Moment of inertia Definition and 1000 Threads

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. C

    How to Calculate Aircraft Pitching Moment of Inertia

    Hi everyone, I have a question which requires me to calculate aircraft pitching moment of inertia. I looked everywhere and can't find much about it, just the basis "what is inertia etc". Can anyone give an example on how to calculate pitching moment of inertia. Thanks a lot
  2. C

    What is the Moment of Inertia of a Cone about its Longitudinal Axis?

    Homework Statement Find the moment of inertia of a solid cone about its longitudinal axis (z-axis) The cone: x^2+y^2<=z^2, 0<=z<=h I_z = \int\int\int_T(x^2+y^2)dxdydzHomework Equations Representing the cone in cylindrical coords: x=zcos\theta y=zsin\theta z=z The Attempt at a Solution...
  3. M

    Moment of inertia of a rod at an angle to the axis

    suppose a uniform rod of mass M and length L is at an angle B to the x axis, one end of the the rod touching the axis. wish to find moment of inertia about x axis. let the rod touches the axis at x=0. let D=density=M/L, and I will integrate along x axis, that means that at a distance x from the...
  4. S

    Polar moment of inertia in a rod?

    How is the polar moment of inertia in a rod calculated? Thanks.
  5. S

    Polar mass moment of inertia *and* simply moment of inertia

    Hi ,everyone… i wanted to clear my one confusion ,that what is actual difference between polar mass moment of inertia and simply moment of inertia…??
  6. P

    Acceleration, Uniform Ball on Incline

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  7. T

    Calculating the moment of inertia of a filled cylinder

    Homework Statement For my IB higher level physics extended essay I will have to calculate the angular momentum of a cylinder rolling down a slope. The cylinder is made out of copper and will be filled with a known mass of car engine oil. I think i can obtain the angular velocity fairly easily...
  8. T

    Tensor moment of inertia -- why is there a "-" sign?

    why there is a negative sign in the tensor moment of inertia??
  9. gracy

    Moment of Inertia: Ignoring Small Bond Axis Molecule

    Why the moment of inertia of a molecule about the bond axis is small enough to ignore ?
  10. 462chevelle

    Moment of Inertia power plant energy

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  11. R

    What is the correct way to find the inertia of a bullet entering a door?

    I am having trouble with this one problem. I have tried multiple times to solve it, but come up with the same wrong answer every time. The mistake that I may be making is in finding the Inertia of the bullet entering the door: I figured that some mass at the given radius from the hinge of the...
  12. S

    Tipping cylinder tube with balls in it

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  13. R

    Moment of Inertia - center of a rod w/ unequal distribution

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  14. andyrk

    Why Can't Moment of Inertia Exceed MR² for Solid Bodies?

    How do we know that moment of inertia cannot exceed MR2 for a solid body like solid cylinder, solid sphere, hollow sphere etc? R is the radius of the object in consideration. I know it has something to do with- that for solid bodies (in some cases hollow also) there are a lot of point masses in...
  15. A

    Torque v. Time Graph -- Find angular velocity

    Homework Statement An object whose moment of inertia is 3.00kgm2experiences the torque shown in the figure (Figure 1) . What is the object's angular velocity at 2.80s ? Assume it starts from rest. http://session.masteringphysics.com/problemAsset/1073771/4/knight_Figure_13_21.jpg Homework...
  16. E

    Finding Rotational Speed once the Moment of Inertia Increase

    Homework Statement A trapeze artist performs an aerial maneuver. While in a tucked position, as shown in Figure A, she rotates about her center of mass at a rate of 6.03 rad/s. Her moment of inertia about this axis is 16.1 kg·m2. A short time later the aerialist is in the straight position, as...
  17. T

    How Do You Calculate the Moment of Inertia for a Door?

    Homework Statement A uniform, thin, solid door has height 2.2 m, width .87m and mass 23kg. Find the moment of inertia for rotation on its hinges. 2. Homework Equations I=∫x^2dm The Attempt at a Solution [/B]The way I set it up was that I made into strips of length H (height) and width dr...
  18. R

    Moment of Inertia about the CoM

    Homework Statement A puck of mass 82.0g and radius 4.09cm slides along an air table at a speed of v = 1.59m/s. It makes a glancing collision with a second puck of radius 6.97cm and mass 130g (initially at rest) such that their rims just touch. The pucks stick together and spin after the...
  19. L

    Non-Uniform Gear Moment of Inertia

    I'm working on a project involving analyzing the motion of a reciprocating saw. This saw converts rotational motion from an electric motor to linear motion by means of a slider crank mechanism. The power transmission goes from motor and attached driving gear, to crank, to connecting rod, to...
  20. Crush1986

    Moment of Inertia Compass Needle

    Homework Statement You place a magnetic compass on a horizontal surface, allow the needle to settle, and then give the compass a gentle wiggle to cause the needle to oscillate about its equilibrium position. The oscillation frequency is 0.321 Hz. Earth's magnetic field at the location of the...
  21. S

    Moment of inertia and angular velocity of a uniform disk

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  22. Q

    Moment of Inertia of a rod and two spheres

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  23. C

    How to Graph Moment of Inertia and Frictional Torque?

    Homework Statement The moment of inertia (I) of the bicycle wheel about the axle, can be obtained graphically using the equation: rm(g-a) = I(a/r)+Frictional Torque. Explain how to form two variables so that, I and Frictional Torque can be obtained from a linear graph. Identify the constants...
  24. S

    Lagrangian mechanics: sphere inside a cylinder

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  25. D

    Calculate the moment of inertia for the system

    Homework Statement Calculate the moment of inertia for the system. The rods of length L are massless. Image here: http://imgur.com/O639ejf Homework Equations /answer[/B] I know that I=ΣMR^2 The correct answer to the problem is 32ml^2 I JUST WOULD LIKE TO SEE HOW THE PROBLEM IS DONE...
  26. L

    The moment of inertia of a rod that is rotating off the end of the rod.

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  27. S

    Proof of disk moment of inertia using area density

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  28. M

    A spinning space station exhibits a change in moment of inertia.

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  29. C

    Moment of Inertia of a solid sphere

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  30. Aristotle

    Question relevant to Moments of Inertia

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  31. K

    Complex Moment of Inertia Calculation

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  32. T

    Unequal tension of a rope in a pulley system

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  33. M

    Calculating moment of inertia for nonuniform sphere

    Homework Statement A sphere with radius R = 0.200 m has density ρ that decreases with distance r from the center of the sphere according to \rho = 3.00 \times 10^3 \frac{kg}{m^3} - (9.00 \times 10^3 \frac{kg}{m^4})r a) Calculate the total mass of the sphere. b) Calculate the moment of inertia...
  34. N

    Moment of Inertia, working backwards

    I've to work out b & d in this equasion. I=bd^3/12 I know that the ratio between b & d in this case is 1.166 and I am given the value of I. It is an engineering problem as the software will only allow the use of square/rectangular or round section but we are using H section and we are allowed...
  35. P

    Moment of inertia about a ceiling fan?

    Homework Statement A ceiling fan consists of a small cylindrical disk with 5 thin rods coming from the center. The disk has mass md = 3.3 kg and radius R = 0.26 m. The rods each have mass mr = 1.2 kg and length L = 0.72 m. What is the moment of inertia of each rod about the axis of rotation...
  36. Hardik Batra

    How is the moment of inertia related to the square of the distance?

    Moment of inertia is I = MR2 object has more mass means more inertia and less mass means less inertia. That means moment of inertia is directly proportional to mass. But how the moment of inertia is directly proportional to the square of the distance.?
  37. M

    How Fast Must a Bullet Travel to Tip a Rotating Cube?

    Homework Statement A solid cube of wood with side 2a and mass M is resting on a horizontal surface. The cube is constraint to rotate about a fixed axis AB. A bullet of mass m and speed v is shot at the face opposite ABCD at a height of 4a/3. The bullet becomes embedded in the cube. Find the...
  38. N

    Moment Of Inertia Of a Weld Profile

    Im needing to calculate the Zxx and Yxx for 2 shapes. The shapes are an H profile and a + profile however its not the basic profiles I am needing but the moment of inertia for the weld profile surrounding them as the theory is they are welded to a vertical metal surface then a load is applied...
  39. Lord Crc

    Rigid body, torque and moment of inertia

    Hi, it's been quite some years since I took some entry-level mechanics at uni, and my book is no longer with me, so I'm struggling a bit here. I'm making a very simple rigid body simulator just for fun, but the angular velocity is not working out right. I suspect I've forgotten something...
  40. J

    Moment of inertia ice skater problem

    Problem: Consider an ice skater spinning in a circle. We will approximate the body as a cylinder with a radius of 0.25 m. The total mass of the skater is 50kg. When the skater has her arms extended we will approximate her body as a cylinder with two rods attached. Each arm is 0.75 m long and has...
  41. S

    Why Is the Mass of a Wheel Concentrated on the Rim?

    A small confusion ! Why is most of the mass of the wheel concentrated on the rim ? I know that it is to increase the moment of inertia , but what is the practical use of doing so ? Thanks in advance !
  42. B

    Moment of Inertia of Hollow Cylinder Derivation

    For a uniform, hollow cylinder, why is this derivation wrong? M = mass of whole solid cylinder m = mass of missing cylindrical piece R = radius of whole cylinder r = radius of missing cylindrical piece moment of inertia = moment of inertia of whole cylinder - moment of inertia of...
  43. B

    Moment of inertia and torque problem

    The Question : A cylinder of mass M , and moment of inertia around its center Io, is initially rotating around its central axis with angular velocity wo. İt is gently placed on a homogeneous thin plate of mass m which rests on a smooth horizontal surface. Assume that the plate is sufficiently...
  44. deep838

    Is the Moment of Inertia strictly different

    I would like to know if there is any proof as to whether the moment of inertia for two bodies (the masses of each body are distributed on a line) about their respective center of masses, is strictly different. If not, can anyone provide me a link to where the work is computed.
  45. stateofdogma

    Moment of inertia of rectangle without dd int

    Homework Statement I found the moment of inertia of a rectangle, with the axis perpendicular and going through the center using double integration. But I also used what i thought to be an equivalent method, but it didn't work.Homework Equations dI=r^2dm \int dI The Attempt at a Solution I was...
  46. B

    Moment of Inertia for Half Disk

    1. Question: What is the moment of inertia of a thin, flat plate in the shape of a semicircle rotating about the straight side (Fig. 12.28)? The mass of the plate is M and the radius is R The diagram is attached for reference. Homework Equations Moment of Inertia: I = ∫R2dm...
  47. Greg Bernhardt

    What is the Moment of Inertia? A 10 Minute Introduction

    Continue reading...
  48. J

    Composite moment of inertia question:

    Good morning PF, I am having some difficulty finding info on calculating a composite moment of inertia. The section I am working with is 8 separate columns (WT5x15) arranged evenly spaced around the circumference of a circle. You can see what I'm taking about in the attached jpg. I found...
  49. K

    Differential moment of inertia

    Homework Statement since it's asking us to find the moment of inertia with respect to y-axis, why did they draw the differential element to x-axis? Homework Equations The Attempt at a Solution because of the initial problem.. i don't know how to start.. so dA = x dy I(y) =...
  50. N

    Need help in proving this Moment of Inertia equation

    Homework Statement Suppose I have a rod with a know length, L, and 2 point-liked mass with know mass, m, each of them are fixed at both ends of the rod. And the rode has an axle connected at its midpoint where the axle is connect to a circular spring. The rod(with the 2 mass) is rotated 180...
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