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. hackhard

    B Can body start spinning without ext cause

    suppose a platform "P" is rotating about the z-axis wrt x-axis . another platform "Q" rotating about z-axis below "P" with same angular velocity wrt x-axis standing on Q , P is at rest wrt Q, After some time rotational inertia of P about z-axis starts changing with time standing on Q , P will...
  2. E

    What is the Difference Between Ixy and Iz in Moment of Inertia for Pipes?

    Hi, I've been always using eq. I=Pi*(D^4-d^4)/64 to find moment of inertia of a pipe. Recently I've seen that moment can be in different directions, and then it is expressed differently. So what is the difference between Ixy and Iz?
  3. Titan97

    Is moment of inertia only for rotating objects?

    Homework Statement A rod attached to a ceiling at one end and a disc on the other end is performing SHM. In case (1) the disc cannot rotate. In case (2) the disc can rotate about its centre. Compare the restoring torque and angular frequency in both cases. Homework Equations...
  4. RoboNerd

    Question on Rotational Motion and Torque Theory

    1. Homework Statement A cylinder of mass M and radius R rolls (without slipping) down an inclined plane whose incline angle with the horizontal is theta. Determine the acceleration of the cylinder's center of mass and the minimum coefficient of friction that will allow the cylinder to roll...
  5. 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...
  6. 2

    Find the moment of inertia of this disk

    Homework Statement A crucial part of a piece of machinery starts as a flat uniform cylindrical disk of radius R0 and mass M. It then has a circular hole of radius R1 drilled into it. The hole's center is a distance h from the center of the disk. Find the moment of inertia of this disk (with...
  7. B

    Question about angular momentum conservation

    Homework Statement Question - Models of global warming predict that large sections of the polar ice caps will melt. Explain what effect this will have on the rotation of the Earth, however slight. Homework Equations L = Iw The Attempt at a Solution Assuming polar ice caps protrude the earth...
  8. J

    Derivation of moment of inertia of a solid sphere

    Homework Statement I need to know why my derivation does not work. I am attempting to derive I=2/5 mR^2Homework Equations I have seen people derive it using disks but my question is why do the shells not work? Where in my set up did I go wrong? Thanks The Attempt at a Solution I am attempting...
  9. A

    How to Calculate the Moment of Inertia for a Half Disk System?

    in this video http://www.physicsgalaxy.com/lectures/1/44/234/Solved-Example-2#12(see only the question) the method illustrated is integration but i thought of an alternate method, moment of inertia of half disc with radius r2 is 1/2mr2^2 and that of half disc with radius r1 is 1/2mr1^2.so...
  10. Arman777

    Moment of Inertia of Half Ring (Half Circle)

    Homework Statement Theres an object which makes a pendulum motion.Lets suppose we hang the mass to the ceiling.We released the object with inital angle 0 to the ceiling.(I mean the angle between the object and the ceiling is zero).Whats the moment of the Inertia to the point A. A is a...
  11. dumbdumNotSmart

    Angular acceleration in moment of inertia

    I'm having a hard time undertanding a concept of moment of inertia and Angular acceleration. Homework Statement We have a closed system above. M1 is a cylinder of 2 Kilograms, moment of inertia of a cylinder ( MR2 /2 ) with a string tightly rolled around it. This string connects to a free...
  12. NinjaChachi

    Physics Lab Help - Moment of inertia and acceleration

    Homework Statement In our physics lab we were to set up a system looking like this: where the mass was released and the velocity/acceleration and of the ring was calculated using MotionLab software. Our goal was to find the predicted total moment of inertia by just calculating the moment of...
  13. vetgirl1990

    Moment of inertia for composite objects

    Homework Statement Find the moment of inertia of these composite objects. I've attached two different composite objects; each rod has length L and mass M. Homework Equations I = Icm + MD2 Long thin rod with rotation axis through centre: Icm = 1/12ML2 Long thin rod with rotation axis through...
  14. I

    Moment of inertia of a cylinder

    Homework Statement Need to find the moment of inertia of a cylinder, about its center of mass, about the three principle axes. Z axis is normal to the circular faces of the cylinder. mass = M, radius = R, height = h Homework Equations ∑miri2 The Attempt at a Solution The Izz for whatever...
  15. vetgirl1990

    Moment of inertia for composite object

    Homework Statement Find the moment of inertia for this composite object: M1=5kg at (3,4) and M2=3kg at (6,8) Homework Equations Moment if inertia for point particle = MR2 The Attempt at a Solution I'm not sure if moment of inertia should be componentized -- broken up into x and y...
  16. T

    How can the moment of inertia of an object be calculated on an arbitrary axis?

    This seems to be a crucial detail that I just glossed over, but when finding the inertia tensor of an object, is the origin always situated at the object centre of mass? For example: In the link (http://hepweb.ucsd.edu/ph110b/110b_notes/node26.html ), is it necessary to do the integral from...
  17. W

    Finding second moment of inertia about x axis

    Homework Statement The cubic is divided into 4 parts , A, B and C, D , each with thickness of 1mm , i am sked to find thesecond moment of inertia about x -axis Homework EquationsThe Attempt at a Solution i' m using the formula Ixx = Ix +A(d^2)[/B] for part CD, Ixx = 50(1^3) / 12 + 50...
  18. faradayscat

    Dynamics/kinematics of rotating sphere

    I'm wondering about how one would describe the dynamics of a rotating sphere. Consider this: a solid sphere of mass "m" and radius "r" is set to rotate about a tangent to its surface. If it is released from the horizontal position such that it swings like a pendulum, what would be the force...
  19. M

    Moment of Inertia: Disk 1 & 2 Homework

    Homework Statement In the diagram, Disk 1 has a moment of inertia of 4.40 kg · m2 and is rotating in the counterclockwise direction with an angular speed of 7.50 rad/s about a frictionless rod passing through its center. A second disk rotating clockwise with an angular speed of 9.50 rad/s falls...
  20. tshah321

    Moment of inertia about center of mass with four holes?

    Homework Statement What is the moment of inertia of the 4 kg disk below when it rotates about the center of mass? The mass of the disk without the holes is 6 kg . Homework Equations Center of mass equation: (1/M)(∫xdm) Moment of inertia of a disk: I = 0.5mr2 Parallel axis theorem: I = Io +...
  21. R

    Total Moment of Inertia of Two Rods

    Homework Statement The rods of length 2 meters and and mass 20 kg are joined at their ends to form a V shape. What is the total moment of inertia measured from the reference point perpendicular to the plane of the paper and at the point where the two rods are joined. (So find total moment of...
  22. C

    Finding moment of inertia of irregular object with calculus?

    For a mathematics project I'm trying to figure out the moment of inertia for a propeller. I'm told that it is possible to find the moment of inertia of irregular objects through calculus, so I'm determined to figure it out using calculus. I plan on using a 3D modelling program (since I don't...
  23. S

    Calculating Moment of Inertia Homework: Soccer Ball

    Homework Statement An object rolls down a ramp starting from rest, demonstrate symbolically the calculation needed to go from time given to a measured κ value here. Pick your favourite object and perform the calculation. d, m, R, t, h are known http://puu.sh/luare/10352d0479.png Soccer Ball...
  24. M

    Period of small oscillations for a pendulum

    Homework Statement A pendulum consists of a light rigid rod of length 250 mm, with two identical uniform solid spheres of radius of radius 50 mm attached one on either side of its lower end. Find the period of small oscillations (a) perpendicular to the line of centres and (b) along...
  25. A

    Moment of Inertia of Rod About Axis XX': ML2/3sin2α

    1. The problem statementα, all variables and given/known data The moment of inertia of a Rod over an Axis XX' passing through the center of mass of the Rod at an angle α is- Homework Equations Moment of Inertia of Rod about the end, I =ML2/3 The Attempt at a Solution I=ML2/3 Answer of the...
  26. dumbdumNotSmart

    What is the acceleration of a cylinder when a torque is applied to it?

    Homework Statement Got a spicy one for you today. There is a cylinder of mass 5kg (M) with no angular velocity and no velocity, on a surface of static friction constant μ=0.6 . It's radius is .1 meters (R). Its baricentric moment of inertia is characterized by I=.5MR^2 . Gravity is 10m/s...
  27. C

    Why Does a Rolling Sphere Climb Higher Than a Sliding Particle?

    Homework Statement A uniform sphere and a particle are sent one-by-one with the same initial speed up the same incline. Each rises to a maximum height before falling back towards the starting point. The sphere rolls without slipping; the particle slides without friction. Use conservation of...
  28. T

    Lab experiment on moment of inertia dynamic

    Homework Statement That is the set up for experiment, and I attached the data I found in the lab. The lab given me the equation to find the moment of inertia of the cylinder Homework Equations Icyl = (2hgM_cylinder) * (1/(W_withcylinder)^2 - 1/(W_nocylinder)^2) The height is 0.9m The Attempt at...
  29. S

    How Do You Calculate the Moment of Inertia for a Tapered Tube?

    Homework Statement To calculate the moment of inertia of a cantilevered tapered tube with mid-thickness large radius RL and mid-thickness small radius RS Homework Equations The Attempt at a Solution Area of the larger end of the tube AL=2*pi*RL*t Area of the smaller end of the tube...
  30. imadrea

    How to devise moment of inertia formula of solid sphere?

    Homework Statement how to divide moment of inertia of solid sphere about its central axis?. Solid sphere has radius R, mass M. Homework Equations I=∫r2dm 2/5 MR^2 The Attempt at a Solution https://photos.google.com/search/_tra_/photo/AF1QipPoXyad0q1Y3yisc0LeeJHGApkIrGbitK6kAk5p i try to...
  31. minimario

    Moment of Inertia of Spherical Shell

    Homework Statement Find the moment of inertia of a spherical shell (hollow) with mass M and radius R. Homework Equations ## I = \int r^2 dm ## The Attempt at a Solution This is method I use to find Moment of Inertia of solid sphere: We use circular cross sections. At some radius r...
  32. K

    Is the moment of inertia matrix a tensor?

    Homework Statement Is the moment of inertia matrix a tensor? Hint: the dyadic product of two vectors transforms according to the rule for second order tensors. I is the inertia matrix L is the angular momentum \omega is the angular velocity Homework Equations The transformation rule for a...
  33. E

    Help with 2nd moment of inertia

    If a beam's y-y axis lies in the plane of the paper, do I use Ixx or Iyy from sections table?
  34. H

    Moment of Inertia and Bike stability

    So is a Bike driver stable when the bike is running because the bike wheels has a certain moment of inertia about the horizontal axis ,which might alter(mi gets lesser) if the direction of the axis changes ? Thanks in advance
  35. H

    Euler's column Formulae application in aluminum cans?

    Maximum axial load is proportional to the second moment of area. Thus can we reason that aluminium cans are cylindrical because they have a high second moment of area(mr^2) compared to other shapes(Which gives it a higher max axial load.)?
  36. W

    Calculating the Moment of Inertia of a loaded flywheel

    Hi, I need to calculate the moment of inertia of this flywheel so I can calculate the torque I need. T= I*alpha Attached is an image of the flywheel the cam follower that will be attached to the shaft (left shaft in this picture). When calculating the Moment of Inertia, what mass do I use...
  37. KastorPhys

    Moment of Inertia of a sphere with different methods

    Hi everyone, I am trying to find out the Moment of Inertia of a sphere which is all known to be 2/5(m)(r^2) I calculate this in 2 ways. One by triple integration and one by disk method. From the textbooks, moment of inertia should be in the form, dI = (r^2) dm, However, the textbook, University...
  38. Jonski

    Reaction Force, find tension, a in the tangential, or alpha

    Homework Statement The diagram shows the instant when a long slender bar of mass 4.9 kg and length 4.9 m is horizontal. At this instant the mass m= 6.5 kg has a vertical velocity of 4.0 m/s. If the pulley has negligible mass and all friction effects may be ignored, what is the magnitude of the...
  39. R

    What is [s]motion[/s] moment of inertia of an equilateral triangle?

    I am unable to find it.
  40. B

    Setting up a Double Integral for Moment of Inertia

    Homework Statement >Problem:<br>Find the Moment of Inertia of a circular disk of uniform density about an axis which passes through the center and makes an angle of $\dfrac{\pi}{6}$ with the plane of the disc. Homework Equations Moment of Inertia ($I$) is $$\int r^2dm$$ where $r$ is the...
  41. 24forChromium

    Moment of inertia of a cylinder in unusual position

    See image. It would be better if you can show me the calculation process. EDIT: the axis of rotation goes right through the centre of the cylinder. NEVER MIND! GOT IT!
  42. J

    Moment of Inertia of multiple bodies in contact

    What is the overall moment of inertia of many cylinders in contact, where each is rotated about an axis through its center of mass. For example, a set of rollers. (See picture)
  43. Z

    Different formulae for moment of inertia

    I know the formula for moment of inertia is but there are I = MR^2 but there are also formulae for different objects as shown in the picture. So, how and when do you use I = MR^2 ? Just in case of (a)? Thanks!
  44. A

    Moment of inertia about tangent of a hoop

    The hoop has radius R. I used the same way to plot the axis for the hoop: ##l^2 = r^2= 4R^2cos\theta## since: ##r=2Rcos\theta## 2\rho \int_{0}^{\frac{\pi}{2}} r^2 r d\theta 2\rho \int_{0}^{\frac{\pi}{2}} 8R^3 cos^3\theta d\theta answer is \frac{32}{3}R^3\rho , and its wrong using the...
  45. L

    Find the angular acceleration of the rod

    Homework Statement Given: ##\mu_B=0.52## ##\theta=30^{\circ}## Weight- ##25## lb ##\omega=0## ## l=6## ft ##r_c=3\sqrt 2## radius of curvature. Homework Equations My Equations of motion are the following: ##\xleftarrow{+}\sum F_x=N_A\sin 60 - F_B=0## ##\downarrow{+} \sum F_y= N_A \cos...
  46. UMath1

    Bicycle Translational Acceleration vs Angular Acceleration

    Homework Statement Given: Wheel radius is 20 CM, Gear radius is 5 CM, Coefficient of Static Friction is .2, Weight on rear wheel is 50 N. What is the minimum force that must be applied to the pedal for the wheel to begin accelerating on a level surface?Homework Equations Net T = I * a T = R X...
  47. M

    Is my procedure correct to compute this moment of inertia J?

    Looking at the image below, we have the function z^2=\frac{r^2}{a} yrevolving about y-axis. We know that y goes from 0 to "a". We also know the mass of the object (of uniform density). Do you think is my procedure correct? Do you get the same result?
  48. L

    Why Are There Different Units for Moment of Inertia?

    Hi, I am a bit confused about the units of MI...I read that MI of rectangle is b*d^3/12,so unit is mm^4..Also i read in wiki that unit of MI is kg.mm^2... Both are correct,but why this difference is?
  49. S

    Moment of inertia (formula problem)

    1. Material 2. Questions: a) (pink) Why does the author use two different values of inertia for the same slender rod ? The Attempt at a Solution [/B] a) I could assume that 1/3 is holding it at the end and 1/12 is holding it in the center. But it's not interchangeable because if I chose 1/3...
  50. K

    Moment of inertia of a non-homogenous cylinder

    Homework Statement What is the moment of inertia round the axis of a cylinder length L, mass M and a linear distribution of mass with the radius R, zero at the center. Homework Equations Moment of inertia: ##I=mr^2#3 The Attempt at a Solution The density ρ=kr. what is k? the total mass is M...
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