What are moment of inertia, mass moment of inertia, and radius of gyration?

In summary, the moment of inertia is a measure of an object's resistance to bending or torsion. It is calculated by integrating over the cross-sectional area and is used in equations for bending stress, vibrations, kinetic energy, and Newton's laws. The mass moment of inertia is similar to mass but for rotation and is calculated by integrating over the mass distribution. The radius of gyration is the distance from an axis at which the mass or area of the body would have to be concentrated to maintain its moment of inertia.
  • #1
ShawnD
Science Advisor
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I'm learning how to find these things in school but I have no idea what they are. Moment of inertia's units are distance^4 such as in^4 or mm^4. Mass moment of inertia has units of mass*distance^2 such as kgmm^2. Radius of gyration is a distance such as mm or in.

So what are they exactly and how are they used.
 
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  • #2
The area moment of inertia [L4] (aka second moment of area) is a measure of the resistance to bending. You'll see it when considering things like the bending stress (due to a bending moment, M) in a cross section ([tex]\sigma = \frac{Mc}{I}[/tex]) or vibrations in continuous systems. To compute the area moment of inertia about, say, the x-axis, you would compute the following:

[tex] I = \int y^2 dA [/tex] (where dA = dxdy)

There is also an analagous polar moment of area that indicates a cross section's resistance to torsion.

Mass moment of inertia [ML2] is the rotational analog of mass. (Some people use J rather than I here to distinguish the to types of moments of inertia, but J is also used for the polar area moment of inertia. As long as you're aware of the context, you shouldn't end up confusing them.) You'll see this moment of inertia in the calculation of kinetic energy or in Newton's laws, for example. The calculation for mass moment of inertia about an axis is [tex] I = \int \rho^2 dm [/tex] where ρ is the distance from the axis.

There is a radius of gyration [L] for mass and area (they are similar concepts). For the mass radius of gyration, it is the number k such that I = m*k2, where I is the mass moment of inertia and m is the mass of the body. It is the distance from a given axis at which the mass of the body would have to be concentrated so that its moment of inertia would remain unchanged. Similarly, the area radius of gyration is the number k such that I = A*k2.
 
  • #3


Moment of inertia, mass moment of inertia, and radius of gyration are all concepts related to the distribution of mass in a rotating object. They are important in understanding how an object will behave when subjected to rotational forces.

Moment of inertia, also known as rotational inertia, is a measure of an object's resistance to changes in its rotational motion. It is calculated by taking into account the mass of the object and the distance of each mass element from the axis of rotation. The units for moment of inertia are distance^4, such as in^4 or mm^4. A higher moment of inertia means that the object will be more difficult to rotate.

Mass moment of inertia, also known as second moment of area, is a measure of an object's resistance to changes in its rotational motion due to its mass distribution. It takes into account the mass of each element of the object and its distance from the axis of rotation. The units for mass moment of inertia are mass*distance^2, such as kgmm^2. It is similar to moment of inertia, but takes into account the object's mass as well.

Radius of gyration is the distance from the axis of rotation at which the entire mass of the object can be concentrated to produce the same moment of inertia as the object's actual mass distribution. It is a measure of how spread out the mass of an object is from its axis of rotation. The units for radius of gyration are distance, such as mm or in.

These concepts are used in various fields such as engineering, physics, and mechanics to analyze the behavior of rotating objects. They can also be used to design and optimize structures and machines that involve rotational motion.
 

1. What is moment of inertia?

Moment of inertia is a property of an object that measures its resistance to changes in rotational motion. It is calculated by taking the sum of all the individual masses in an object and their respective distances from the axis of rotation.

2. How is moment of inertia different from mass moment of inertia?

Moment of inertia and mass moment of inertia are often used interchangeably, but they are slightly different. Mass moment of inertia specifically refers to the moment of inertia of a solid object, while moment of inertia can also refer to the moment of inertia of a fluid or a system of particles.

3. What is the equation for calculating moment of inertia?

The equation for moment of inertia is I = Σmiri2, where mi is the mass of each individual component and ri is its distance from the axis of rotation.

4. How is the radius of gyration related to moment of inertia?

The radius of gyration is defined as the distance from the axis of rotation to a point where the entire mass of an object can be concentrated to produce the same moment of inertia as the object itself. It is mathematically related to moment of inertia as I = mk2, where m is the mass of the object and k is the radius of gyration.

5. Why is moment of inertia an important concept in physics?

Moment of inertia is an important concept in physics because it helps us understand how objects behave when they are rotating. It is used in many applications, such as calculating the stability of structures, predicting the behavior of spinning objects, and designing mechanical systems. It is also a key factor in the conservation of angular momentum, an important principle in physics.

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