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.

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  1. lc99

    Moment of Inertia in Planet System

    Homework Statement Two planets are masses stuck to end of long thin rods. These model systems (Rod + mass) will be rotated by a motor at their other end of the rod, as shown in the figure. the length of the rod is proportional to the orbital distance of the planet, and the mass at the end is...
  2. lc99

    Quick question about moment of inertia

    Homework Statement Homework Equations I = mr^2 (point mass) The Attempt at a Solution I wasn't sure how to calculate the moment of inertia of this triangle if the axis is going through one vertice perpendicularly. Do i calculate the inerta of 2 sides with Inertia of a rod and the 3 side as...
  3. I

    Moment of Inertia Formula Query

    Hello, this formula was given to me in an exam to find the moment of inertia of an I beam, https://ibb.co/jY7ZKG However this formula seems to give a different answer to the standard bd^3/12 method, is the formula in the image wrong ? Thanks
  4. K

    Moment of inertia -- a lump of clay hits and adheres to a rotating stick....

    Homework Statement A thin stick of mass M = 3.9 kg and length L = 1.8 m is hinged at the top. A piece of clay, mass m = 0.3 kg and velocity V = 2.9 m/s hits the stick a distance x = 1.60 m from the hinge and sticks to it. What is the angular velocity of the stick immediately after the...
  5. J

    How to calculate the moment of inertia for complex 3D shapes

    <<<moved from another sub forum, no template>>Hi, I need to calculate the moment of inertia for the component in the attached image so that i can calculate the angular momentum. Is it possible? Overall i am trying to calculate the forces on this lug as it passes around a 3" radius at 2M a...
  6. Arup Biswas

    Is Calculating Moment of Inertia for a Disc with (x+dx)^2 Valid?

    In the picture the question marked spot is confusing me. Should the MI be calculated in this ugly way? That we do not consider the points in between and on dx! Why only x is taken as the distance? You may say take x+dx, resulting dx in the exapnsion and the dx from the mass to square and being...
  7. T

    HW question w/ rotational kinetic energy

    Homework Statement : [/B] A stationary horizontal platform is free to rotate about its vertical axis. The radius of the platform is R=1.6m and its moment of inertia is 660 kgm^2. A 43 kg boy jumps on the rim of the platform with the velocity 2.2 m/s tangential to the rim. What will be the...
  8. Z

    Question about calculating moment of inertia.

    Homework Statement I was learning how to calculate moment of inertia on Youtube, and I'm confused about when to use 1/2 mr2 instead of mr2. This video says that he used 1/2 mr2 instead of mr2 because the cone is solid But in this video, while he was calculating a solid sphere, he used mr2...
  9. A

    Calculating Moment of Inertia for Outstretched Arms: 1/3 vs 1/12 Formula

    Homework Statement Kindly see the screenshot attached below for the question. Homework Equations I=1/3ML^2 1/12ML^2 The Attempt at a Solution In the solution to this question, the moment of inertia of the hands (when outstretched) is taken to be 1/12ML^2 (combined). I think that it should be...
  10. Z

    How is the moment of inertia of gears calculated?

    If two gears are joined to together, how are their moment of inertia calculated? Separately or combined?
  11. Jefffff

    Moment of Inertia of a Disk

    Homework Statement A disk of radius R has an initial mass M. Then a hole of radius (1/4) is drilled, with its edge at the disk center. Find the new rotational inertia about the central axis. Hint: Find the rotational inertia of the missing piece, and subtract it from that of the whole disk...
  12. E

    Rotating spool on table with friction

    Homework Statement I am referring to this thread and question: https://www.physicsforums.com/threads/rotation-of-a-spool-about-rough-ground.295666/ Here is the problem, restated: Homework Equations ##\tau_{net} = I\alpha## ##\tau = Frsin(\theta)## ##F_{net} = Ma_{cm}## ##\alpha = a_{cm} /...
  13. yavorh

    Mass moment of inertia of an airplane about its main gear

    Homework Statement Okay, so this isn't so much a problem as a design issue while following an example given in a book. (Relevant material is here: http://www.aero.us.es/adesign/Slides/Extra/Stability/Design_Control_Surface/Chapter%2012.%20Desig%20of%20Control%20Surfaces%20(Elevator).pdf ) Now...
  14. lichenguy

    Accelerating a car including the moment of inertia of the wheels

    Homework Statement A car accelerates from rest on a horizontal surface. The engine provides a torque of τ = 200 Nm on each of the two front wheels. Each of the four wheels on the car weigh m = 15 kg, have radius R = 0.35 m and can be considered solid, uniform discs. The rest of the car (not the...
  15. T

    How to solve for the moment of inertia and angular acceleration?

    A girl is climbing up a set of stairs and has her femur horizontal. Her thigh weighs 12 pounds, acting at a center of mass 15 cm from her hip joint. She places 50 pounds of weight on the lower portion of that leg, vertically through her knee. Her gluteus maximus is applying a force directed 30...
  16. D

    Calculating Moment of Inertia for a Rotating Cylinder | Physics Tutorial

    Hi I am struggling with the following question. A cylinder of radius 20cm is mounted on a horizontal axle coincident with its axis and is free to rotate. A light chord is wound onto it and a 50g mass hung from it. After release the mass drops 1m in 12seconds. What is the moment of inertia...
  17. CollinsArg

    How to calculate the moment of inertia of a system with multiple bodies?

    Homework Statement Calculate the moment of inertia in the OO' axis of the following figure (image attached).Homework Equations The relevant equations are in the image, except: I = Icm + m.h2 (formula of moment of inertia to change the axis from the centre of mass to a parallel axis). The...
  18. 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...
  19. A

    What is the angular acceleration of the rod?

    Homework Statement A thin, uniform, 18.5 kg post, 2.10 m long, is held vertically using a cable and is attached to a 5.00 kg mass and a pivot at its bottom end (as shown below). The string attached to the 5.00 kg mass passes over a massless, frictionless pulley and pulls perpendicular to the...
  20. J

    Understanding MOI and Weight in Golf: Exploring the Optimal Balance

    At what point will weight have a diminishing effect on increasing MOI with a given size remaining the same?
  21. S

    About torque and moment of inertia

    I know I just posted a topic, but I have another question. From my understanding, torque is force times the length of the lever arm, and the longer the lever arm or the more force there is, the more torque is applied. For moment of inertia, it is mass times the square of the distance from the...
  22. J

    Moment of inertia of a charged spinning top

    How can I calculate the moment of inertia of a spinning top with charge Q submerged in an electromagnetic field? I've already calculated it for the case with no charge, but I don't know how to do this.
  23. S

    Moment of Inertia of a Rotating Disk

    Homework Statement Given a non magnetic rectangular plate with dimensions: L & B. The plate is of uniform density and thickness.There are 2 points equidistant from the center of the rectangular plate along the line at B/2. Both of these pivot points have 2 disks of radius R and mass M that are...
  24. G

    Moment of inertia changes during rotation—Calculate work?

    Hi everyone, i have a question Moment of inertia changes during rotation. Calculate the work done that changes kinetic energy? Angular moment (along the axis of rotation) L = I * w A point mass M rotates along an axis attached to a mass-negligible rod, of length r. If someone moves the mass M...
  25. G

    Moment of inertia changes during rotation -- Calculate work?

    Hi everyone, i have a question Moment of inertia changes during rotation. Calculate the work that changes kinetic energy? Angular moment (along the axis of rotation) L = I * w A point mass M rotates along an axis attached to a mass-negligible rod, of length r. If someone moves the mass M at...
  26. J

    How do you calculate moment of inertia for circle?

    Hey, not sure if this is the right place to post this, but here it goes. how do you calculate the moment of inertia for a circle that is not at its center of gravity. I am trying to find the moment of inertia for a complex shape made of many circles, and this seems like a good place to start...
  27. S

    Moment of inertia of a gearbox

    I just want to check to see if I am on the right path, I have to calculate the moment of inertia of a gearbox for an application. I was thinking of breaking down the input shaft/worm into different size cylinders since the shaft has different diameters for the the input and bearings. And then...
  28. A

    Moment of inertia of a rod with balls attached

    Homework Statement Attached. Homework Equations I=mr^2, I=(1/12)m*l^2 for a rod. The Attempt at a Solution Part A I got by doing I=(1/12)3.5*(2.6)^2+2*.7*1.3^2 (I added the moment of inertia of the rod and the balls). Part B since the axis is on one of the balls, I thought we don't include...
  29. Z

    Moment of Inertia for a blower

    I have a blower within a system, and I'm trying to calculate the system inertia. I'd like to get the moment of inertia for this component as would be seen by the shaft. It's shape has me a little stumped (considered perpendicular axis theorem) on what method to use for finding the inertia. Any...
  30. M

    Determine Mass moment of inertia about any axis given Ixx...

    Hello, I am analyzing human motion. for each body segment, I have measured values for Ixx, Iyy, and Izz in local coordinate system. I want to transfer all body properties from one coordinate system to another using a transformation matrix. My question is that, is there a way to calculate mass...
  31. T

    Calculating Torque on a Rod: Net Torque, Mass & Distance

    So I've learned that the torque on an object is just: (perpendicular force)(radius) and that has worked well for things like seesaws but it doesn't take into account the torque of the object itself! In the picture the rod has a total mass of 5kg, and by definition the torque on the right is...
  32. MrDickinson

    Question about torque and center of mass

    Supposing that we have a hollow sphere of mass m and radius R. Furthermore, suppose that we attach the sphere to a rod of length L and mass M. Now, suppose we rest the rod-sphere contraption on a pyramid a distance (X+R) from the center of mass of the sphere and a distance [(L/2)-X)] from the...
  33. S

    Have a problem with this Rotational Motion question

    Homework Statement [/B] A wheel, of radius 200mm, rolls over the top of a hill with a speed of 20m/s and negligible friction losses. (I = 1/2mr^2) Homework Equations [/B] Find the speed of the wheel when it is 10m below the top. The Attempt at a Solution [/B] mgh = 1/2mv^2 + 1/2IW^2 W=...
  34. G

    Moment of Inertia: Definition & Physical Meaning

    What is the physical meaning and definition of moment of inertia?
  35. Isaac0427

    Moment of inertia for a fidget spinner....

    Hi, I was just curious, how would you go about calculating the moment of inertia for a fidget spinner? I was thinking about it, and I don't know how to calculate moment of inertia for weird shapes like that. Thanks!
  36. Pbaseball

    Conservation of momentum and Moment of Inertia

    Hello everyone, I'm needing help understanding how to take the conservation of momentum equation and substitute moment of inertia inplace of mass in the equation. I understand this is a linear equation and I'm trying to incorporate angular momentum. I just want to determine the speed of the ball...
  37. M

    Moment of inertia of a cylinder?

    Homework Statement the moment of inertia of A cylinder of height 2h radius (a) and uniform mass density ρ about a line x=y=z using multiple integration. Homework Equations I=ρ∫s^2*dV where the integral is over the volume V of cylinder and s is the perpendicular distance to the axis of...
  38. fisher garry

    A Calculate the moment of inertia of H with the probability from wavefunction

    the spin, S, for an electron is $$\frac{\hbar}{2}=5.27 \cdot 10^{-35} $$ $$\frac{2MR^2 \omega}{5}=\frac{2MRv}{5}$$ It is said that the speed of an electron is 2200 km per second and can be calculated in classical manners from electrostatic and accelerating forces on the electron from (1.11)...
  39. DevonZA

    Centre of gravity, moment of inertia and radius of gyration

    Homework Statement Homework Equations Centre of gravity: X=m1x1-m2x2/m1-m2 MOI rectangle: 1/3ml^2 MOI triangle: 1/18md^2 Radius of gyration: Ixx=mk^2 The Attempt at a Solution Mass of body 1: b*l*p = 0.8*1*10=8kg Mass of body 2: 1/2b*h*p = 1/2(0.4)*0.6*10=1.2kg1.1 X=m1x1-m2x2/m1-m2...
  40. harpazo

    MHB What is the concept of moment of inertia?

    In simple terms, what exactly is meant by moment of inertia as taught in Calculus 3?
  41. I

    Calculating the moment of inertia of a rod as a point mass

    The moment of inertia of a rod with consistent density rotating about its end is M(L^2)/3. However, if you treat the rod as a point mass and try to calculate this using mr^2, you get M(L/2)^2=M(L^2)/4. Why doesn't this work?
  42. W

    Torque with 2D center of mass

    Homework Statement A cube of mass M rests tilted against the wall as shown (see below). There is no friction between the wall and the cube, but the friction between the cube and the floor is just sufficient to keep the cube from slipping. When ##0\lt\theta\lt 45^\circ## find the minimum...
  43. I

    Stuck on separable equation relating to moment of inertia

    Homework Statement (a) Consider a cylindrical can of gas with radius R and height H rotating about its longitudinal axis. The rotation causes the density of the gas, η, to obey the differential equation dη(ρ)/dp = κ ω2 ρ η(ρ) where ρ is the distance from the longitudinal axis, the constant κ...
  44. B

    How to measure the moment of inertia of a motor's rotor?

    Hello, I need to measure the moment of inertia of a DC motor's rotor. Any suggestion? Thanks
  45. 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...
  46. Const@ntine

    Error Transmission (Moment of Inertia)

    Homework Statement Compute the moment of intertia Io of the two cylinders, per their axis of symmetry, and then, using Steiner's formula, the moment of inertia Ib, as per the axis of rotation. Homework Equations Io = 1/2*m*R2 Ib = 2*(Io + m*x2) The Attempt at a Solution Now, this is...
  47. Arisylia

    Deriving the moment of inertia of solid sphere

    So i was going through derivations of moments of inertia of objects. For objects like the disk and rod, i was able to assume a relationship between mass and volume and integrate From there like $$ \frac{d_m}{m} = \frac{dl}{l} \\ d_m = \frac{dl*m}{l} \\ \int_{0}^{L}r^2\frac{dl*m}{l} \\...
  48. top12eaper

    Moment of inertia of cylinder with a rectangel shaped hole

    i know how to calculate the inersia for the cylinder, than i have to take away the inersia of the rectangel, but i can't find an equation for the inersia for a rectangel with the plane spinning perpendicular to the rotation axis, please help !
  49. R

    Calculating the moment of inertia of a solid sphere

    Homework Statement To calculate moment of inertia of a solid sphere of uniform density[/B]Homework Equations $$ I = \int r^2 dm$$ The attempt at a solution I consider an elemental disk of small thickness ##d\theta## ##dm = \frac{M}{4/3 \pi R^3}*\pi R^2\cos^2\theta* Rd\theta##...
  50. R

    Moment of inertia of a non uniform disc

    Homework Statement A non uniform disc of radius R has a mass of M. Its centre of gravity is located at a distance x from the centre. Find the moment of inertia of mass (moi) around the axis perpendicular to the surface passinf through the centre of gravity. Homework Equations Parallel axis...
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