What is the difference between mass moment of inertia and area moment of inertia and why mass moment of inertia alone used for kinetic rotation? Is there a possibility to use them interchangeably?
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$\begingroup$ Why not expand your question with 2 examples of use. $\endgroup$– Solar MikeFeb 28, 2022 at 9:43
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2 Answers
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The two terms are very different in practical applications. Confusingly very often the term moment of inertia is used for both of them, and also the same symbol is commonly used ($I$).
| Mass moment of inertia | second moment of area | |
|---|---|---|
| Equation | $\int y^2 dm$ | $\int y^2 dA$ |
| units in SI | $ kg\cdot m^2$ | $m^4$ |
| shows | Distribution of mass | distribution of surface area |
| Usually encountered | dynamics problems | strength of materials or mechanics of materials |
Both of them show the distribution of a quantity (mass or area) about a plane. However, IMHO, the similarities end there.
- The mass moment of area is relevant in dynamics problems e.g. $\sum M = I\cdot \alpha$
- the second moment of area is is relevant in strength of materials problems. E.g. when calculating the stresses of the bending of beam $\sigma= \frac{M}{I} \frac{y}{2}$
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Mass moment of inertia has to do with the distribution of mass on a system from its neutral axis. the more spread this distribution the more torque is needed to turn this system about its neutral axis.
To get an idea let's say you have a thin sheet of steel 6 inches wide by 12-inches length, and call the axis going through the width X and the axis gong through the length Y and set the origin at its center.
The mass moment of inertia of this rectangular about X-axis is
$$I_X=\int y^2 dm = \rho_{steel} 6*12^3/12$$
And the mass moment about the Y-axis is,
$$I_Y=\int x^2 dm = \rho_{steel} 12*6^3/12$$
So we see
$$I_X=4* I_Y$$
And if the sheet of steel is not the same density across, the integral has to count for that.
The mass moment of inertia is related to angular acceleration and angular momentum. It is the counterpart of mass in linear dynamics, $F=ma$, but in rotational dynamics:
$$\tau=I\alpha$$
The second moment of the area looks similar to the mass moment of inertia but is totally different. It is a measure of how the section area is dispersed about the neutral axis that we are interested in.
$$I_X=\int y^2 dA $$
It is related to the stiffness of a beam and its strength for resisting bending moments.
And they can not be used interchangeably.
