What is Moment inertia: Definition and 16 Discussions

The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation.
It is an extensive (additive) property: for a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis.
For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3 × 3 matrix, with a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.

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  1. Chimaera987

    Inertia of the drivetrain components of a 4x4 vehicle

    Hi, I’m making a vehicle simulator and I’m not entirely sure if my inertia calculations are right or are completely wrong. What I currently have now is something like this: 1. Engine: engine_Out_Inertia = engine_Inertia; this should contain the inertia of the engine and all of it’s auxiliaries...
  2. curiousPep

    I Resolve moment of inertia at an angle

    Initially, I calculate the moment of inertia of of a square lamina (x-z plane). Thr this square is rotated an angle $\theta$ about a vertex and I need to calculate the new moment of inertia about that vertex. Can I split the rotated square to two squares in the x-z plane and y-z plane to find...
  3. Anak Soleh

    Does friction torque in speeding up = speeding down?

    On speeding up: τload - τfriction = Iαup On speeding down: τfriction = Iαdown If i substitute τfriction from speeding down to speeding up equation, i get moment of inertia: I = (τload)/(αup+αdown) But, is this allowed? Does friction torque in speeding up is equal to friction torque in...
  4. C

    A basic question about rotation

    Problem Statement: Let there be a ring of mass m and radius r. Let 3 masses be attached to the ring and named as O,P and Q. Mass of O and Q is 2m and mass of p is M. The angle between 2 masses is 15 degrees as shown in the figure. Find the maximum velocity the ring must roll so that it doesn't...
  5. D

    Experimental determination of the moment inertia of a sphere

    Hello, I was recently given the task to find experimentally the moment inertia of a sphere. I thought of rolling the sphere down an inclined plane and applying conservation of energy to the sphere. The equations i came up with are: mgh = 1/2mv2 + 1/2Iω2 solving for v^2 we come up with the...
  6. Mind----Blown

    General approach to find principal axes of rotation?

    Suppose i have an equilateral triangle and i want to find the principal axes of rotation passing through one of the vertex. How can i do that? I am thinking along the following lines but I'm not too sure: 1)Since the equilateral triangle has symmetry about a median, that definitely is one...
  7. Jewelz

    Moment of Inertia of a Spinning Record with a Hole in Center

    Homework Statement A 45 RPM record is a disk with a wide hole in the center. The mass of such record is 45g. It is 7 inches in diameter, and the hole in the center has a 1.5 inch diameter. (Sorry for the odd units, this was the way it was given). Find the moment of inertia of the 45 RPM...
  8. Kumaken

    Application of integrals to find moment Inertia

    I need to make a project that integrates physics with math, involving the use of integrals to find moment inertia of areas. The theory could be read here :http://www.intmath.com/applications-integration/6-moments-inertia.php I need to make an object that applies the theory above. Can anybody...
  9. S

    Moment inertia - theoretical problem

    Moment inertia -- theoretical problem Homework Statement Show that the moment inertia of a hollow cylinder with inner radius a and outer radius b is (1/2)*M*(a^2+b^2), calculated for the center axis. Homework Equations I know that the moment inertia of a non-hollow cylinder is I =...
  10. F

    Moment Inertia w/ density F(x)

    Homework Statement Calculate the moment of inertia of a uniform rigid rod of length L and mass M lying along the x-axis which rotates about an axis perpendicular to the rod (the y axis) and passing through it’s center of mass. The rod has a line density that is a function of location such...
  11. P

    Mass moment inertia of object displaced from axiz

    Hey y'all, A plea for general advice here as I embark on a project. Say I have an object rotating around an axis but *displaced* some distance "r" from that axis. It's length, a dimension perpendicular to the axis, is "a." I'm interested in finding the mass moment of inertia for the object...
  12. C

    Moment Inertia of a cube

    To which of the two cubes has a larger moment of inertia? I think it's the right one because We know that the minimal moment of inertia is throw the principal axes that goes throw the center of mass. in the right one , the rotation isn't throw the principal axes . there is also the following...
  13. R

    Find the Moment Inertia of a Cross Section: y = 30 & 35

    Hi.. I've got a problem about moment inertia. I don't understand about looking the distance of centroid (y). I am still confucing about the formula of "y" in Moment Inertia. Can you shows me about the formula of y or if any thread for this before, I am glad to see it too.Thanx's for your help...
  14. Q

    Derivation of moment inertia formula

    How do I derive the formula 1/12 Ml^2? Derive the formula for moment of inertia of a uniform thin rod of length l about an axis through its center perpendicular to the rod.
  15. E

    So, what's the problem?

    Hi Ho! I know that many books show the way to derive the moment inertia of a solid and a hollow sphere in many ways, each according to the lines of reasoning of their authors. I also have my own line of reasoning that I have successfully applied in finding the moment inertia of a solid...
  16. S

    Moment Inertia Tensor: Difference Explained

    what is the difference between moment of inertia and moment inertia tensor?
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