quantities.
W = Fr = FrDq since t = rF then W = tDq
Power = W/Dt = tDq/Dt = rw
Angular momentum and its conservation
In a like manner, the linear momentum, p=mv, has a
rotational analog. It is called angular momentum, L. For
an object rotating about a fixed axis, it is defined as;
L = Iw
Newton’s second law for rotation can also be written as;
St = D L/ Dt
Angular momentum is an important concept in physics
because, under certain conditions is a conserved
quantity. We can see if the net torque on an object is
zero, then the sum of the torques equals zero. That is,
L does not change. This is the law of conservation of
angular momentum for a rotating object – the total
angular momentum of a rotating object remains
constant if the net torque acting on it is zero.