moment of inertia
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What is Moment of Inertia?
Moment of inertia, also known as rotational inertia, is
analogous to the inertia of linear motion.
It is necessary to specify a moment of inertia with respect to an axis of rotation.
According to Newton's first law of motion "A body maintains the current state of motion
unless acted upon some external force". The moment of inertia is related to the
distribution of mass throughout the body, not just mass of the body alone.
That is why two bodies of the same mass may have different moments of inertia.
If we consider a rigid body as a system of particles and the relative position of these
particles does not change, then the moment of inertia of a point
mass is equal to:
I = m.r ^{2},
for each particle so that the moment of inertia is equal to the mass multiplied by the square of the distance,
where m = the mass of the particle, and r = the distance from the axis of rotation
to the particle.
Moment of inertia is also referred to as the second mass moment.
The first mass moment is equal to mass multiplied by distance, m.r.
The centre of mass of a system of particles or a rigid body can be derived using
the first moment concept.
The moment of inertia of an object is a measure of how difficult it is to change the
angular motion of that object about the axis. For example we can take two discs of
equal mass but the diameter of one is greater than the other. As a result its mass would
be distributed further from the axis of rotation and it would take more effort to
accelerate the first disc as compared to the second. This is because the first disk has
a larger moment of inertia.
If the shape of object changes so would the moment of inertia of that object.
When a skater pulls in her oustretched arms her rotation speeds up because
her moment of inertia is reduced and consequently the angular momentum increases.
Rotational Kinetic Energy
The rotational kinetic energy of a body can be expressed in terms of its moment of
inertia. Given masses m moving with speeds v, the rotational energy T for each is mass
T=1/2mv ^{2}=1/2m(wr) ^{2}=1/2mr2w2=1/2Iw ^{2}
where w is the angular velocity
When the angular momentum vector is parallel to the angular velocity vector, they can be related using the equation
L=wIwhere the angular momentum is L andthe angular velocity ω.
If the moment of inertia is constant the torque on an object and its angular acceleration are related by
T= I αWhere t is the torque and α is the angular acceleration.
Parallel Axis Theorem
Once the moment of inertia has been calculated for rotation about the centre of mass
of a rigid body, the moment of inertia for any parallel
rotation axes can be calulated as well, without needing to go back to the formal definition. If the axis of
rotation is displaced by a distance R from the centre of mass axis of rotation (e.g.
a disc being spun around a point near its edge rather than the centre) the
displaced and centre-moment of inertia are related as follows:
We can calculate the moment of inertia for parallel rotation axes easily once we have
calculated the moment of inertia for rotations about centre of mass. If the axis of rotation
is a distance R from the centre of mass, then we get
the following equation.
I(displaced) = I(centre) + M.R^{2}
Formulae for Moment of Inertia
Thin cylindrical shell with open ends, of radius r and mass m:
I = m.r^{2}
Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2 and mass m :
I = 1/2m.(r1^{2}+r2^{2})
Solid cylinder of radius r and mass m :
I = m.r^{2}/2
Thin, solid disk of radius r and mass m :
I = m.r^{2}/2
Thin circular hoop of radius r and mass m:
I = m.r^{2}
Solid sphere of radius r and mass m:
I = 2m.r^{2}/5