What is Moments of inertia: Definition and 83 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.

View More On Wikipedia.org
Recent contents Top users
  1. H

    Moments of Inertia of a Flat Body

    How do you take the moments of inertia of a flat body? I know howto take it if it's a 3d body. And the 2d case should be really simpe,but I'm too stupid to figure it out. Can you help me? For example.. Say we have a body that's a rectangle of mass m on |x| < a, |y| < b..? Thanks so much.
  2. A

    Find the principal moments of inertia of a flat rectangular plate

    Hello everyone; here is the problem that I’m currently working on: =============================================== a- Find the principal moments of inertia of a flat rectangular plate (mass = 30g, a = 80 mm, b = 60 mm) that rotates about a diagonal with velocity ω = 15 rad/s. b- What are...
  3. A

    Beam Bending and Moments of Inertia

    Homework Statement This is a two part question. In the first part you are asked to determine the mam moment that can be applied to the beam if it is bent about the z axis. They then ask you to redo the question bending about the y axis. This is probably a really simple question but what is the...
  4. L

    Moments of Inertia, Kinetic Energy and rotations

    Homework Statement A bicycle racer is going downhill at 11.0 m/s when, to his horror, one of his 2.25 kg wheels comes off when he is 75.0 m above the foot of the hill. We can model the wheel as a thin-walled cylinder 85.0 cm in diameter and neglect the small mass of the spokes. How fast is...
  5. M

    Relationship of Moments of Inertia

    Hey guys, I need some help. I have an old fortran program that needs axial moment of inertia as well as polar moment of inertia. I have the part in Unigraphics NX5.0, where I can get the mass, volume, radius of gyration, and moment of inertias about each axis. However, I cannot find where to...
  6. P

    Moments of inertia in image processing

    Hi, I'm currently working on an imaging problem that oddly requires some physics. Basically, I'm given a set of gray scale pixels and I have to determine whether they're a fiber or just random noise. My question is, how do I calculate the moment of inertia of the pixels while considering...
  7. R

    Moments of Inertia, Centres of Mass help

    Moments of Inertia, Centres of Mass... help please! Homework Statement A rod of length l rotates about an axis that runs perpendicularly to the rod through one end. The linear density p(x) of the rod in terms of the distance x from the axis is given by: p(x) = p0(1 + (x^2/l^2)) where...
  8. S

    Prob. 51 Physics: Moment of Inertia for 10 kg Rolling Cylinder

    Hi I have a few problems from 'Physics for Scientists and Engineers' by Serwat and Jewett, chap 10 Prob. 51 A cylinder of mass 10 kg rolls without slipping on a horizontal surface. At the instant its centre of mass has a speed of 10 m/s. Determine (a) the translational kinetic energy of...
  9. J

    Tension of String in Yo-Yo Motion

    Homework Statement A yo-yo consists of two uniform heavy discs, each of mass M and radius R, connected by a light axle of radius a around which one end of a string is wound. One end of the string is attached to the axle and the other to a fixed point P. The yo-yo is held with its centre of mass...
  10. A

    Moment of Inertia: 4 Spheres Connected by Rods in Square

    Homework Statement Four small spheres, each of which you can regard as a point of mass 0.200 , are arranged in a square 0.400 on a side and connected by light rods. The picture has 4 spheres connected by rods in the shape of a square. Theres a point O in the middle of the square and a...
  11. A

    Find the Moments of Inertia question

    Homework Statement The four masses shown in the figure below are connected by massless, rigid rods. http://i241.photobucket.com/albums/ff4/alg5045/p13-17.gif a.) Find the coordinates of the center of mass if Ma=100g and Mb=Mc=Md=230g. b.) Find the moment of inertia about an axis that...
  12. M

    Superposition of moments of Inertia

    Homework Statement Consider a thin rod of length L which is pivoted at one end. A uniform density spherical object (whose mass is m and radius is r = 1/6L) is attached to the free end of the rod. The moment of inertia of the rod about an end if I = 1/3 mL^2. The moment of inertia of the...
  13. M

    Moments of Inertia object problem

    Homework Statement Consider 3 objects of equal masses but different shapes: a solid disk (radius R), a thin ring (radius R), and a thin hollow square (side 2R). The ring and the square are hollow and their perimeters carry all the mass, but the disk is solid and has uniform mass density over...
  14. T

    Physics - Moments of Inertia, Angular Momentums etc HELP

    Homework Statement A large disk of mass M and radius R is spinning like a CD of merry-go-round - that is, about an axis perpendicular to the plane of the disk. It is rotating at an angular velocity (omega initial). At some instant, a sphere of mass M/4 , which is initially not rotating, is...
  15. T

    Moment of Inertia: Solving Point A from 3r & 1r Up

    So I have a point A on a graph with x and y axes. From the origin, the point is 3r to the right, and 1r up. I'm to find the Moment of Inertia along the x, y, and z axes. I've found the x and y moments, but not the z and I'm not really sure on how. I'm not getting any ideas from the graph...
  16. M

    What is the relationship between the moment of inertia and the length of a rod?

    The moment of inertia about an axis along the length of a rod is zero, correct?
  17. B

    Calculating Moment of Inertia of a Sphere

    Hi, I'm in the middle of programming an inertia system and am only really just starting to appreciate what the heck inertia is :) I have been taught inertia, but I haven't actually applied it in a real situation (all my exams and tutorials have resulted in formulae giving answers like...
  18. T

    Moments of Inertia: Finding Center Point of Semicircle

    can anyone tell me how to find moments of inertia for the centre point of a semicircle??
  19. A

    Deriving Common Moments of Inertia: Sphere I=\frac{2}{5}mr^{2}

    Could someone direct me to a site that explains how the common moments of inertia were arrived at? My physics professor put up on the board today that for a uniform sphere: I=\frac{2}{5}mr^{2}. He said it was just the anti-derivative of something, but he didn't want to go into it because...
  20. N

    Using primitives to integrate moments of Inertia

    using "primitives" to integrate moments of Inertia My classmates are lost and I just can't think the way my proffessor does when approaching these problems. He gave us about 50 shapes to find moments of inertia for this weekend and I'm not having any trouble doing them... my way. I can use a...
  21. M

    Question on moments of inertia

    I have a mechanics question that I can't seem to figure out. I've spent quite a bit of time on it but don't have much of an answer. If anyone can help I would appreciate it. Round objects are rolling without slipping down an inclined plane of height H above the horizontal. The box is...
  22. J

    Adding/subtracting moments of inertia

    Hi i was wondering, if the sphere is divided into 5 spherical shells, each shell has a different density and each shell is of equal thickness My question is: Can I calculate the moments of inertia for each shell by subtracting the moment of inertia of the sphere inside it, from the entire...
  23. J

    I beams and moments of inertia

    I recall that the shape of an i-beam is near optimal because of its moment of inertia. Does any have a reference that shows this, with explanation?
  24. Y

    How do I find moments of inertia for different shapes?

    Hi, can anyone help me understand how to find the moments of inertia for the following: 1- A triangular lamina (isosceles) of mass M, base 2B and height H. about line of symmetry. 2- A uniform lamina of mass M, bounded by the curve with equation y²=4ax and the line x = 4a about the x-axis...
  25. C

    Moments of Inertia for a right circular solid cone of mass

    Hi there, I was hoping that someone here could maybe give me a hand with a couple of issues I'm having to do with moments of inertia. For a right circular solid cone of mass m, height h and base radius a, we have to show that its moment of inertia about a line through its vertex and...
  26. W

    How Do You Calculate the Initial Acceleration of a Rod's Center of Mass?

    Pivot point on a Uniform Rod and Acceleration I'm stuck on this question, which seems like it should be fairly simple: A uniform rod of length 1.15 m is attached to a frictionless pivot at one end. It is released from rest from an angle theta = 21.0° above the horizontal. Find the magnitude...
  27. B

    Calculation of Moments of Inertia

    A uniform thin solid door has height 2.20 m, width .870 m, and mass 23.0 kg. Find its moment of inertia for rotation on its hinges. Is any piece of data unnecessary? So far, I don't understand how to calculate moments of inertia for things like this at all. I can do a system of particles...
  28. C

    The moment of inertia of the stool with respect to an axis at its center?

    Hi! I am working on this problem: A solid circular disk has a mass of 1.2 kg and a radius of 0.17 m. Each of three identical thin rods has a mass of 0.16 kg. The rods are attached perpendicularly to the plane of the disk at its outer edge to form a three-legged stool. Find the moment of...
  29. S

    Moments of Inertia, almost got it

    The problem is: Three identical thin rods, each of length L and mass m, are welded perpendicular to one another. The assembly is rotated about an axis that passes through the end of one rod and is parallel to another. Determine the moment of intertia of this structure. Ok what I know is...
  30. C

    Calculating Moments of Inertia for Arbitrary Sections

    Hi all, Why are there so many moments of inertia for a given section? That is, i come across moments about the origin, x-axis, y-axis, centroid, xy-axis...etc... What are the differences? are there typical applications? Please point me in the correct direction Thanks in advance, any...
  31. S

    Moments of Inertia: Practical Applications & Integration

    I am currently learning about how to calculate moments of inertia of various shapes. I can calculate them fine using the Parallel Axis Theorem, but I am having some difficulty trying to understand the overall concept. Can somebody give me some practical applications of MOEs? I am also...
  32. S

    Moments of Inertia: Calculating R & Summing Values

    In general, I=[inte]R2dm. First of all, what is R? R is the component of the position vector that is perpendicular to both the angular and linear velocity. Is this correct? EDIT: One more question. If we know the moments of interia of say two bodies, how do we find the moment of inertia...
  33. enigma

    Understanding Moments of Inertia & Tensors

    This may be a sort of odd question. I know what a moment of inertia is. I know what they represent, I know how to use them, I even know how to calculate them from scratch if need be. But I don't know why they are what they are. Why is it that this weird value with seemingly random d^2...
Back
Top