What is Moment of inertia: Definition and 1000 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. M

    Derivation for Moment of Inertia of Rectangle rotated through center

    Homework Statement I need to know how to derive the equation for the moment of inertia of a rectangle rotated about an axis through its center. The rectangle has sides a and b. I know the equation to be (1/12)M(a2+b2), but I am having trouble deriving it. I have searched all over the...
  2. J

    How to Calculate Moment of Inertia for a Disk Using Calculus?

    I know that the moment of inertia of a disk about the axis perpendicular to the plane of the disk is I=1/2 MR^2 and by the perpendicular axis theorem the moment of inertia about the other axis is I=1/4 MR^2 I want to get the 2nd result with calculus. I can't get the right answer...
  3. R

    Moment of Inertia of a molecule of collinear atoms

    Homework Statement Landau&Lifshitz Vol. Mechanics, p101 Q1Find the moment of inertia of a molecule of collinear atoms Homework Equations The Attempt at a Solution I defined the origin alone the orientation of the molecule. I_3=0 obviously. For I_2 I wrote I_2=Ʃm_b[x_b-\frac{Ʃm_a x_a}{μ}]^2...
  4. C

    Dynamics question regarding moment of inertia

    Hi, I have a question regarding a dynamics question that be found here:http://by.genie.uottawa.ca/~mess/files/MCG_2107/MCG2107_M2005_p2.pdf, It is the first problem on that link (labeled number 2.) I did ƩFx, ƩFy, ƩMG which allowed me to find angular acceleration, frictional force, and...
  5. P

    Moment of Inertia of a Rectangle

    I have been trying to do the moment of inertia of a rectangle and I have it figured out when we have the center of the rectangle as the center of the rotation. The equation is ∫∫ρ(x^2+y^2)dy dx where the first integral is from -b/2 to b/2 (if b is the height) and the second integral is -a/2 to...
  6. L

    Finding the moment of inertia of a neutron star

    Homework Statement The Crab Nebula is a cloud of glowing gas about 10 light-years across, located about 6500 light years from the Earth (the figure (Figure 1) ). It is the remnant of a star that underwent a supernova explosion, seen on Earth in 1054 a.d. Energy is released by the Crab Nebula...
  7. G

    Expressing moment of inertia in terms of m,a,r and g

    Homework Statement Find an expression for the moment of inertia (I) of the platform in terms of acceleration of the mass (a), the value of the hanging mass (m), the radius of the spindle (r) and the constant (g). Diagram: http://i.imgur.com/rv3zFYG.jpg Homework Equations F=ma t=Iα...
  8. J

    Moment of Inertia of Curves and Surfaces

    Greetings! I enjoyed the definition of moment of inertia for a volume and for an area in the form of matrix. It's very enlightening! I = \int \begin{bmatrix} y^2+z^2 & -xy & -xz\\ -yx & x^2+z^2 & -yz\\ -zx & -zy & x^2+y^2 \end{bmatrix}dxdydz '->...
  9. S

    Torque and Moment of Inertia of a black ring

    Homework Statement A black ring is placed concentrically on a turntable and is being pulled by a constant force of 12.5 N and the force is applied at a distance of .01m from the spindle/pulley. It's known that the ring accounts for about 8.50% of the total moment of inertia. Find moment of...
  10. C

    Why Does the Calculation of Moment of Inertia Differ in This Problem?

    Hi friends the problem is as: https://fbcdn-sphotos-d-a.akamaihd.net/hphotos-ak-ash4/1379445_1440214886205617_1276077735_n.jpg While finding moment of inertia of the system about the point from where the rod is fixed, the book says that M.I. of the system will be, I = Irod + Idisc I =...
  11. K

    What Is the Moment of Inertia About a Point One-Fourth the Length from One End?

    Homework Statement Small blocks, each with mass m, are clamped at the ends and at the center of a rod of length L and negligible mass Compute the moment of inertia of the system about an axis perpendicular to the rod and passing through a point one-fourth of the length from one end...
  12. W

    Moment of Inertia Using Parallel Axis Theorem

    Homework Statement Q is the edge length of the cube. The moment of inertia of the cube about an axis passing through its center and the center of two opposing faces is (1/6)mQ^2 (PCM in the diagram). Use the parallel axis theorem to find the moment of inertia if the axis is along one face of...
  13. D

    Why is the inertia tensor calculated about a point instead of an axis?

    Generally, when we talk about moment of inertia, we talk about rotation and inherently, we talk about moment of inertia about an axis. But when we talk about inertia tensor, we calculate about a point. Is there a reason for this difference? Am I missing something? I am new to tensors.
  14. M

    What is the Polar Moment of Inertia and How is it Calculated?

    I know some of you may have given up on me understanding moment of inertia/second moment of area, but here is another problem. I am using the this table to get the equations for the polar moment of area of a hemicircle around the x-axis, then I am applying the parallel axis theorem to find the...
  15. M

    Moment of inertia with T shape

    ]1. Homework Statement [/b] Homework Equations The Attempt at a Solution On paper it looks like not much of an attempt has been made, but I have tried a few different ways to find I_y. I thought to take the entire rectangle and subtract the parts on the upper left and right...
  16. C

    General: Calculating the Moment of Inertia

    Hey, sorry for not using the exact template; I just have a general question about how to calculate the moment of inertia that I will have to apply to a number of instances. In this particular case, I need to calculate the moment of inertia for a rod pendulum. Of course, I could just use...
  17. G

    Moment of inertia of a point mass

    Hello, This is a very basic question. Wikipedia says moment of inertia I of a point mass m is calculated as: I = m * r^2 Where, r is distance from the axis of rotation. What this actually mean? I mean what is an r?
  18. G

    How to find the moment of inertia of a cube?

    Hello, If I have a rigid cube rotated about an axis with dimensions L x W x H centimeters, how can we find the moment of inertia? Thank you.
  19. I

    Increase moment of inertia, slipping down an incline

    Hi, I'm trying to find out what effect an increase in moment of inertia of a cylinder would have on slipping when the cylinder rolls down an incline. Starting with: ma=mgsin(theta)-f, where f is the friction force, which is less than or equal to µmgcos(theta), normal force times µ. We already...
  20. PsychonautQQ

    Moment of Inertia of mud thrown on door

    Homework Statement A solid wood door 1m wide and 2m high is hinged along one side and has a total mass of 40kg. Initially open and at rest, the door is struck at its center by a handful of sticky mud with mass .5kg. What is the original and final moment of inertia for the door? What is the...
  21. O

    The moment of inertia of a rectangular plate about its diagonal

    Homework Statement Find the moment of inertia of a rectangular plate of mass m, sides 2h and 2k , rotates about its diagonal. Homework Equations I= Ʃmr^2 The Attempt at a Solution I=Ʃ m(x^2 +y^2) , let 2h parallel to x-axis, 2k parallel to y-axis. I= m(x cosθ)^2 + m (y cos ∅)^2 where...
  22. J

    Moment of inertia by connecting two identical rods

    Homework Statement Four T-shaped objects are constructed by connecting two identical rods of equal mass and equal length. Rank in order, from LARGEST to SMALLEST, the moments of inertia for rotation about the axis indicated by the dashed line. Homework Equations The Attempt at a...
  23. S

    Finding Moment of Inertia Using Integration

    Homework Statement A cylinder with radius R and mass M has density that increases linearly with distance r from the cylinder axis, ρ = αr, where α is a positive constant. Calculate the moment of inertia of the cylinder about a longitudinal axis through its center in terms of M and R. This is...
  24. S

    How does the moment of inertia scale?

    Homework Statement If we multiply all of the design dimensions by a scaling factor f, it's volume and mass will be multiplied by f^3. (a) By what factor will its moment of inertia be multiplied? (b) And if a 1/48 scale model has a rotational kinetic energy of 2.5J, what will be the kinetic...
  25. N

    Calculating Moment of Inertia (2D rectangles)

    Hi everyone. I'm presently improving a 2D physics engine i wrote, with angular dynamics. Moment of inertia is becoming a big sticking point for me. However, given that my applications are fairly simple, I'm hoping someone can help me come up with a simple way of calculating the moment of...
  26. D

    Calculating Moment of Inertia for Square Particle System

    Homework Statement There are four masses (particles) with a mass of 30gr each and they are at the vertices of a square that has each side with a length of 90cm. What is the moment of inertia of this system through a perpendicular axis that is at the center of the square?The Attempt at a...
  27. D

    Moment of inertia of a bar with two spheres

    Homework Statement A dumbbell has two spheres of radius 0.10m and mass 10kg (each). The two spheres are connected using a bar with length 1.0m and mass 12kg. What is the moment of inertia of the dumbbell about an axis perpendicular to an axis at the bar's center of mass? Moment of inertia...
  28. P

    How can I accurately calculate the moment of inertia for random 3D shapes?

    Hello, I am trying to find moment of inertia of various random 3D shapes and I have few general questions. I take very thin 'plates' of the shape, calculate moment of inertia of that area, multiply it by thickness of one 'plate unit', also I add a square distance it is from the origin...
  29. M

    Is this a correct derivation of Moment of Inertia?

    Hello everybody! This is the derivation (for a single particle). \tau = F_{\perp }r \ = ma_{\perp}r \ = \alpha mr^2 \\ \text{if }\ \tau = I\alpha \text{ where } I \text{ is resistance to accleration then } \\ I = mr^2 I'm curious what the problem with this is because I haven't...
  30. P

    Calculating volume & moment of inertia

    Hi there, I am trying to calculate both volume and moment of inertia of shapes which surface coordinates are depicted by following equation: |xn| + |yn| + |zn| = Rn When n is 1 it is simple octahedron, when it is 2, it is sphere and then choosing any n, it becomes something else. I...
  31. K

    Minimizing Moment of Inertia, keeping Moment constant

    Minimizing the Moment of Inertia while keeping the Moment constant Hi there. I am dealing with a mathematical problem which seems to be much harder than I initially expected: Minimize the functional J(\Omega) = \frac{1}{\rho} I_{z} = \int \!\! \int \!\! \int_\Omega \left( x^{2} + y^{2}...
  32. K

    Area moment of inertia calculation without standard formula

    Hi, I read in a textbook that area Moment of Inertia (M.I.) of any shape would be Ʃ of (y2 χ A)1,2...n where, A represents the Area of strip no. 1, 2... n y is the distance between globlal centre of gravity (C.G.) of the shape to th local C.G. of strip no. 1, 2... n (pl. look at the image)...
  33. E

    Moment of inertia of a square lamina through a diagonal

    Homework Statement Square lamina (of side a) of uniform density. Find I about a diagonal. Homework Equations I = ∫ dm*l^2 The Attempt at a Solution So I drew a square and its diagonal and I imagine a differential mass drawn somewhere on the lamina. The distance squared to that...
  34. E

    Moment of Inertia through center of mass

    Homework Statement A rod is 10 ft long and has a density that goes from 4 to 24 lb/ft. a) Find the mass. b) Find the center of mass. c) Find the moment of inertia through the center of mass. Homework Equations I = Ml^2 I = I(cm) + Md^2 (parallel axis theorem) The Attempt at...
  35. A

    Rotational Dynamics / Moment of Inertia Question

    Homework Statement An oxygen molecule consists of two oxygen atoms whose total mass is 5.3x10^-26 kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is 1.9 x 10^-46 kg*m^2. From this data, estimate the effective distance between...
  36. D

    Finding the moment of inertia of a 2D polygon.

    Hi everyone, Is there a general method for finding out the moment of inertia of an irregular convex 2D polygon given the coordinates of its vertices? I have thought of one possible method: Split the polygon into multiple triangles and find the moment of inertia of each triangle around the...
  37. andyrk

    Why can't moment of Inertia be never greater than MR2

    Why can't moment of Inertia be never greater than MR2 for uniform bodies with simple geometrical shapes?
  38. I

    Moment of inertia of combined cylinders

    Hi, I've attached a word document with my problem since I've used the Mathtype program (sorry, I didn't quite know how to use the tools on this forum), hope you don't mind :)
  39. R

    Kinetic Energy and Moment of Inertia

    Homework Statement A hoop with mass m= 10 kg, radius r=0.6 m, and moment of inertia I=mr^2 is rolling without slipping with a linear velocity of 8.5 m/s. What is the total kinetic energy (in J). Homework Equations K=1/2 m v^2 Kr=1/2 I ω^2 The Attempt at a Solution The kinetic...
  40. liometopum

    Example of moment of inertia constant K, greater than one?

    Regarding the moment of inertia constant, k, as in I = kmr^2, k is often a fraction less than one, and sometimes is equal to one. http://en.wikipedia.org/wiki/List_of_moments_of_inertia Is there an example of k being greater than one?
  41. CrimsonFlash

    Moment of Inertia of a regular hexagonal plate?

    Homework Statement How do you calculate the moment of inertia of a regular hexagonal plate of side a and mass M along an axis passing through its opposite vertices?Homework Equations Moment of inertia for a right triangle with an axis running along its base would be I = m h2 /6 where h and m...
  42. V

    Moment of inertia of an ellipse formula

    hello! I have to verify the formula of the moment of inertia of an ellipse about its' centroidal axis, is it M*(a^2 + b^2)/4.This is the one I got by myself. But in a webpage it was given as Pi*a*(b^3)/4. NOTE: don't ask for the proof of what I did.Its' a bit longer.I just want to know...
  43. R

    Moment of inertia tensor about y-axis of a cylinder.

    Homework Statement What must the ratio of height to radius of a cylinder be so that every axis is a principal axis (with the CM as the origin)? Homework Equations Moment of inertia tensor. I need I_yy = \sum m *(x^2 + z^2) The Attempt at a Solution I calculated I_zz = MR^2 /12...
  44. C

    Moment of inertia and work problem

    Homework Statement A 4.00-m length of light nylon cord is wound around a uniform cylindrical spool of radius 0.500 m and mass 1.00 kg. The spool is mounted on a frictionless axle and is initially at rest. The cord is pulled from the spool with a constant acceleration of magnitude 2.82 m/s2...
  45. N

    Find the Mass Moment of Inertia about the x,y,z axes

    Homework Statement The two small spheres of mass m each are connected by the light rigid rod which lies in the x-z plane. Determine the mass moments of inertia of the assembly about the x, y, z axes. I have attached an image of the question Homework Equations The Attempt at a...
  46. B

    Two free standing pillars holding beam, moment of inertia problems.

    Hi, I'm new here and didn't really know where to put this so decided to put it here as it is to do with my course work. I'm currently doing my Final year project on a lifter for lifting fertilizer bags its going okay and do know how to calculate moment of inertia but the problem is that I have...
  47. M

    How Do You Calculate Moment of Inertia for Different Axes?

    Homework Statement a)Find the moment of inertia about an axis that passes through mass A and is perpendicular to the page. In kg m^2 b)Find the moment of inertia about an axis that passes through masses B and C.In kg m^2 http://session.masteringphysics.com/problemAsset/1070524/4/12.EX16.jpg...
  48. E

    Moment of Inertia - Swinging arms

    My workings are below : Following the calculation of w I use v=wr where r= 3 and calculate the tangential velocity. It's then easy to see that a speed of 340 (Mach 1) will be attained in 128 s Is this correct ?
  49. S

    Calculating Moment of Inertia for a Tyre: Methods and Formulas

    how could i calculate the moment of inertia of a tyre if i needed the moment of inertia about the central access of the tyre could i add the moment of inertia of the tyre+ the moment of inertia of the rim and how do i calculate the moment of inertia of the tyre, i can't find any assosiated...
  50. S

    Moment of Inertia: Wheels Accelerated to 100 km/hr

    what is the moment of inertia about the wheels , when the vehicle is accelerated upto 100 km/hr and the foot is lifted off the accelerator
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