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. K

    Moment of inertia of the body Question

    Homework Statement A hole of radius R/2 is cut from a solid sphere of radius R. If the mass of the remaining plate is M, then moment of inertia of the body about an axis through the centre is____. Answer:57/140*M*R^2 Homework Equations Moment of inertia of solid sphere is...
  2. Z

    Moment of inertia of a music triangle

    Moment of inertia of a "music triangle" Homework Statement Find the moment of inertia about an axis thru the centre of mass, perpendicular to the plane of an equilateral triangle consisting of identical rods of length 'l' and mass 'm' connected at their ends. Homework Equations...
  3. O

    Clarify about the moment of inertia

    hey, I want to ask you something about to calculate the moment of inertia for any shape. My problem is that I can not calculate that moment of inertia values for any shape. Such as rectangle or stick and the others. I have tried to calculate for stick and I have found the solution of...
  4. A

    What is the Moment of Inertia of a Book Pivoting at One Corner?

    Homework Statement Suppose I have a book with dimensions 24 by 27 cm and its being pivoted at one of its corners so that it can swing along a plane parallel to itself. I'm supposed to find the period. But in order to do that I need to know its moment of inertia, would I have to write an...
  5. D

    Does the Moment of Inertia Change When a Gyroscope is Lifted?

    Hey I just finished an experiment in my physics lab where we examined the precession and nutation of a gyroscope. The gyroscope was built with a shaft which had a pivot in the middle, on one end of the shaft the large spinning disc was placed and on the left side of the shaft...
  6. A

    Moment of Inertia for Rotational Kinematics

    Homework Statement A thin uniform rod of mass (M) and length (L) is bent at its center so that the two segments are now perpendicular to each other. a. Find its moment of inertia about an axis perpendicular to its plane and passing through the point where the two segments meet...
  7. L

    How Do You Calculate Equivalent Moment of Inertia for Complex Systems?

    Hi Guys, I'm having trouble finding the equivalent moment of inertia for a system. Basically it's a mass attached to a string, which is attached to a shaft. As the mass drops, the string unravels imparting some rotation on the shaft, this shaft rotates a small flywheel, and also rotates a...
  8. T

    Where does the kinetic energy go when a man moves on a spinning roundabout?

    If you have a roundabout spinning with a man standing on it close to the centre, and then he walks out towards the edge of the roundabout, angular momentum is conserved, but kinetic energy is not (the roundabout rotates with a smaller angular speed). I'd like to know where the kinetic energy in...
  9. B

    The moment of inertia of a solid sphere

    Homework Statement A solid ball of mass M and radius is connected to a thin rod of mass m and length L as shown. What is the moment of inertia of this system about an axis perpendicular to the other end of the rod? Image: http://imageshack.us/photo/my-images/35/helpfy.jpg/ Homework...
  10. X

    Question involving moment of inertia, rotation about a horizontal axis.

    Homework Statement A uniform cylinder, of radius 2a and moment of inertia $2Ma^2$ is free to rotate about its horizontal axis. A light, inextenzible string is wound round the cylinder and a particle of mass m is suspended on its free end. If the system is released from rest, find the...
  11. S

    Moment of inertia of an object with uneven distribution of weight

    I'm doing an experiment and I have to calculate calculate rotational kinetic energy of this can rolling down an inclined plane. According to the equation, Er=1/2Iw^2, I=mr^2, and w=v/r (no slipping effect) right? but doesn't I=mr^2 apply only when the weight is evenly distributed throughout...
  12. K

    Moment of Inertia and Torque problem

    Homework Statement A wheel free to rotate about its axis that is not frictionless is initially at rest. A constant external torque of +44 N·m is applied to the wheel for 23 s, giving the wheel an angular velocity of +520 rev/min. The external torque is then removed, and the wheel comes to rest...
  13. C

    Derivation of the moment of inertia eqn for a thin spherical shell

    So I've been trying to derive the moment of inertia equation for a thin spherical shell and I've slammed into a dead end algebraically. I was able to derive an equation for a hollow sphere: I = (2/5) M (Ro^5 - Ri^5)/(Ro^3 - Ri^3) where Ro is the distance to the very outside of the sphere...
  14. L

    Find support developing a moment of inertia calculation software

    I got the algorithm to deal with every section profile. The software would calculate the area and the moment of inertia of any section or any combination of sections in different materials. Further more, I would develop it to deal with non-linear analysis finding the stress and strain on every...
  15. A

    Moment of Inertia Homework: Parts C & D

    Homework Statement part c and part d And the ans. a.)140 b.)130/21 c.)6.92M d.)150M/7 Homework Equations The Attempt at a Solution It seems the ans is wrong, I am not sure. Can you help me?
  16. Femme_physics

    Static moment VS Moment of inertia - what's the difference?

    Static moment VS Moment of inertia -- what's the difference? What's the difference between moment of inertia and static moment? How does it differ in calculations?
  17. Femme_physics

    Moment of inertia: Answer without calculations

    To this shape http://img10.imageshack.us/img10/2301/nocalculations.jpg Uploaded with ImageShack.us I'm asked to calculate the estimate of the static moment of inertia of the planar shape towards the x axis, that goes through its center of gravity. And then I'm told that I'm asked for...
  18. G

    Moment of inertia of a solid of rotation

    Homework Statement The first quadrant area bounded by the curve x^3+y^3=8 is rotated around y-axis to give a solid of rotation. The question asks for an integral which represents the solid's moment of inertia around the axis. My answer is: I_y=M\frac{\int_0^2x^2ydx}{\int_0^2x^2dx}...
  19. E

    Rotated Section - Moment of Inertia

    Hi everybody: I can see that there are formulas to calculate the moment of inertia of a 2D Area (Second moment of area) here: http://en.wikipedia.org/wiki/Second_moment_of_area" In the same link, there is a formula to calculate the Inertia moments about a ROTATED AXIS (Axis Rotation)...
  20. G

    Moment of Inertia for a machine

    Guys i need a little help in finding the moment of inertia of a part of a machine which looks like the part of a torus as the arc of a circle.Can somebody help me out with it please?
  21. J

    Moment of Inertia using Triple Integral

    Homework Statement Compute the moment of inertia around the z-axis of the solid unit box [0,1]x[0,1]x[0,1] with density given by \delta=x^{2}+y^{2}+z^{2}. Homework Equations I=\int\int\intr^{2} \delta dV The Attempt at a Solution I know that the distance r^{2} from the z-axis would...
  22. J

    How do you find the moment of inertia for wierdly shaped objects?

    i need to find the moment of inertia for a cardboard boomerang. what equation could i use?
  23. P

    Triple Integral moment of inertia

    set up a triple integral for the moment of inertia Iz for the region inside the sphere x^2+y^2+z^2=4a^2 and inside the cylinder x^2+y^2-2ax=0 so I draw my picture and convert to cylindrical coord. and i get an integral from 0 to sqrt(4a^2-r^2) an integral from 0 to 2acostheta and an...
  24. K

    Finding the moment of inertia of a uniform square lamina

    http://img20.imageshack.us/img20/9443/ssssnm.png parallel and perpendicular axis theorem for moment of inertias So i solved the Moment of inertia for the large square through the perpendicular axis through a, (1/3)*M*(l^2), where l is 4a/2=2a, using the perpendicular theorem, Ixx+Iyy=Izz, we...
  25. K

    Moment of inertia of a wire shaped into a semi circle

    Find the moment of inertia of a wire, AB, of mass M and length pi*a, which is bent into a semicircle, about AB. Mr^2/b] [b]The mass of the wire is M=pi*a*m, where m is the mass per unit length of the rod. Then a small element, ds is regarded, of the circumference of the semicircle as being...
  26. Femme_physics

    What shapes have the greatest/smallest moment of inertia

    I'm basically just trying to find a list of all the shapes and their relative moment of inertia to each other. I want to see what shapes have more and what less, just to get a more intuitive sense to this subject. Does anyone show such a list, or maybe can just tell me of basic geometrical...
  27. D

    Moment of inertia of wheel and frictional torque.

    Homework Statement A wheel free to rotate about its axis that is not frictionless is initially at rest. A constant external torque of +43 N·m is applied to the wheel for 20 s, giving the wheel an angular velocity of +610 rev/min. The external torque is then removed, and the wheel comes to...
  28. A

    Unraveling the Mystery of Rotational Motion and Moment of Inertia

    This a question that has been haunting me for some time now. Regarding the rotational motion of rigid bodies why wasn't the moment of inertia defined as the integral sum elements of infinitesimal mass time the radius from the axis of rotation rather than the radius squared. In this case the...
  29. S

    What is the Proper Method for Calculating Moment of Inertia for a Disk?

    Not a homework question per se, but I'm having some issues with moments of inertia. Say I wanted to calculate the I for a ring. What I would do is: I = \int r^2dm m = \lambda L dm = \lambda dL I_{ring} = \int_{0}^{L}\lambda r^2dL And that would give the requiside mr2. My question...
  30. K

    Problems with finding moment of inertia of triangle

    Hi guys I am trying to find the moment of inertia of a thin triangle sheet as shown in the attached file, I couldn't so I looked at the solution and it said that x1 = ay/h and x2 = by/h. That is the only part I don't get, it seems to be some sort of ratio but I can't work out how or why its like...
  31. L

    Finding the moment of inertia of a skeletal frame

    Homework Statement This isn't a homework problem but rather a problem I need to solve for a personal project of mine. Basically I need to find the moment of inertia of the following: http://img860.imageshack.us/img860/3952/roboframe.jpg Uploaded with ImageShack.us The rotation is with...
  32. P

    Proving moment of inertia of hollow sphere with y^2+x^2=R^2 method

    Could anyone prove the moment of inertia of a thin walled hollow sphere using the y^2 + x^2 = r^2. I have only studied up to single variable calculus. I can take regular integrals but not multivariable integrals. I don't want to use the angle method or any polar coordinate systems except in the...
  33. M

    Relation between moment of inertia and acceleration

    Homework Statement I am trying to find an equation relating the acceleration (not angular acceleration) to the moment of Inertia. I have a question that says a coin and a ring have same mass and same radius, which one would reach the bottom first if they were released from the top of an...
  34. P

    Proving the moment of inertia of a thinwalled hollow sphere

    I was wondering if there is any way to prove the moment of inertia of a thin walled hollow sphere by using y^2 + x^2 = r^2 instead of using the angle method. I want to use a Cartesian coordinate system not a polar coordinate system.
  35. S

    Radius of Gyration and moment of inertia

    Homework Statement Homework Equations i guess its r=sqrt(I/A) where A is the area of the circle thing and I is the moment of intertia. The Attempt at a Solution I guess I'm just having trouble getting I. A is 33pi
  36. Femme_physics

    Units for moment of inertia of a circle

    We've always been using mm^4, but I see in my answer book (answer written below) that one solution says R^4. Is it the same thing, just that R^4 is used for a circle? The measurement in my exercise are in mm. I did get the answer, I just wasn't aware I was supposed to write it in terms of R^4.
  37. U

    Moment of inertia of a spinning disc pendulum

    Homework Statement Find the moment of intertia of a pendulum, consisting of a disc free to spin attached to a rod that is hinged at one end. Homework Equations Moment of intertia of rod hinged at end = (1/3)Ml2 Moment of intertia of disc = (1/2)mR2 + ml2 The Attempt at a Solution...
  38. Femme_physics

    Statics, moment of inertia simple square-shape (no calculus)

    Statics, moment of inertia simple square-shape...need help (no calculus) Homework Statement Calculate the moment of inertia to the central axes Xc and Yc for the sketched cross-section. Xc = 33.9 mm Yc = 150 mm http://img822.imageshack.us/img822/1303/97823499.jpg Uploaded with...
  39. S

    Moment of inertia for rectangular plate

    Hi! I've got a problem with this: Count moment of inertia for rectangular plate a x b, if you know that moment of inertia of thin rod is \frac{1}{12}ml^2 . Do not use integrals!, others mathematical functions required (I can proof this moment by integrals, but this is not issue). I know that...
  40. T

    Equilibrium and Moment of Inertia Question

    Homework Statement See attachment or here http://i.imgur.com/zBDa0.png Homework Equations Torque = FR Moment of Inertia of Rod at End = I = 1/3 ML ^2 The Attempt at a Solution http://i.imgur.com/3Avqh.png
  41. Femme_physics

    Moment of Inertia, difference between Ixc, Ix, and Io?

    So I understand that Ix resistance to rotation around the X axis, Ixc is resistance to rotation around the center of gravity of the shape on its X axis, and Io I was told is also resistance to rotation around the object's center of gravity. So, I'm completely confused as to the difference...
  42. S

    How Do You Calculate the Moment of Inertia for a Disk Rotated Off-Center?

    Hi, I just got of a test that had a question about moment of inertia on it. The question "Calculate the moment of inertia of a thin uniformed disk that is being rotated about an axis of rotation". This axis is halfway between the center of the disk and the outer perimeter. The mass of the disk...
  43. I

    Moment of Inertia: Pendulum Problem

    1. 2. I = (x/x) M L ^ 2 3i honestly have no idea where to go with this. i do have an attempt so please don't laugh. I took the (L) for the moment for the rod = 3.4 m and divided it into 2 parts. R sub 1 and R sub 2. With R sub 2 on half of R sub 1. The equation i got using the R...
  44. E

    Last moment of inertia problem I have

    Homework Statement Four small spheres, each of which you can regard as a point of mass 0.200kg , are arranged in a square 0.400m on a side and connected by light rods . Find the moment of inertia of the system about an axis through the center of the square, perpendicular to its plane (an...
  45. E

    Another moment of inertia problem. help

    Homework Statement Find the moment of inertia Ix of particle a with respect to the x-axis (that is, if the x-axis is the axis of rotation), the moment of inertia Iy of particle a with respect to the y axis, and the moment of inertia Iz of particle a with respect to the z axis (the axis that...
  46. E

    Calculating Moment of Inertia for a System of Small Blocks on a Clamped Rod

    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...
  47. J

    Moment of Inertia of a solid uniform sphere

    Hello, Calculating the moment of inertia of a solid uniform sphere about it's center I get (3/5) Ma2. I know I am supposed to be getting (2/5) Ma2. I am using a differential volume dm as 4*pi*rho*r*r*dr, where rho is density, and r is the distance from the center to the differential volume...
  48. A

    Moment of Inertia for Three Point Masses

    Homework Statement Three particles of mass m are placed at A=(-a, -a), B=(a, -a) and C=(0, a) Find the moment of inertia for an axis along the z-axis through the origin Homework Equations I = m((rA)2 + (rB)2 + (rC)2) The Attempt at a Solution I calculate that: (rA)2 =...
  49. A

    Moment of Inertia of a drum with center axle

    I am confused with how to find moment of inertia of composite example. Like a hollow shell having one disc at each end and a shaft in the centre. How to find out Moment of Inertia if Force is applied tangentially on the shell and if force is applied tangentially on the shaft. Should I sum up...
  50. P

    Using centroid of object to find moment of inertia

    Hey, I was doing some problems involving finding the moment of inertia of a spun spandrel, and I came across the idea of using the centroid to find the moment. For example, if you have a parabola, find the centroid. If you're rotating around the x-axis (y=x^2), then find y_bar and multiply...
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