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.

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  1. D

    Addition of moments of inertia

    Hi To calculate the moment of inertia of a large sphere , I can calculate I for a smaller sphere and then add to it , I for the spherical shell that added to the smaller sphere makes the larger sphere. Does this general process apply to all shapes ? If so , does this process have a name ie. is...
  2. FluffCorgi

    Rotational Motion and moments of inertia

    I have the moment of inertia for the core(initial) and full body(final) but my answer for the moment of inertia for the arms(initial) was incorrect. Arms(initial) moment of inertia:(1/12)(6)(1.7^2)=1.445 this is incorrect for some reason Core(initial) moment of inertia: .9558 Full...
  3. F

    How to add area moments of inertia for varying cross sections

    Hello I am doing a shaft design layout for a spindle to minimize deflection and I have to calculate the area moment of inertia for a shaft overhang.The shaft overhang is the distance from a bearing to the end of the shaft. Here is a picture detailing the location of the overhang. The black...
  4. J

    Subtracting object MOI from assembly to get other parts MOI?

    Homework Statement It's been ages since I have done Moments of Inertia of complex objects! :( Can anyone help?? Say you have an assembly like an airplane and you want to just get the moment of inertia(MOI) of the tail section and you are given the Mass of whole plane,mass of tail section...
  5. J

    Three principal axis moments of inertia for hexagonal prism?

    Homework Statement I'm trying to figure out the mass moments of inertia for a hexagonal prism, with the z-axis being longitudinal. I'm trying to recall my calc 3 from 2 years ago and am failing miserably. I know the height of the prism is h. Each hexagonal side is length a. The prism has...
  6. lc99

    Objects with Different Moments of Inertia Rolling Down an Inclined Plane

    Homework Statement Homework Equations inertia equations The Attempt at a Solution I think the answer for this is B) or D) but I am not sure what the sentence "a special solid cylinder in which the density is proportional to the radius" means... The solid sphere has little inertia because...
  7. harpazo

    MHB Moments of Inertia and More....2

    Verify the given moment(s) of inertia and find x double bar and y double bar. Assume that the lamina has a density of p = 1, where p is rho. The diagram given is a circle with radius a. The entire circle is shaded. We are also given I_o = (pi•a^4)/2, where I_o is the moment of inertia about...
  8. harpazo

    MHB Moments of Inertia and More....1

    Verify the given moment(s) of inertia and find x double bar and y double bar. Assume that the given lamina has a density of p = 1, where p is rho. I_x = (bh^3)/3 I_y = (b^3h)/3 I found the mass to be bh. x double bar = sqrt{(bh^3)/3 ÷ bh} x double bar = sqrt{b^2/3} x double bar =...
  9. harpazo

    MHB Moments of Inertia & Radius of Gyration

    Find the moments of inertia about the x-axis, y-axis and the origin. Also, find the radius of gyration about the x-axis and y-axis. y = 0, y = b, x = 0, x = a Rho = ky 1. Is ky the density function? 2. Do I integrate over dxdy or dydx? 3. Are the limits of integration y = 0, y = b, x = 0...
  10. G

    Moments of Inertia: Hi, Need Help Understanding?

    Hi, I need some assistance understanding moments of inertia. I am doing some review for an upcoming exam, however I am slightly stumped by this question. I have already tried googling "moments of inertia" to try and understand the concept better, but I am having issues knowing when to apply...
  11. R

    Tension, Spring and Moments of Inertia

    Homework Statement A given pulley has a radius of 20 cm and moment of inertia 0.2 kgm2. The string going over it is attached to a vertical spring of spring constant 50 N/m on one end and a 1 kg mass on the other end. The system is released from rest with the spring at natural length. Find the...
  12. tarchuletta

    How to use Moments of Inertia to find Max Acceleration (sportbike pre-wheelie)

    So in this problem there's a lot of leeway as far as parameters go. The numbers I give for weights, dimensions, etc, may not be super realistic, but that's not too important to me because I can plug in realistic numbers later. I more of need help understanding the concept of moments of inertia...
  13. E

    Superposition of Moments of Inertia about Different CGs

    Hi all, You can use superposition to add moments of inertia when they're calculated about the same center of gravity (cg), but let's say you calculate the moments of inertia of several elements of a system about one cg and then use the Parallel Axis Theorem to then reference the total moments...
  14. B

    Understanding Moments of Inertia for Solving Homework Problems

    Homework Statement Attached in a picture! Homework Equations I = mr^2 m = (x*y) cm = (x/2, y/2) The Attempt at a Solution I was able to solve Part 1 fairly fast, but Part II has drawn a blank for me overall. I was thinking of that rotational thing from Calculus where you spin a 2D figure...
  15. C

    Where can I start learning about moments of inertia?

    Homework Statement My question doesn't follow the guidelines for asking for homework help because I'm looking for sources where I can learn about moments of inertia, not help for a specific homework problem. I'm going to try to figure out the moment of inertia of a propeller, but I don't know...
  16. CivilSigma

    Moments of Inertia by Integration

    Homework Statement I am having trouble understanding the formula for Moments Of Inertia by direct integration. Homework Equations I understand the following (which is the definition) : $$ I_x = \int y^2 dA $$ $$I_y = \int x^2 dA $$ However come to application on a problem, my book doesn't...
  17. K

    Moments of inertia of a rectangular plate

    Homework Statement Show that the moment of inertia of a rectangular plate round it's diagonal, line B-B is equal to the one round a line parallel to one of the edges and passing through the center, line C-C Homework Equations Moment of inertia: ##I=mr^2## The Attempt at a Solution For the...
  18. jdawg

    Moments of Inertia for Composite Area

    Homework Statement Determine the moment of inertia of the cross sectional area of the T beam with respect to the x' axis passing through the centroid of the cross section.(I'm attaching a diagram) Homework EquationsThe Attempt at a Solution Hi! I'm having a little trouble with this problem...
  19. S

    How Do You Calculate the Mass of a Rotating Square Plate?

    Homework Statement A square plate has a side length of 1.2 m. An applied torque of 300 N · m gives the plate an angular acceleration of 4.8 rad/s2 about an axis through the centre of the plate. Determine the plate’s mass. Homework Equations t = lα l = 1/12M(l^2 + w^2) The Attempt at a...
  20. S

    Moments of Inertia Explained: Helping Scott Understand Statics

    Hey there, In my statics class we're just finishing up moments of area or moments of inertia as they're being called. I can do the math and figure them out just fine but I have no clue what I'm actually finding! What is it? I am having trouble actually conceptualizing what it is that I'm...
  21. J

    What force is needed to lift a wheel over a bump with given mass and radius?

    Homework Statement The question is as follows: How much force should I apply if i want to have a wheel of mass 2kg and radius R to pass over a step with height h, where R is twice as big as h. I attached a picture of the problem for reference. Homework Equations I am not sure but I think I...
  22. S

    Moments of Inertia of two particles

    Homework Statement Two particles, each of mass m, are attached one to each end of a diameter PQ of a uniform circular disk, of mass 4m, radius a with its centre at O. The system is free to rotate about a horizontal axis through A, a point on PQ such that OA = b as indicated in the diagram...
  23. Aristotle

    Question relevant to Moments of Inertia

    Homework Statement Homework Equations I= ∑ miri^2 ? The Attempt at a Solution I'm not sure on how to start this problem. How exactly do I know which one is IA, IB, IC since it didnt specify? Can somebody please hint me in solving this problem?
  24. B

    Calculating Principle Moments of Inertia

    Hi Folks, Is it possible to calculate the principle moments of inertia acting along the principle axes of inertia given the moments of inertia and their directional vectors. Ie , I have the following information Moments of inertia ##J_1, J_2,J_3=18kgm2,15kgm2,6kgm2## and the directional...
  25. T

    Moments of inertia using parallel axis theorem

    Homework Statement The mass of a homogenous thin plate is 36kg. Determine it moment of inertia about the x axis Homework Equations Parallel axis theorem: Ix = Ix' + d^2*m The Attempt at a Solution p = m/a = 36/0.36 = 100 Because the density is homogenous, I found the mass of the first...
  26. K

    MHB Regions and moments of inertia

    Hi Guys, It has been a while since my last post but it's great to be back. I am having some trouble with part b) of this question. Don't fully understand the concept and what I'm meant to do. Any guidance or assistance would be greatly appreciated. Thanks in advance you legends!
  27. S

    Moments of Inertia of non-uniform rod

    Homework Statement a rod of mass M and length L is supported by a smooth horizontal floor and leans against a smooth vertical wall, the mass density increases linearly with p=kr where r is the distance from the wall and k is a positive constant. a.finf the moment of inertia of the rod with...
  28. F

    Inertia - Moments of Inertia of a rigid body (different axes)

    Inertia -- Moments of Inertia of a rigid body (different axes) Here is the problem http://imgur.com/pL6Bdgw So I missed class today because I was studying for a genetics test. I don't need the answer or anything but I was wondering what the general rule for inertia that I would use for...
  29. D

    MHB Calculating moments of inertia

    What formula do I need here?Starting from the sum $I = \sum\limits_{\alpha = 1}^Nm_{\alpha}\rho_{\alpha}^2$ and replacing it by the appropriate integral, find the moment of inertia of a uniform thin square of side $2b$, rotating about an axis perpendicular to the square and passing through the...
  30. S

    Solving the Stick of the Gods: Mass, Center of Mass & Moments of Inertia

    Homework Statement So the problem starts out like this Stick of the Gods! You hold one end of a stick, but it has no other end. It simply extends into infinity. Its one-dimensional density distribution is given by: λ=(λinitial)times(e^(-x/L)) λ is the density The problem doesn't state what...
  31. J

    Moments of Inertia for a Cylinder and Disk

    What are they? I have some poeple telling me that for a cylinder it's I = (1/4)MR2 + (1/12)ML2 and that for a disk it's the same. Other people have told me that it's I = (1/2)MR2 for a disk and that it's the same for a cylinder. In the past I've used the former and have gotten the right answers...
  32. C

    What is the Moment of Inertia of a Sphere with Different Densities and Radii?

    [b]1. Sphere inner core radius 3490km and density 13000kg/m3. Outer radius 6400km and density 4000kg/m3. Show that the moment of inertia is approximately 8 x 1037kg m2. [b]2. I know that the moment of inertia of a sphere is 2/5xmr2. [b]3. I have calculated the volume for each part...
  33. R

    How Is Moment of Inertia Calculated for Arbitrary Axes in Dynamics?

    Homework Statement Hi, I've got a Dynamics question on moments of inertia and I'm not really sure what the question is trying to get at. The question is let R=(O,B) be a rest frame of a rigid body and let the entries in J(R) be Jij. Show that the moment if inertia about an axis through O...
  34. L

    Moments of inertia by integration

    Homework Statement A mass distribution in the positive x region of the xy-plane and in the shape of a semicircle of radius a, centred on the origin, has mass per unit area k. Find, using plane polar coordinates, its moments of inertia about the x and y axes. Homework Equations The...
  35. J

    Finding Acceleration and Coefficient of Friction for Rolling Disc on Rough Plane

    A disc of mass 'm' rests on a rough horizontal plane. A light string that is wound onto the disc applies a horizontal force 'T' to the disc, which causes it to roll without slipping. a) Show that the acceleration from rest of the centre of the disc is = 4T/3m b) When T = mg, calculate...
  36. H

    Moments of inertia in flywheel

    I have managed to rearrange an equation give to me to make I (moment of inertia) the subject. I am now having problems working out the values for parts of it. I = moment of inertia m = mass g = gravity h = height v = translational speed of the mass falling Tf = work done against...
  37. U

    Product moments of inertia- what are they intuitively?

    Product moments of inertia-- what are they intuitively? Hi, I don't understand what the off-diagonal terms in the moment of inertia (tensor) matrix are intuitively...they are called the product moments of inertia...but if Ixx is the moment of inertia about the x axis, which axis is the Ixy...
  38. J

    Hamilton operator with moments of inertia : time - independence

    Homework Statement The Hamilton-operator is given as \hat{H} and describes the movement of a free rigid object that has the moments of inertia I_{i} Under what circumstances is <\Psi|\hat{L_{1}}|\Psi> time-independent? Homework Equations...
  39. fluidistic

    Classical mechanics, principal moments of inertia of a rigid body

    Homework Statement Determine the principal moments of inertia of a circular cylinder with radius R and height h. Homework Equations Not sure. The Attempt at a Solution This is the first problem of this kind I attempt to solve. From what I've read on wikipedia, the tensor of...
  40. L

    Area Moment of Inertia & Stability of Floating Bodies

    Just totally confused which value to use to get the area moments of inertia with this question The ship was initially looks like this the mass of the ship is 5000 tonnes two heavy cargocontainers each of mass 73,370 Kg each assume that the ship has a rectangular waterline area, with length...
  41. H

    Moments of Inertia Homework: Find Max Height H

    Homework Statement part a) A block of mass m slides up an incline with an initial speed v0. The incline is on an angle, &theta;, and block stops sliding at height H. -If the incline is frictionless, determine the max height H to which the block will rise, in terms of given quantities and...
  42. G

    Why Do Principal Moments of Inertia Values Change with Different Origins?

    Consider 4 equal masses at the 4 corners of a square of side b. First I took one of the corners as the origin and found the principal moments of inertia to be Ixx=mb^2, Iyy=3mb^2, Izz=4mb^2 after solving the secular equation. Again, I found the principal moments of inertia but now with respect...
  43. G

    Why do different origins result in different principal moments of inertia?

    Homework Statement When I try finding the principal moments of inertia with respect to different origins for any arbitrary configuration(assuming that the inertia tensor is diagonalized), I end up getting different values. Intuitively, this is quite acceptable because the mass distribution is...
  44. G

    What could be wrong with my approach to finding principal moments of inertia?

    Homework Statement I have a question on finding the principal moments of inertia of a discrete set of mass points on a plane. If I choose one of the mass points as the origin, I always end up having a diagonal matrix for the inertia tensor for any configuration. Isn't that weird? I sense...
  45. U

    Moments of Inertia Derivation,

    Homework Statement I have attached the problem in one file and I have attached my attempt in the second file. I only need help deriving the moment of inertia for the first (1) and fourth (4) objects but I have attached my solutions to the other objects in case it helps jog someones memory...
  46. S

    Inertia Tensors for Fractional Volumes: How to Calculate?

    Homework Statement How do you find the tensor of inertia for fractions of the usual volumes? (like half of a sphere, half of a cylinder, half of a disk, quarter of a disk, etc.) I already know the tensor of inertia for the whole volume. Take M as the fraction of the volume's mass, not the...
  47. P

    Calculate Max Angular Velocity of Spring: Moment of Inertia Homework

    Homework Statement I linear spring has one end attached to a ceiling, the other end hanging freely. It was noted that the spring was extended by 120mm when a mass of 2.5kg was attached at the bottom. If the moment of inertia of the mass is 10kg.m2 about its vertical axis, calculate the...
  48. S

    Principal moments of inertia ellipsoid

    Problem: Find the principal moments of inertia of an ellipsoid. I began to label moments of intertia about minor axises. But I'm not sure where to go after that.
  49. D

    Homework help on Moments of inertia of system

    Homework Statement Four masses are held in position at the corners of a rectangle by light rods.Find the moment of inertia of the system about the x axis.Find the moment of inertia of the system about the y axis.Find the moment of inertia of the system about an axis through O and...
  50. M

    Rolling hemispheres and moments of inertia

    Homework Statement Hi there - I'm working through a bit of mechanics revision and have been looking at some of my uni past exam papers from a good few years back when the syllabus was different, so I'm not familiar with a lot of the material covered - it's not strictly relevant to my current...