What is Inertia tensor: Definition and 64 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. L

    Inertia tensor around principal Axes part 2

    Hi, it's about the task e) Since the density is homogeneous, I have assumed the following for ##\rho=\frac{M}{V}##. I then started the proof of ##I_{23}##, the integral looks like this: $$ I_{23}=\int_{}^{} -\frac{M}{V}r'_2r'_3 d^3r$$ Now I apply the transformation $$ I_{23}=\int_{}^{}...
  2. L

    Inertia tensor around principal Axes

    Hi, unfortunately, I am not getting anywhere with the following task The inertia tensor is as follows $$\left( \begin{array}{rrr} I_{11} & I_{12} & I_{13} \\ I_{21} & I_{22} & I_{23} \\ I_{31} & I_{32} & I_{33} \\ \end{array}\right)$$ I had now thought that I could simply rotate the...
  3. Math Amateur

    I The Inertia Tensor .... Determining Components of Angular Momentum ....

    I am reading Tensor Calculus for Physics by Dwight E. Neuenschwander and am having difficulties in following his logic regarding proceeding to derive the components of Angular Momentum and from there the components of the Inertia Tensor ... On page 36 we read the following: In the above text...
  4. George Keeling

    I Time derivative of the moment of inertia tensor

    I am completely stuck on problem 2.45 of Blennow's book Mathematical Models for Physics and Engineering. @Orodruin It says "We just stated that the moment of inertia tensor ##I_{ij}## satisfies the relation$${\dot{I}}_{ij}\omega_j=\varepsilon_{ijk}\omega_jI_{kl}\omega_l$$Show that this relation...
  5. warhammer

    Question on Moment of Inertia Tensor of a Rotating Rigid Body

    Hi. So I was asked the following question whose picture is attached below along with my attempt at the solution. Now my doubt is, since the question refers to the whole system comprising of these thin rigid body 'mini systems', should the Principle moments of Inertia about the respective axes...
  6. Like Tony Stark

    Determine tensor of inertia of a rod

    I have to find the inertia tensor of these rods and I don't have the concept that clear... I mean, I know the formulas like: ##I_{xx}=\int y^2 + z^2 dm## ##I_{xy}=\int xy dm## But I don't know what ##x, y, z, dm## stand for. In other words, I don't know what I should replace in the formula...
  7. sams

    Questions Regarding the Inertia Tensor

    In Chapter 11: Dynamics of Rigid Bodies, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, pages 415-418, Section 11.3 - Inertia Tensor, I have three questions regarding the Inertia Tensor: 1.The authors made the following statement: "neither V nor ω...
  8. Raihan amin

    Determining the precession radius

    Homework Statement A stick of length 2L and of mass M spins with angular velocity ω around the axis perpendicular to the COM of the stick.It is rotating with making an angle Θ,with the axis parallel to Z axis.The COM has an initial velocity v=υ(to the y axis).Find it's precession radius R...
  9. N

    Inertia tensor of cone around its apex

    Im trying to calculate the principals moments of inertia (Ixx Iyy Izz) for the inertia tensor by triple integration using cylindrical coordinates in MATLAB. % Symbolic variables syms r z theta R h M; % R (Radius) h(height) M(Mass) % Ixx unox = int((z^2+(r*sin(theta))^2)*r,z,r,h); % First...
  10. kasoll

    Inertia tensor v.s pincipal axes moment of inertia

    Is there a method to calculate inertia tensor form principal axes moment of inertia? Like now we have moment of inertia: (Ix,Iy,Iz)=(20,18,25), and hot to calculate the inertia tensor like (Ixx,Ixy,Ixz Iyx,Iyy,Iyz, Izx,Izy,Izz)? I have read about this page several times, but still have no idea.
  11. JTC

    What is the Transformation Rule for the Moment of Inertia Tensor?

    (Forgive me if this is in the wrong spot) I understand how tensors transform. I can easily type a rule with the differentials of coordinates, say for strain. I also know that the moment of inertia is a tensor. But I cannot see how it transforms as does the standard rules of covariant...
  12. physicsdude101

    Inertia tensor for point masses

    Homework Statement Three equal point masses, mass M, are located at (a,0,0), (0, a, 2a) and (0, 2a, a). Find the centre of mass for this system. Use symmetry to determine the principle axes of the system and hence find the inertia tensor through the centre of mass. (based on Hand and Finch...
  13. S

    Inertia tensor of a body rotating about 3 axes

    Homework Statement Hello, I know about the inertia tensor about one axis, but how about a body that rotates around 3 axis x,y and z such as a spacecraft with changes in the attitude. Thanks for you help. Homework EquationsThe Attempt at a Solution
  14. T

    I Substituting a solid body with mass points

    I ran into the following problem, and stuck for a couple of days now. I have a solid body, rigid and and has uniform density. Its mass M, the location of the center of gravity x_M, y_M, z_M and its inertia matrix is known: Jx Jxy Jxz Jyx Jyy Jyz Jzx Jzy Jz I have to write an algorithm...
  15. BiGyElLoWhAt

    Moment of inertia tensor calculation and diagonalization

    Homework Statement Not sure if this is advanced, so move it wherever. A certain rigid body may be represented by three point masses: m_1 = 1 at (1,-1,-2) m_2 = 2 at (-1,1,0) m_3 = 1 at (1,1,-2) a) find the moment of inertia tensor b) diagonalize the matrix obtaining the eigenvalues and the...
  16. L

    Inertia tensor combination point mass-sphere

    Homework Statement hello, i want to calculate the inertia tensor of the combination of a point mass and a sphere in the object's frame, the center of mass is at the origin. The point mass remains at the surface of the phere The sphere is uniform, radius r and mass M, and the point mass has mass...
  17. Q

    Inertia Tensor.... Help please

    Hi! I am new at robotics, can you guys please help me what is the principal moment of inertia?? how to define the pose of axis about center of mass of my robotic link of a legged robot?? please guide me with some visual representation and also how to calculate Inertia Tensor?? I will really...
  18. B

    Relationship between inverse matrix and inertia tensor?

    Seems exist some relationship between the inverse of a matrix with the inertia tensor, looks: This relationship really exist?
  19. Q

    Rotational EOM's with non diagonal inertia tensor

    I'm having difficulties understanding how I should calculate the angular velocities of a rigid body when the inertia tensor is given in body coordinates and has off diagonal elements. Let's assume I have an inertia tensor ## I = \begin{bmatrix} I_{xx} & -I_{xy} & -I_{xz} \\ -I_{yx} &...
  20. D

    Eigenvectors of Inertia tensor

    Hi, I've written a little fortran code that computes the three Eigenvectors \vec{v}_1, \vec{v}_2, \vec{v}_3 of the inertia tensor of a N-Particle system. Now I observed something that I cannot explain analytically: Assume the position vector \vec{r}_i of each particle to be given with respect...
  21. R

    Impose Uniqueness on Diagonalization of Inertia Tensor?

    Given an inertia tensor of a rigid body I, one can always find a rotation that diagonalizes I as I = RT I0 R (let's say none of the value of the inertia in I0 equal each other, though). R is not unique, however, as one can always rotate 180 degrees about a principal axis, or rearrange the...
  22. L

    Inertia tensor punctual masses

    Homework Statement Hi everyone, I need some help to know how to find the components of the inertia tensor matrix of a rigid body formed by a gruop of point masses attached to bars with no mass. I have 3 masses with cartesian coordenates: 1 (a,a,0), 2 (a,0,0) and 3 (-a,-a-0). The...
  23. P

    Inertia tensor of a crosshanger

    I've done many exercises about inertia tensors of 3D bodies and sticks but now I have this exercise and I got stuck without any idea of how to do the integration to compute the inertia tensor. The statement is this: "Compute the inertia tensor of a cross-hanger consisting of 3 thin and linear...
  24. T

    Finding rate of change of moment of inertia tensor

    Homework Statement The Wikipedia article on spatial rigid body dynamics derives the equation of motion \boldsymbol\tau = [I]\boldsymbol\alpha + \boldsymbol\omega\times[I]\boldsymbol\omega from \sum_{i=1}^n \boldsymbol\Delta\mathbf{r}_i\times (m_i\mathbf{a}_i). But, there is another way to...
  25. W

    Calculating the Inertia Tensor of cone with uniform density

    Homework Statement Calculate the moments of inertia I_1, I_2, and I_3 for a homogeneous cone of mass M whose height is h and whose base has a radius R. Choose the x_3 axis along the axis of symmetry of the cone. Choose the origin at the apex of the cone, and calculate the elements of the...
  26. S

    How do I convert an inertia tensor from body space to world space?

    I've been trying to solve the following problem involving Angular Acceleration and the inertial tensor for about 2 weeks now. I know it's bad ask for a question to be solved, but I'm really at a loss here folks. I'm a high school student who has taken a physics class. What I'm Trying To Do...
  27. binbagsss

    Moment of Inertia Tensor Cylinder.

    I am computing the \hat{I} - moment of inertia tensor - of a cylinder with height 2h and radius R, about its axis of symmetry at the point of its centre of mass. I am working in cartesian coordinaes and am not sure where I am going wrong. (I can see the cylindirical coordiates would be the...
  28. J

    Moment of inertia tensor for a laminar

    Hi, Consider a 2D laminar only rotating about the z axis, with the axis origin at the bottom left hand corner and adjacent sides coinciding with the z and x axes. so ω = (0,0,ωz) y = 0 I don't understand how the IXZ component is 0 to just leave the IZZ component?
  29. binbagsss

    Moment of Inertia tensor - displaced axes theorem:

    Ok, so the system consists of two massive spheres, m1 and m2, of radii a and b respectively, connected by a massless rod of length R, as seen in the diagram attached. The question is to calculate the moment of inertia tensor. Sol: Set the origin at the centre of mass . So that we are in...
  30. D

    Moment of Inertia Tensor

    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.
  31. 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...
  32. D

    Point Mass Inertia Tensor

    Homework Statement A)Find the moment of inertia tensor for a system of 4 particles of mass 1kg at A=(1,1,0), B=(1,-1,0) C=(-1,1,0) D=(-1,-1,0) in Cartesian coordinates. B)Rotate the coordinates 30 degrees around the z axis and find the tensor in the new coordinates. Homework Equations...
  33. D

    Moment of inertia tensor of hollow cone

    I am having trouble right now with the same problem (finding Ixx and Iyy). \begin{equation} I_{yy} = \int(x^2 + z^2)dm \end{equation} where \begin{equation} dm = \frac{2M}{R^2 + H^2} q dq \end{equation} and q is my generalized coordinate that is measured from the origin down the length of the...
  34. M

    Find rotation matrix that diagonalizes given inertia tensor

    Homework Statement In a particular coordinate frame, the moment of inertia tensor of a rigid body is given by I = {{3,40},{4,9,0},{0,0,12}} in some units. The instantaneous angular velocity is given by ω = (2,3,4) in some units. Find a rotation matrix a that transforms to a new coordinate...
  35. S

    Calculating the Inertia Tensor of a Homogeneous Sphere

    Homework Statement Calculate the moments of Inertia I_{1}, I_{2}, I_{3} for a homogenous sphere Homework Equations I_{jk}=\intx^{2}_{l}\delta_{ik}-x_{i}x_{k}dV The Attempt at a Solution For I_{x} i set up the equation using the above equation in cartesian coordinates and then i...
  36. A

    Find torque in the x direction given moment of inertia tensor for plate.

    This is for prelim study. Just wondering if this solution is correct. Problem A thin homogeneous plate lies in the x-y plane. Its moment of inertia tensor in the x,y,z basis is given by \textbf{I}=σl^{4}\begin{pmatrix} 2 & -2 &0 \\ -1 & 2 & 0 \\ 0 & 0 & 4\end{pmatrix} If the plate...
  37. P

    Mathematica Mathematica: Inertia Tensor w/ 3-d Rectangle

    Hey All, I'm trying to create a 3-D rectangle in Mathematica with the following measurements: Mass M=1.5 kg, and sides of length a=10 cm (parallel to the x-axis), 2a (parallel to the y-axis), and 3a (parallel to the z-axis). Let one corner be at the origin, and let the three adjacent edges...
  38. 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...
  39. R

    Have Inertia Tensor for Object. Find It for Object after Rotation

    Homework Statement Let's say I have a coordinate system that has (0,0,0) at the CM of an object, and I know the object's inertia tensor for that coordinate, T. (T is a 3x3 inertia tensor where (1,1) is moment of inertia about x-axis, (2,2) is moment of inertia about y-axis, and (3,3) is moment...
  40. A

    Moment of inertia tensor

    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...
  41. 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...
  42. S

    Moment of inertia tensor of three spheres

    Hi everyone Homework Statement I want to find out the moment of intertia tensor of the graphic below. Homework Equations parallel axis theorem The Attempt at a Solution We know the moment of inertia for one sphere, that's given, so I don't have to calculate it...
  43. L

    Center of Mass and Inertia Tensor Experimentation

    Hello All, I am trying to create an experiment to determine the center of mass and inertia tensor of an aribtrary object. This object is small (softball size), and non symmetrical on all axis. I have some thoughts concerning gyroscopes and accelorameters, but am curious to see what thoughts...
  44. U

    Inertia Tensor as a Transformation

    Hi everyone, I was thinking about the relationship between angular velocity and angular momentum for a rigid body: I \omega = L. In particular, I'm trying to gain a little bit of intuition as to what transformations I can perform on \omega. Let's use a reference frame at the center of mass...
  45. B

    Inertia Tensor of a Hollow Sphere and of a Slender Rod

    Homework Statement I need to find the Inertia Tensor of a Hollow Sphere and of a Slender Rod with center of mass set at the origin for my calculus 2 final project. I know how to do the triple integrals I am just having trouble figuring out what the limits should be for each of these shapes...
  46. W

    Calculating Moment of Inertia Tensor for Rod Along x Axis

    Homework Statement A thin rod has mass M and length L. What is the moment of inertia tensor about the center of mass if placed along the x axis. Homework Equations I would write the inertia tensor in component notation, but I don't know how to use Latex. The Attempt at a Solution...
  47. A

    Interpretation of Moment of Inertia Tensor in generalized coordinates

    I've been wondering what the interpretation of the moment of inertia tensor in generalized coordinates is, and whether there is a way to derive it from first principles, similar to the integration we do in a Cartesian coordinate system. Specifically, I've been given the inertia matrix for a...
  48. S

    Inertia Tensor - Rotating Cone

    Homework Statement From Goldstein's Classical Mechanics (Chapter 5 - Exercice 17 - Third Edition) A uniform right circular cone of height h, half angle A, and density B rolls on its side without slipping on a uniform horizontal plane in such a manner that it returns to its original position...
  49. S

    Inertia tensor for multibody system(Robot)

    Hi, I'm trying to make a simulation of a legged robot. The Robot consists of a body, with 6 legs attached, all with three DOF. This entails that i have a total of 19(18 from the legs + 1 from the body) moving parts. To be able to make a simulation, i need the centre of mass and the inertia...
  50. F

    Reducing 3x3 Inertia Tensor to Scalar Value

    How to reduce a 3 by 3 inertia tensor to a scalor value? Hi I have the following tensor and i need to reduce it to a scalor quantity: 3 by 3 matrix 4150470.48 , 317.64, -353.42 317.64, 2047101.07,-1407556.61 -353.42, -1407566.61, 2284136.55 Please its urgent and any help would be...