What is the difference between polar moment of inertia, mass moment of inertia and simply moment of inertia?
Its not that I dont know anything. I am confused where and why do we use different moment of inertias at different places. I cant understand the concept of radius of gyration either.
Please help me in clarifying this also.
Answer:
Hi
i will clear ure doubts...
make sure ure basics are clear...
i will explain u with example and drwings..k
POLAR MOMENT OF INERTIA
The Polar Area Moment Of Inertia of a beams cross-sectional area measures the beams ability to resist torsion. The larger the Polar Moment of Inertia the less the beam will twist.
The following are the mathematical equations to calculate the Polar Moment of Inertia:
Jz = INTEGRAL of (x^2 + y^2)dA
x is the distance from the y axis to an infinetsimal area dA.
y is the distance from the x axis to an infinetsimal area dA.
for formulae (click on)
http://www.efunda.com/math/areas/momento...
MASS MOMENT OF INERTIA
The Mass Moment of Inertia of a solid measures the solid's ability to resist changes in rotational speed about a specific axis. The larger the Mass Moment of Inertia the smaller the angular acceleration about that axis for a given torque.
The mass moment of inertia depends on a reference axis, and is usually specified with two subscripts. This helps to provide clarity during three-dimensional motion where rotation can occur about multiple axes.
Following are the mathematical equations to calculate the Mass Moment of Inertia:
CLICK
http://www.efunda.com/math/solids/massmo...
FORMULAE
Ixx = INTEGRAL (density)^2 dm=INTEGRAL of (x^2 + y^2)dm
similarly , foe xx , yy, zz axis ...k
where
x is the distance from the yz-plane to an infinitesimal area dA.
y is the distance from the zx-plane to an infinitesimal area dA.
SIMPLE MOMENT OF INERTIA
Moment of inertia is the name given to rotational inertia, the rotational analog of mass for linear motion. 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 = mr^2... 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.
FORMULA...
I = Integral of (r^2)dm...limit from M to 0...k
click on
http://hyperphysics.phy-astr.gsu.edu/hba...
RADIUS OF GYRATION
The radius of gyration Rg describes the distribution of particles (or infinitesimal elements) in a D-dimensional space by relating it to an equivalent distribution in a D-dimensional sphere, usually a circular (D=2) or spherical (D=3) distribution.
CLICK ON (for formulaes)
http://www.efunda.com/math/areas/radiuso...
hope this is the best answe...
all the best
I'm more able to answer why than the derivations.
Personally, I know normal moment of inertia is what I use when an object is bent. Putting more material as far from the center (really the centroid, but it is close enough for now) is a good way to stiffen a beam and keep if from moving. So a good beam is in an I shape with most of the material at the top and bottom and very little material (the web) connecting the top and bottom, but if it is left or right of the center isn't very important.
When you try to twist something - torsion - a different property comes in to play -polar moment of inertia. The more material that is a radius away from the center, the better, material at the center is pretty useless, so a hollow pipe is better than a solid pipe, and a big hollow pipe is better than a small hollow pipe.
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