Moment of inertia Definition and 1000 Threads

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. S

    Moment of inertia: vector derivation

    We have the representation of torque attached. The components of r are (x,y,z). Where did the matrix come from and how did we get the stuff in the matrix? (Basically I understand all the steps except the step from the 3rd line to the 4th line.) Thank you very much!
  2. F

    Are These Equations for Moment of Inertia Correct?

    I'm working on a lab where I have to go into some detail about moment of inertia. I understand the concept and everything but am a little confused by the equation that I found on wikipedia. I've seen only two equations for this: momentum=torque divided by angular acceleration and one...
  3. Nono713

    Substituting the angular velocity and moment of inertia

    Hello, I need some help verifying whether what I did is correct. Homework Statement Attached is a screenshot of the problem. Homework Equations Angular momentum L = \vec{r} × m\vec{v} = mvr \sin{\theta} Conservation of angular momentum. The Attempt at a Solution This is my...
  4. R

    How to find the moment of inertia K from rolling an object down a ramp

    Homework Statement I am teaching a class where we are trying to find the K of I=1/2Kmr^2 from rolling a very thin cylinder down a ramp. Homework Equations Linear Forces: mgsinθ-F(friction)=ma angular forces: τ=Iα where τ=F(friction)r, I=Kmr^2 and a=αr distance=1/2*a*t^2 since the...
  5. R

    Hey im so confused about moment of inertia. ?

    I=mr^2 I know moment of inertia depends on how for a object how far a bit of mass of that object is from the axis of rotation. i.e. farther from the axis = higher moment of inertia-= more torque is needed to cause rotation. this just doesn't make any sense for the seesaw example, on the seesaw...
  6. A

    Moment of inertia for inclined plates

    What is the moment of inertia of a rectangular plate making an angle except right angle or multiple of right angle with it's axis which is passing through the centroid of the plate?
  7. R

    Moment Of Inertia Of A Semi-Circular Disk?

    Homework Statement http://dl.dropbox.com/u/630750/Screen%20Shot%202012-04-26%20at%2010.03.36.png Homework Equations The Attempt at a Solution http://dl.dropbox.com/u/630750/Photo%2026-04-2012%2010%2001%2034.jpg I know this is wrong but I can't really see why, can anybody...
  8. J

    Determine the mass moment of inertia of a quarter of an annulus

    Homework Statement B.1 The quarter ring has mass m and was cut from a thin uniform plate. Knowing that r1 = 1/2r2 determine the mass moment of inertia of the quarter ring with respect to A. axis AA' B. The centroidal axis CC' that is perpendicular to the plane of the quarter ring. CC' is...
  9. Z

    Moment of inertia of a squARE lamina

    a square lamina is made of 4 uniform thin rods each of which was a moment of inertia Ml^2 /12 about an axis perpendicular to their length and passing through their centres. My way of understanding it is the following : the two rods to whom the axis of rotation is perpendicular and passes...
  10. R

    New mass moment of inertia of an object after rotation

    Homework Statement I hope I can explain this properly: 1. Let's say you're given the mass moment of inertia of some non-uniform 3-dimensional object. 2. The moment is relative to coordinate system "CS1", with origin at the object's center of mass and about axes x, y and z. 3. Say...
  11. L

    Wheel pivoted about a frictionless axle - moment of inertia, force of axle

    Homework Statement A wheel has 4 spokes, see sketch ( http://i44.tinypic.com/213hp52.png ), and is pivoted about the indicated frictionless axle which is perpendicular to the page. The wheel is lifted such that θ = 90° (above the negative y axis) and released from rest. What is the force...
  12. L

    Moment of inertia of disk about off centre axis

    Homework Statement Find the moment of inertia of a uniform disk of mass m and radius r about an axis normal to the disk, through a point x from the centre. Homework Equations The Attempt at a Solution Let ρ be the density. I=ρ∫r^2 dA=ρ∫∫r^3 dr dθ in polar coordinates. However I...
  13. mishima

    Moment of Inertia in Equilateral Triangle

    The problem is to find the moment of inertia for a solid equilateral triangle about an axis at the triangle's vertex which is perpendicular to the plane of the triangle. Mass is given as M. I was wanting to treat it like 3 point masses at each vertex with 1/3 mass. Then I could use the sum...
  14. U

    Moment of Inertia and Frequency of Oscillation

    How do you calculate the moment of inertia given frequency of oscillation?
  15. T

    How Do You Calculate Basic Moment of Inertia?

    Homework Statement Question #4Homework Equations The formula for step 1 is supposed to be general equation for moment of inertia.The Attempt at a Solution as seen. The answer to the question is 5.870 My attempt: 1/3pi^3 * -cos(pi) still wrong.
  16. U

    What is the Moment of Inertia for a Non-Uniform Cane with Attached Device?

    I have a cane with non-uniform density and I'm trying to find its moment of inertia. There is no equation for its mass/density vs. length. If you need to picture it, it's a folding White Cane used by the blind. https://www.amazon.com/dp/B000VB0CG0/?tag=pfamazon01-20 I found the center of mass...
  17. D

    Moment of inertia tensor for two particles

    Hi, I need to compute the inverse of the moment of inertia (MOI) tensor of a bunch of point particles in a simulation algorithm. The number and location of the particles differs at each evaluation. In all cases, I'm taking the particle coordinates with respect to their center of mass...
  18. C

    Moment of Inertia of an Infinite Rod with Non-Uniform Density?

    Homework Statement A thin rod extends along the x-axis from x= +b to infinity. It has a non-uniform linear mass density of A/x^{4} where A is a constant with units of kg m3. Calculate the moment of inertia of the rod for rotation about the origin. Homework Equations The Attempt at a Solution...
  19. F

    Moment of inertia - displaced axis

    Hi, I need some help, I have a system with 3 cylinders (I got the center of mass for the system). Now I need to calculate the moment of inertia for the system. I have tried to find some examples or general information, but they all show when the axis either is centered within the object or...
  20. P

    Finding the moment of inertia for a specific figure (illustrated)

    Homework Statement http://img577.imageshack.us/img577/2069/inertia.gif The following figure with these known lengths are given. I need to find the moment of inertia.Homework Equations According to my book's facit, this is the solution...
  21. A

    What is the role of tensors in classical mechanics?

    When I took classical mechanics we were given definitions for an object's moment of inertia, which I understand to be a scalar quantity that describes that objects tendency to resist rotation about a fixed axis either about, or some distance from its center of mass. I was recently...
  22. B

    Finding moment of inertia of a pulley

    Homework Statement Two masses, one of m_{1}= 3kg, the other of m_{2}=4kg, hang from opposite sides of a pulley of radius .15m. When released from rest, the heavier mass falls .34m in 4s. What is the rotational inertia of the pulley? Homework Equations Newton's second law and \tau =...
  23. P

    Moment of Inertia of a cylinder with varying density

    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. I have I = ∫r^2dm and ρ...
  24. K

    Pulley, Ramp with Moment of Inertia

    Homework Statement http://imgur.com/ZkjXv The given answers are (b) .309 m/s^2 (c) T1= 7.67N, T2=9.22N Homework Equations I_disc = .5(mass_disc)r^2 Sum of Torque = I α a = r α The Attempt at a Solution We will define the system moving right and down as positive to get a positive...
  25. M

    Moment of Inertia in Cylindrical Coordinates

    Homework Statement What is the mass moment of inertia about the y-axis of the object shown if the density is 4 slugs/ft3? Homework Equations I_y = \int (x^2 + z^2) dm dm = \rho dV The Attempt at a Solution Not really too sure how to set this one up, my prof said it should be...
  26. E

    Calculating Mass Moment of Inertia for a Roller Door

    This is not a question from any textbook or assignment. It is a concept question in order to get a task done. I need to size a motor but just need to confirm my suspicion. I am sizing a motor needed to open/close a roller door that curls around a drum. Not one of those modern ones. I need to...
  27. R

    Rotation about CM - Can I use Moment of Inertia?

    Rotation about CM -- Can I use Moment of Inertia? Homework Statement I have a program I coded to simulate the movement of a some point masses connected to each other rigidly. Each mass is the same. I am trying to code in the correct equations for rotation, but I am having some difficulty with...
  28. B

    Angular velocity, acceleration. and moment of inertia

    Homework Statement A grinding wheel is a uniform cylinder with a radius of 8.70 cm and a mass of 0.400 kg. a)calculate the moment of inertia about the center. b)Calculate the applied torque needed to accelerate it from rest to 1950 rpm in 6.00 s if it is known to slow down from 1250 rpm to...
  29. O

    How Do You Calculate the Moment of Inertia for a Wheel with Accelerating Mass?

    Homework Statement So basically there is a wheel with a radius of 0.3 m. A light cord wrapped around the wheel supports a 0.75-kg object that accelerates at 3 m/s^-2 downwards. What is the moment of inertia of the wheel? (no friction) Homework Equations I=mr^2 T=I(alpha) (alpha)=ra...
  30. T

    Why Is a Minus Term Used in the Moment of Inertia Calculation for a T-Beam?

    Hi i tried to understand an old report about a beam, in the report maximum normal force due to bending is investigated , the normal procedure is applied nothing special actually they calculate the moment of inertia of a t-beam part by part and from m.c/I they solve the problem. I want to ask...
  31. B

    Moment of inertia - curved flat bar

    I am trying to calculate the moment of inertia of a formed (uniform curve) flat bar. I have attached a sketch . The bar is 5/8" thick x 1" wide. I'm stumped.
  32. A

    How Do You Calculate the Moment of Inertia Tensor for a Plate?

    Homework Statement Find the moment of inertia tensor of the plate attached below Homework Equations σ = area density The Attempt at a Solution So the main problem I'm having is solving for Ixx and Iyy: 1) Ixx = σ∫∫(y^2 + z^2)dydz, since there are no dz components, I don't see how...
  33. L

    Moment of inertia tensor of hollow cone

    Homework Statement Find the inertia tensor for a uniform, thin hollow cone, such as an ice cream cone, of mass M, height h, and base radius R, spinning about its pointed end. Homework Equations I_{zz} = \sum x^{2}+y^{2} \rho = \sqrt{x^{2}+y^{2}} The Attempt at a Solution I first...
  34. I

    Moment of Inertia of Bike Wheel

    Hi, I have conducted a few simple experiments involving a very ordinary, road going bike wheel. I came up with a value for the moment of inertia of 0.17kgm2 I really have no idea if I am in the right ballpark with that value, would anybody be able to enlighten me as to if that is a feasible...
  35. S

    To calculate the polar moment of inertia of a fan

    1.The motors driving the fans of a large cooling tower must be represented in a dynamic simulation of a power plant auxiliary system. Each fan can be described as follows: Fan diameter = 4.5 meter No of blades/fan = 6 Fan RPM = 90 The fan blades have a tapering cross section. The fan...
  36. Q

    Mass Moment of inertia of a cylinder with 4 holes?

    Homework Statement Hi I was reviewing for my final and I came across this problem: Problem: Basically there is a 6in long cylinder with dia = 24in Given weight density: 490 lb/ft^3 Each Hole is drilled symmetrically, each has 6in dia and equally space around a 10in dia circle concentric with...
  37. Q

    Mass Moment of inertia of a cylinder with 4 holes

    Hi I was reviewing for my final and I came across this problem: Problem: Basically there is a 6in long cylinder with dia = 24in Given weight density: 490 lb/ft^3 Each Hole is drilled symmetrically, each has 6in dia and equally space around a 10in dia circle concentric with the cylinder...
  38. Demon117

    Calculating the moment of inertia for this configuration

    Homework Statement I need to find the rotational kinetic energy of this system in the figure I have provided. Here the spring is rotating with the system at angular speed \omega. The two beads on the wire move vertically, and the third bead is held by two massless rods of equal length L...
  39. D

    What is the moment of inertia of a disc and rectangular plate system?

    q) A disc of mass 'm' and radius 'R' is attached to a rectangular plate of the same mass, breadth R and elngth 2R as shown in the figure. Find the moment of inertia of this system about the axis XX' passing through the centre of the disc and along the plane...
  40. T

    Moment of inertia for non homogeneous density.

    Ok so I am trying to figure how I would find the moment of inertia for a special case. I have a 55 gallon barrel that is almost half way full and I am suppose to spin it roughly 5-10 rpm. I know that to find the momement of interia of a hollow cylinder with thick walls is simply...
  41. teroenza

    Rings Vs Disks In Moment of Moment of Inertia Integral

    Homework Statement Not really homework, but putting it here to be safe. While doing a center of mass calculation of a cone centered about the origin (tip touching origin, z axis through it's center) my teacher put on the board that the volume element was dv= (pi*r^2)*dz using the area...
  42. M

    Find Moment of Inertia: Z-Axis

    Homework Statement Find the moment of inertia of the regoin bounded by z = x^2 + y^2 and z=1 if the density is propertial to the distance from the z axis Homework Equations Moment of interia about the z axis = the double integral R of (x^2+y^2)*(density)dV The Attempt at a Solution...
  43. C

    Comparing moment of inertia to mass in linear motion

    Homework Statement Compare the role played by the moment of inertia in rotational motion, to that played by mass in linear motion. Homework Equations I have no idea how to tackle this question. I tried looking up the defenition of moment of inertia as a start, but I only found anything...
  44. K

    Moment of Inertia and kinetic energy question

    Homework Statement The flywheel of a gasoline engine is required to give up 550 of kinetic energy while its angular speed decreases from 720 to 400 . What moment of inertia is required? Homework Equations K=(1/2)Iw^2 The Attempt at a Solution K=550J I'm not sure how to get the...
  45. T

    Oscillation and Moment of inertia

    Homework Statement A pendulum is constructed of a solid metal sphere of mass M = 4.00 kg, attached to a thin metal rod of mass m = 1.00 kg and length L = 40.0 cm. The pivot point for the pendulum is at the upper end of the thin rod. The pendulum oscillates through a small angle with a period...
  46. A

    Moment of inertia for a hollow ball calculation

    Homework Statement A uniform solid sphere of mass m and radius a has moment of inertia 2/5ma2 about any diameter. Material is removed from the sphere to make a concentric spherical cavity of radius a/2. What is the mass of the resulting hollow ball ? Show that its moment of inertia about a...
  47. K

    General Atwood system with moment of inertia

    Homework Statement I have taken this image from http://farside.ph.utexas.edu/teaching/336k/lectures/node79.html Hello, I am currently doing a project on Lagrangian mechanics. As part of my project, it was advised that I should redo each of the problems done using the Lagrangian method in...
  48. S

    Moment of inertia about different axes

    I found this problem in a book and am trying to answer it by myself. 1. Can you think of a body that has the same moment of inertia for all possible axes? If so, give an example, and if not, explain why this is not possible. Solution: I can't think of any example or disprove the...
  49. M

    Moment of Inertia Explained: Definition & Examples

    what is a moment of inertia? I know it's used for angular mechanics (like angular velocity and stuff like that), but what is a moment of inertia? A good definition with some examples would help tremendously. Thank you much in advance :)
  50. C

    Torque, angular acceleration, and moment of inertia

    Homework Statement We did this question today in class, but looking back at it, I'm kind of confused. Two identical dumbbells are formed by placing equal point masses at either end of two identical light (ie. massless) rods. The rods are pivoted so that dumbbell A rotates around the centre...
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