Origin of Moment of Inertia

hello

very fundamental question, and the concept being mathematical in nature, you need to appreciate the mathematical "meaning".

That apart, you may try to "google" and find more info.

In engineering the moment of inertia is visualized as the "resistance of a cross section against bending"

Not only bending, torsion as well.

For bending the inertia moment with respect to an axis will be the "resistance" and for torsion it is the "polar" inertia moment which playsthe same role.

GO DUCKS!

Nothing is "mathematical in nature" you have to understand it from an engineering practical case ! ( please go to my answer # 15 in reply to # 3 )

We get always confused when we use " I = Area moment of inertia " it is better to use the "I = Second moment of an area around an axis " because inertia is more related to a mass resisting acceleration ( Newton's 2nd Law or D'Alembert principle )

In bending a bar having a rectangular section ( a ruler for example !) the distribution of the area around the axis (the smaller or the larger ? ) normal to the possible rotation of the bar affects the deflection ( I is the dominant factor )

The term "Moment of Inertia" is used commonly in two related but seemingly different ways (and your distinctions re mass and area are good, but not universally applied). You will very often find it in structural calculations of deflection and strength. In these cases the material is thought to be under stress but not moving appreciably -- so "inertia" then seems like an odd word to be using. The term is used to describe how a material is distributed in space: so, for example, you can calculate the difference in beam deflection given the same overall material weight, but two different beam sizes. If your beams were tubes, you'd find (given two tubes of the same length and weight but different diameters and wall thicknesses) that the large diameter tube with thinner walls would be much stiffer than a smaller tube with thicker walls.

Used in the other sense, "inertia" seems more appropriate: in this case you are wondering about the moment (torque) required to accelerate a body in rotation. As you can guess, the further the mass is from the center of rotation, the more torque is required to accelerate that mass.

Here's another thread on the topic, and there have been several other threads here also, which you can find by searching with the "Search all of CR4" function.

Thanks

I understand but my questions are,

1. why mass moment of inertia is defined as "I=MR²",why it cant be defined as "I=RM²", or otherway.
2. Can u give me the derivation of this mathematical equation means how "R" is in squared why not single ,tripple or otherwayelse.

Hello

You seem to be at sea with the "mathematics" of the concept. The "R²" is the result of an integration in a function where the Mass/ Area is a 'function" of "R".

Ah Ha! Excellent question. I think about this sort of thing all the time (such as: "Why is it E=MC² rather than E=MC^2/3 ?) The answer is fairly simple, but the explanation is a little long, so I'll come back to this when I have a little more time -- probably late today or early tomorrow.

BTW, if you often have questions of this type, you should register. In some of these discussions, there can be may "guests", and it gets more than a little confusing -- with (seemingly) one guest taking different sides of an issue, etc.

Hi Ken,

"I think about this sort of thing all the time (such as: "Why is it E=MC² rather than E=MC^2/3 ?)"

You too? I thought I was maybe the only one around with such questions. I would be very interested in reading your thoughts on this (powers of C, etc. and why we do that).

How does C² fit in to E's equation? a²+b²=c² is elementary but C² in E=MC²??? What relationship does "²" contribute to the equation?

-John

This is actually related to Newton's second law of motion...As in case of linear motion we have F = ma where as in angular motion we have τ=I.α.....

if we convert that angular motion into linear motion we have

τ = F . R

α = α_{t ^{/ R.}}

_{^{F = M .}}α

NOW SOLVE THIS EQUATION FOR I IN τ=I.α ... you will find your answer

First, mass moment of inertia is a mathematical contrivance to enable people to calculate things like how a flywheel will accelerate with a given torque. Flywheels can have almost all their mass in the rim, or evenly distributed, or mostly near the center. Moment of inertia quantifies that distribution of mass, and applies only to a particular axis of rotation: a broom handle spins much differently when spun across it length vs in line with its length.

You could make a wheel out of a bunch of sledge hammers, with the heads forming the rim, and the handles forming spokes. If we apply a torque to the hub, the wheel will experience an angular acceleration. At any instant we could think of the acceleration of any given hammer head as being in a straight line (a tangent). So, the force applied at the hammer head (and therefore its acceleration) for a given torque at the hub, will depend on the handle length (radius): If you double the handle length you'd need twice the torque for a given tangential force on the hammer head. So… if you wanted to calculate the torque required to accelerate the wheel, radius would have to show up as a factor at least once.

But, when you double the radius you also double the distance each hammer head travels tangentially per degree of rotation (and therefore double its speed... and acceleration). So the radius comes into play twice (R²): once to double the torque required for a given accelerative force at the hammer head, and once to double the acceleration itself (which then means that torque must be doubled again to supply that force, per F=MA.)

I understood little ur example

please give me a descriptive of this example.

You might look at Mr Nalawade's explanation of the math.

Considering a flywheel: A change in radius does 2 things: it changes the force at the rim, and it changes the speed at the rim (which changes the acceleration at the rim). So, it shows up as a factor twice, thus r^2.

If your question is more along the lines of "Why is F=MA and not F=2MA?" then the answer has to do with the somewhat circular definitions of these values. To "weigh" an astronaut, you apply a force, and measure the acceleration, and calculate the mass. To weigh someone on earth, you apply an acceleration (gravity), measure the force, and calculate the mass.

This is actually related to Newton's second law of motion...As in case of linear motion we have F = ma where as in angular motion we have τ=I.α.....

if we convert that angular motion into linear motion we have

τ = F . R

α = α_{t ^{/ R.}}

_{^{F = M .}}α

NOW SOLVE THIS EQUATION FOR I IN τ=I.α ... you will find your answer

You cannot understand the meaning of I without considering a practical example

As an example if you talk about kinetic energy of a rotating body

KE = 1/2 m v² for a particle of the body mass

but v = ω x r (where ω is the rotational speed of the body)

thus, KE = 1/2 m r².ω² for an element of the body

The total KE of the body will be = 1/2 ω² .( ∑ m r² )

The sum in the above equation is called The mass moment of inertia I ( second mass moment around the rotating axis )

I depends on how far the mass is distributed around the axis ( shape of the body ! )

Go to Google Books and look up Mechanics by James Ellsworth Boyd. It has the whole story and you can download it.