What is Moment of inertia: Definition and 1000 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
  1. A

    Moment of Inertia of a Disc: Is it paradoxical?

    In deriving the MOI of a ring about its center perpendicular to plane, our teacher said think of ring as made up of ##dm## units each at ##R## distance from the axis therefore the MOI becomes: $$I=R^2\int{dm}=MR^2$$ If disc is considered to be made up of rods of ##dx## thickness, in this manner...
  2. Z

    Calculation of moment of inertia of cylindrical surface

    Here is the homogenous paper rectangle And if we roll it we get a cylinder with base radius ##a##. It is not clear to me what "an axis through a diameter of the circular base means". Let's imagine such as axis is ##\alpha## in the following figure Then we have...
  3. giodude

    MIT OCW 8.01 PS11.3: Elastic Collision Between Ball and Pivoted Rod

    Given that we're working with an elastic collision we want to populate the following system: ##k_{i} = k_{f}## ##p_{i} = p_{f}## Solve for kinetic energy just before and after the collision: ##k_{i} = \frac{1}{2}mv_{i}^{2}## ##k_{f} = \frac{1}{2}mv_{f}^{2} + \frac{1}{2}I_{P}^{sys}...
  4. al4n

    B Moment of Inertia of Rectangles

    I often encounter the formula: I = (1/12)Mbh^2 when dealing with moment of inertia of rectangles and got confused when I was unable to get the same result when figuring it out with integration. It seems that the axis of rotation used is a line perpendicular to one of the bases and on the plane...
  5. S

    Moment of inertia of a thin, square plate

    I don't really understand what the 2 integrals (dx and dxdy) for I_x represent. Could I get some explanation here please? Thanks in advance.
  6. S

    Wrong moment of inertia?

    so calculated, the moment of inertia for a rod about an axis at the end of the rod is I = 1/3 * M * L^2 here for case 1: arms to the side I is calculated to be ##I = 0.224## for case 2: arms stretched ## I = 1 / 3 * M * L^2 + M * d^2 ## with L = 0.6 m (length of rod) and d = 0.2 (dinstance from...
  7. M

    Calculating Moment of Inertia for Hollow Cylinder w/ Water

    I was able to derive an equation for acceleration for the case of a fully solid cylindrical shell and then used law of conservation of energy to determine equations for the velocity of a solid and hollow cylinder and I understand that the moment of inertia's of the aforementioned cans can also...
  8. G

    I Moment of inertia and contact force

    Two rotating cylinders are held in contact by a force F1. The force is applied through the center of one of the cylinders. One cylinder is the driving cylinder and the other is the driven cylinder . Does the moment of inertia of the system depends on the force contact force F1? Why? And...
  9. A

    Moment of inertia problem involving a cylinder rolling down an incline

    a=2/3*g*sin(25*(pi/180))=>a=2.8507 m/s^2 vf=vi+at=>vf=0+2.8507*1.50=>vf=4.2760 m/s So the translational motion of the cylinder is 4.2760 m/s. 4.2760=R*w w=134.04 rad/s PE=mgh=>PE=215*9.8*.108=>PE=227.56 J PE = KE at the end of the roll because of energy conservation. 227.56 =...
  10. J

    Moment of inertia question for two plates welded together

    If two plates of Aluminum (both are 2" tall, and 1/2" thick) were placed over each other (see photo 1) and then welded together all around the perimeter of the second plate (the blue one) as indicated by the yellow lines in photo 2, then would the moment of inertia be calculated as if the local...
  11. Aaron Mac

    Torque needed to rotate a rack

    The weight of the rack is supported on an axial bearing as seen in the attached pdf below. I have made an attempt to calculate the torque by taking a look at the chain traction force and the required shaft power to make the plates rotate. For the moment of inertia case i don't know how to treat...
  12. S

    Rotating Rod in Plane: Kinetic Energy & Moment of Inertia

    hello guys, I wanted to ask whether I can just consider/think about this as being rotation around a fixed axis in a plane representing it as if it was 'just' a rod. This is mainly so that for the kinetic energy in the second position is where if we think about it in just a plane. Is this...
  13. L

    Moment of Inertia of something that rotates

    Hi, unfortunately, I am completely confused about the task It is about the task part a I have now defined the two rotations as follows: The thin disc rotates around the ##z## axis, red in the picture, and then the rod to which the disc is attached rotates around the ##z_I## axis, in the...
  14. H

    Solving for 'a' with Torque, Force, and Mass Moment of Inertia

    What I did was plug in the outer radius time the force into the torque and then the mass moment of inertia is equal to m*ro^2 so then I plugged in the mass times the radius of gyration squared into I and solved for a but this is not right.
  15. N

    I Moment of Inertia about an axis and Torque about a point

    Angular Momentum and Torque are defined about a point. But Moment of Inertia of a body is defined about an axis. There are equations which connect Angular momentum and Torque with Moment of Inertia. How will this be consistent? When I say that the torque of a force acting on a body about a point...
  16. S

    Moment of inertia of T bar about 3 axes

    Using the equation above I get Xcm = 0.022 m. I set the origin be at the left of the vertical rod parallel to its centre of mass as in the diagram. But I’m not sure if the equation is correct for 3d. for the moments of inertia I am using I = Icm + md^2 = (mr^2)/2 + md^2 where d is the...
  17. AF Fardin

    Moment of inertia of a double physical pendulum

    I am having trouble to find the moment of inertia of the second rod! Is it related to the first rod?? At the beginning I thought It's not! But when took those as constant,the equation had become way much simpler and there is nothing about chaos! My approach is given below
  18. L

    Moment of inertia of a uniform square plate

    I placed my Oxy coordinate system at the center of the square, the ##x##-axis pointing rightwards and the ##y##-axis pointing upwards. I divided the square into thin vertical strips, each of height ##h=2(\frac{L}{\sqrt{2}}-x)##, base ##dx## and mass ##dm=\sigma h...
  19. V

    Moment of inertia of a disk about an axis not passing through its CoM

    I have come up with two different approaches, but I'm not sure which one is correct since they give different answers. We use the following equation to get the total moment of inertia. ##I_o## = moment of inertia of disk about O axis + moment of inertia of road about O axis Approach 1...
  20. V

    Determining $L_{o}$: Finding Angular Momentum of System

    I think the the time given doesn't matter since no torque is acting on the system, but not sure. Therefore, all we need is to determine the angular momentum about the axis passing through O and perpendicular to the plane of disk. This will involve finding the moment of inertia of smaller disk...
  21. GopherTv

    B How to find the Moment of Inertia for a Sphero robot?

    What kind of experiment can I design to determine the actual value of the moment of inertia. What should I instruct the sphero to do and what data should I collect?
  22. V

    What is the "r" in the moment of inertia?

    We solved this problem in class as follows: Net torque about the center of the pulley taking counterclockwise rotation to be positive = m1gR - m2gR = I_tot α, where I_tot is the moment of inertia of the full system. My professor said that I_tot = I + m1R^2 + m2R^2, where m1R^2 is the moment...
  23. 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...
  24. M

    What's wrong with my solution? -- Area moment of inertia

    I used the parallel axis theorem to solve the question but my answer is still wrong. Any ideas where I slipped? I can't seem to figure out the problem?
  25. L

    Disk with rod attached rotating about the center of the disk

    1) Since the rod is uniform, with mass m and length l, it has a linear mass density of ##\lambda=\frac{m}{l}##, so ##I_{rod_O}=\int_{x=r}^{x=r+l}x^2 \lambda dx=\frac{\lambda}{3}[(r+l)^3-r^3]=\frac{\lambda r^3}{3}[(1+\frac{l}{r})^3-1]=\frac{1}{3}mr^2[3+\frac{3l}{r}+\frac{l^2}{r^2}].##...
  26. curiousPep

    I Resolve moment of inertia at an angle

    Initially, I calculate the moment of inertia of of a square lamina (x-z plane). Thr this square is rotated an angle $\theta$ about a vertex and I need to calculate the new moment of inertia about that vertex. Can I split the rotated square to two squares in the x-z plane and y-z plane to find...
  27. A

    Moment of Inertia of a 4 rod system

    This was the question (The line below is probably some translation of upper line in different language) For disc it was ma^2/2 For ring it was ma^2 For square lamina it was 2ma^2/3 For rods It was different Please explain Thank You🙏
  28. Tapias5000

    How to determine the same moment of inertia in two different ways?

    My solution is now I am asked for the same result but in this form but I don't know where to start.
  29. L

    Disk hit by two masses

    1) By conservation of linear momentum: ##m_1 v_1-m_2v_2=(m+m_1+m_2)v_{cm}\Rightarrow v_{cm}=\frac{m_1}{m+m_1+m_2}v_1-\frac{m_2}{m+m_1+m_2}v_2=\frac{3}{8}\frac{m}{s}##; 2) By conservation of angular momentum: ##-Rm_1v_1-Rm_2v_2=I_{total}\omega=(I_{disk}+m_1R^2+m_2R^2)\omega## so...
  30. L

    Tangential velocity of rotating rod

    1) ##LT\sin(\frac{\pi}{2}-\theta)-\frac{L}{2}mg\sin\theta=0\Rightarrow T=\frac{mg}{2}\tan\theta##. ##N_{x}-T=0, N_{y}-mg=0\Rightarrow N=\sqrt{N_x ^2+N_y ^2}=mg\sqrt{(\frac{\tan\theta}{2})^2 +1}## 2) ##E_{k_{fin}}=mg\frac{L}{2}[1+\cos\theta]## 3)...
  31. M

    Torque due to moment of inertia (?)

    Hi! I would like to calculate (roughly) how much torque is needed bringing the blue plateau in movement. Assume the blue plateau is loaded with 7.5 kg. The radius of the blue circle is 100 mm.
  32. Ang09

    Moment of Inertia with varying distance from Centre of Mass

    h = d1 + 0.08 d1 = h - 0.08 d2 = h + 0.08 I of the vertical portion = 1/12 m (l^2 + b^2) + md1^2 = 1/12 m (0.28^2 + 0.04^2) + m(h - 0.08)^2 I of the horizontal portion = 1/12 m (l^2 + b^2) + md2^2 = 1/12 m (0.28^2 + 0.04^2) + m(h + 0.08)^2 The moment of inertia for the whole T-shape about...
  33. P

    Lagrangian mechanics - rotating rod

    Hello, It might sound silly, but when I try to calculate the kinetic energy of a rotating rod to form the Langrangian (and in general), why it has both translational and rotational kinetic energy? Is it because when I consider the moment of Inertia about the centre I need to include the...
  34. antonov

    Moment of inertia composite body

    I have this moment of inertia problem and is a little confused on the semicircle part and if the rest is really right? I get over 10 if I calculate it in crew CAD but by hand I get 7,568032142. What is right and what am I doing wrong?
  35. Hamiltonian

    Moment of inertia of a disc using rods as differential elements

    I know there are more convenient differential elements that can be chosen to compute the moment of inertia of a disc(like rings). the mass of the differential element: $$dm = (M/\pi R^2) (dA) = (M/ \pi R^2) (2\sqrt{R^2 - y^2})(dy)$$ the moment of inertia of a rod through its COM is...
  36. PiEpsilon

    Analyzing an Angular Impulse Problem

    What we know: The ball is dropped at the tip A with some speed ##v_0## and rebounds with speed ##v##. This collision produces an angular impulse, changing the angular momentum of the bar with the flywheels. Solution inspired by an answer provided by @TSny in the similar question. Angular...
  37. TheBigDig

    Determine the moment of inertia of a bar and disk assembly

    I have been given an answer for this but I am struggling to get to that point $$ANS = 0.430\, kg \cdot m^2$$ So I thought using the moment of inertia of a compound pendulum might work where ##I_{rod} = \frac{ml^2}{12}## and ##I_{disc} = \frac{mR^2}{2}## (##l## is the length of the rod and ##R##...
  38. P

    Moment of Inertia of a sphere about an axis

    I = 2/5M R^2 + Md^2 This is analagous to Earth's movement about the Sun. Is the moment of inertia of Earth about the centre of mass of the Earth Sun system = 2/5MR^2 + Md^2, where: M = Mass of earth, R = Radius of Earth, d = distance from Earth to centre of mass of earth-sun system.
  39. PiEpsilon

    Freely hinged rods on a table - Linear velocities of CM after the impulse

    We know that impulse is $$\vec J = \vec F \Delta t = \Delta \vec p$$ Let ##l, m## be the length of single rod and its mass respectively. Analyzing torques and forces on each rod separately we have: Rod ##AC##: $$F\Delta t +N_x\Delta t = mV_{ac,x} \space\space\text{ eq. }(1)$$ $$F\Delta t\cdot...
  40. 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...
  41. Who_w

    Moment of inertia of a regular triangle

    Please, I need help! I need to calculate the moment of inertia of a triangle relatively OY. I have an idea to split my triangle into rods and use Huygens-Steiner theorem, but after discussed this exercise with my friend, I have a question: which of these splits are right (picture 1 and 2)? Or...
  42. Leo Liu

    Computing the polar moment of inertia (calculus)

    Question: Diagram: So the common approach to this problem is using polar coordinates. The definition of infinitesimal rotational inertia at O is ##dI_O=r^2\sigma\, dA##. Therefore the r. inertia of the triangle is $$I_O=\int_{0}^{\pi/3}\int_{0}^{\sec\theta}r^2r\,drd\theta$$ whose value is...
  43. E

    Moment of inertia alphabet soup :)

    We start from the definition$$I_{ij} = \int_V \rho(x_k x_k \delta_{ij} - x_i x_j) dV \iff \dot{I}_{ij} = \int_V \rho (2 x_k \dot{x}_k \delta_{ij} - \dot{x}_i x_j - x_i \dot{x}_j ) dV$$Now since the rigid body rotation satisfies ##\dot{\vec{x}} = \vec{\omega} \times \vec{x} \iff \dot{x}_i =...
  44. S

    Tubular Column Moment of Inertia

    Can anyone explain why the moment of inertia for a tubular column in that textbook is like so? (take a look at the attachments). It should be (I = MR^2), as far as I know.
  45. C

    Moment of Inertia of a Solid Cylinder With a Wedge Removed

  46. Ugnius

    Rotating body moment of inertia

    I have done some lab work , and now i have to answer some theoretical questions , but i can not find any data about this on the web or atleast i don't know where to search , i will add some pictures of experiment for you to better understand it. I was wondering can someone share their knowledge...
  47. MattGeo

    Torque and Moment of Inertia of a Lever Arm

    I never really considered this back when I was taking physics in college but imagine for the sake of thought experiment that you have an extremely and impractically long wrench and it is fixed to the bolt you wish to tighten. Now the longer the lever arm the greater the torque so if you double...
  48. C

    Confused by Unexpected Results: Acceleration & Moment of Inertia

    Like I said, objects with the higher acceleration are giving me the lowest values. For a hoop, I got I=0.1*MR^2 For a cylinder, I got I=0.7*MR^2 this seems backwards, no?
  49. T

    Moment of Inertia with pulley and two masses on a string (iWTSE.org)

    Note: the working (taken from iWTSE website) refers to inertia as the symbol ‘J’ (in-case there was any confusion). I found equations of motion for mass m and 2m which were ‘T1 = ma + mg’ and ‘T2 = 2mg – 2ma’, respectively. I know they are connected particles with the same acceleration ‘a’...
  50. wcjy

    Moment of inertia of a uniform 2D triangular plate

    Answer is 37.8 g cm^2 new to latex