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. D

    Moment of Inertia: Calculate Centroid & Axis Inertia

    a. Determine the coordinates and of the centroid C of the area shown in the figure. b. Calculate the moments of inertia of the area shown about a horizontal and a vertical axis going through the centroid. See attachment for the object area !
  2. M

    Determining Mass of Moment of Inertia for Flywheel in IC Engine

    i have diagram of the turning moment in flywheel in internal combustion engine , i want to determine the mass of moment of inertia of the flywheel needed to keep speed fluctuation < or equal 1.5% at engine speed 560 rpm ? http://img534.imageshack.us/img534/1765/57306264.jpg
  3. J

    Finding the moment of inertia and torque for a spinning ball

    Homework Statement A ball is going around in a circle of radius 5 m. It starts from rest, and goes through an angle change of 26 rad in 5 seconds. The ball weighs 0.3 kg. Find its moment of inertia, and find the torque about the origin. Homework Equations The Attempt at a...
  4. e2m2a

    A Contradiction in Moment of Inertia Formulae by reductio ad absurdum?

    In physics the moment of inertia of a thin rod which rotates around an axis through its center of mass is : I cen = 1/12 m L sq (1) Where: m is the mass of the rod, L is the length of the rod. The moment of inertia of a thin rod which...
  5. H

    Did I do this right or get lucky? Deriving spherical shell moment of inertia

    Homework Statement Basically just to find moment of inertia of a spherical shell Homework Equations Moment of Inertia = Int(r^2 dm) The Attempt at a Solution I am in 12th grade and don't know much about integrals? Did I get lucky or am I right? I = Int(r^2dM) dM= (phi)(dA)...
  6. B

    Moment of inertia of a rectangular prism

    I was trying to derive the formula for the moment of inertia of a rectangular prism at its centre of mass. And ended up with this...
  7. M

    Finding Center of mass and Moment of inertia Experimentally.

    Hi, I am doing a science experiment. For that, I have made a small structure (sort of, almost pyramid like) using metal strips. I have calculated mathematically: its center of mass/gravity, moments of inertia matrix and principle MI axis ( usual formulae). My job first is to find these same...
  8. P

    Torque from moment of inertia

    I have an assignment that asks to find a torque from given angular velocity, mass and moment of inertia. For a turntable disk, mass is known as 0.04452199 kilograms The Angular velocity should be 106.81 (rads/sec) These values are from solidwork file (anyone familiar with Solidworks?) Center...
  9. C

    To which of the two cubes has a larger moment of inertia?

    To which of the two cubes has a larger moment of inertia? I think it's the right one, is it correct? How can I explain that without using the parallel axis theorem?
  10. A

    Moment of Inertia: Definition & Fluids | Explained

    Is moment of inertia defined for fluids?
  11. R

    How Do You Calculate the Moment of Inertia for a System of Particles?

    Hey everyone, Im solving a group of questions based on a system of particles, A(2,0) with a mass of 4kg, B(0,1) with a mass of 6kg and C(1,1 ) with a mass of 3kg. I have to find the centre of mass and also calculate the moment of inertia. To find the centre of mass I am attempting the...
  12. M

    Find Moment of Inertia of 4 Rods Constructing a Square: Explained

    Homework Statement When we put together 4 rod in case they will construct a square and when we hang it from the corner. How can we find its total moment of inertia? Length of rods: L Mass of rods: M Homework Equations Paralel Axis Theorem: I = I + Md^2 The Attempt at a...
  13. N

    Moment of inertia of a solid sphere calculation

    How would one calculate the moment of inertia of a solid sphere (of uniform density, rotating about the axis through its center)? I know it's 2MR^{2}/5 but I got 3MR^{2}/5 when I attempted to derive it (just for fun, because I find rotational dynamics rather interesting). I would prefer a...
  14. U

    Moment of Inertia of Solid Sphere

    This is not homework, just my practise question for an exam. I keep getting an answer of 3/5MR^2 when the correct one is 2/5MR^2, and that was only because I fluked it by deciding to change my intergrand a little bit. I've some sources including one from hyperphysics which suggests...
  15. D

    Finding moment of inertia

    a uniform solid block has mass 0.172 kg and edge lengths a=3.5cm, b=8.4cm, and c=1.4cm (c is the height of the rectangular solid).Calculate its rotational inertia about an axis through one corner and perpendicular to the large faces. i know the formula is integral of r^2 dm, but i have no...
  16. H

    Calculate Moment of Inertia for Sphere with Thick Walls

    1. Derive the moment of inertia of a sphere with thick walls.Homework Equations I=(.5)(M)(R^2+R^2) ? one is for the inner radius the other is for the outer radius. The Attempt at a Solution I believe this can be found by integrating a series of infinitely thin cylinders with different size...
  17. E

    Moment of Inertia of helicopter blades.

    I have built a model of rotating blades of a helicopter and want to mathematically calculate it's moment of inertia. I am unsure on how to do that. Its a simple model: a square wooden base with uniform thickness and 4 equally long blades made of styrofoam from the midpoints of the 4 sides of the...
  18. Pengwuino

    Moment of Inertia, relation to other moments?

    Is there a reason the moment of inertia is called the "moment" of inertia? A while back, for whatever reason, I remembered how the moment of inertia is formulated and I realized it had similarities to the e/m moments; that is the integration of the mass density multiplied by coordinates. In...
  19. J

    Gyroscopes, moment of inertia, rotational energy and friction.

    Homework Statement A gyroscope consists of a solid disk of diameter 1m and mass 100kg mounted on an axis passing through the center of mass of the disk. (a) Calculate the moment of inertia of the disk about its rotational axis. (b) What is the kinetic energy of rotation of the disk if its...
  20. D

    Moment of inertia for T shaped object

    h have a T shaped object (2-d) for which i need to find the moment of enertia, i used a parallel axis theorem, then to check myself i found i site which had a direct equation for it http://www.efunda.com/designstandards/beams/SquareTbeam.cfm what i did b1=100mm h1=12mm b2=12mm h2=75mm...
  21. D

    What is the Moment of Inertia of a T-Shaped Object?

    h have a T shaped object (2-d) for which i need to find the moment of enertia, i used a parallel axis theorem, then to check myself i found i site which had a direct equation for it http://www.efunda.com/designstandards/beams/SquareTbeam.cfm what i did b1=100mm h1=12mm b2=12mm h2=75mm...
  22. R

    Area moment of inertia and second moment of inertia

    Hi PFians What is the significance of area moment of inertia and second moment of inertia? Well, I know that it signifies resistance to bending but how can you explain practically. I mean if we see mass moment of inertia, we can say that it tells us how mass is distributed in 3d about an axis...
  23. E

    Moment of inertia of a disc

    Homework Statement A) a horizontal disc of diameter 12.0cm is spinning freely about a vertical axis through its centre at an angular speed of 72 revolutions per minute. a piece of putty of mass 5.0g drops onto and sticks to the disc at a distance of 4.0cm from the centre. The angular speed...
  24. K

    Moment of inertia and conservation of energy problem

    A uniform sphere is rolling without slipping on a horizontal surface at 2m/s. It then rolls without slipping up a hill to height h, where it briefly comnes to rest. Find the vertical height h. [The moment of inertia, I, of a sphere is 2/5 mR^2] where m is the mass and R is the radius of the...
  25. R

    Calculating Moment of Inertia of Grinding Wheel - 45 N Brake Applied

    Here is the problem I am working on. Here is the work I have so far, does this look right. A grinding wheel that has a mass of 65.0 kg and a radius of 0.500 m is rotating at 75.0 rad/s. (The carousel can be modeled as a disk and assume it rotates without friction on its axis) a) What is...
  26. G

    Hollow Sphere Moment of Inertia Help

    Homework Statement A hollow sphere has a mass of 15 kg, an inner radius of 12 cm and an outer radius of 18 cm. What is the rotational inertia (moment of inertia) of the sphere about an axis passing through its center? Homework Equations The Attempt at a Solution I = 2/3 MR^2 for a...
  27. U

    Moment of Inertia of a Quarter Disc

    1. Find the Moment of Inertia of a Quarter Disc which has mass M and radius R about the axis passing through the center (of original disc) and perpendicular to the plane. 2. The attempt at a solution I found the Moment of Inertia (I) of a disc about the axis passing through the center...
  28. D

    What is the Moment of Inertia of a Flywheel?

    Homework Statement The work done in accelerating a flywheel from rest to an angular speed of 14.5 revolutions per second is 22.0 kJ. What is the moment of inertia of the flywheel?Homework Equations ½Iω² = 22.0 kJThe Attempt at a Solution ½Iω² = 22.0 kJ I = (2)(22.0 kJ)/ω² ω = (14.5 revs/sec) =...
  29. M

    Moment of inertia and Kinetic Energy (Rotational Motion)

    Homework Statement A skater spins with an angular speed of 8.3 rad/s with her arms outstretched. She lowers her arms, decreasing her moment of inertia by a factor of 8.9. Ignoring the friction on the skates, determine the percent of change in her kinetic energy. Answer in percent...
  30. D

    What is the moment of inertia of Earth about its center?

    Homework Statement The density of Earth, at any distance r from it's center is approximately p(r) = [14.2 - 11.6 (r/R)] * 10^3 kg*m^-3 where R is the radius of the Earth. Show that the density leads to a moment of inertia I = 0.330 MR^2 about an axis through the center, where M is the...
  31. M

    Moment of Inertia and torque?

    Homework Statement The combination of an applied force and a frictional force produces a constant total torque of 39.9Nm on a wheel rotating about a fixed axis. The applied force acts for 4.5s, during which time the angular speed of the wheel increases from 3 rad/s to 12 rad/s. The applied...
  32. C

    Rotational kinetic energy and moment of inertia

    Homework Statement A 16 g CD with a radius of 6.0 cm rotates with an angular speed of 42 rad/s. (a) What is its kinetic energy? Homework Equations k=1/2Iw^2 The Attempt at a Solution k=1/2(0.016x0.0036)(1764)=0.05 J
  33. E

    Acceleration of a pulley. Torque, and moment of inertia.

    Homework Statement There are two blocks, one of mass m1 and the other of m2. They are connected by a rope that runs over a pully of radius R and interia I. m2 slides down a frictionless incline at an angle theta, and m1 hangs virtically while being pulled up by the tension in the rope. Find...
  34. J

    Moment of Inertia 2: Kinetic Energy, Speed & Acceleration

    Homework Statement a) A disk with radius R and mass M is rolling without slipping on a level surface, as shown. The moment of inertia about an axis perpendicular to the page and through the center of the disk is I. If the center of the disk is moving at speed v, what is the kinetic...
  35. J

    Calculating Moment of Inertia of Particles & Cubes

    Homework Statement a) Four point particles, each of mass m, are arranged in a square and attached by massless rods that run along the sides of the square, and along the diagonals. The length of each side of the square is a. The construction is then pivoted about an axis perpendicular to...
  36. A

    Center of Mass and Mass Moment of Inertia

    Homework Statement Using masses, springs a stop watch and ruler, determine the location of the center of mass and mass moment of inertia about an axis passing through the center of mass of a bicycle crank. The Attempt at a Solution I've been staring at this problem for a while now...
  37. D

    Simple Moment of Inertia Question

    Here is the picture of the problem: http://yfrog.com/3uwtfup I understand every other step except for the equation for moment of inertia. In my book the most basic equation for moment is ΔI = (r)^2 Δm but for this problem ΔI = (1/2)(r)^2 Δm. Why is that? Does it have something to do with the...
  38. D

    Moment of Inertia of a Disk and mass

    Homework Statement Figure (a) shows a disk that can rotate about an axis at a radial distance h from the center of the disk. Figure (b) gives the rotational inertia I of the disk about the axis as a function of that distance h, from the center out to the edge of the disk. The scale on the I...
  39. L

    Moment of Inertia Problems and Conservation of Energy

    I have this situation here: I'm given the masses for the objects, as well as the radius and mass of the pulley. I have to use conservation of energy laws to find out how fast the mass on the right is going the instant it hits the ground. I can find out what the alpha of the pulley is...
  40. K

    Calculating Moment of Inertia of Sphere w/ Mass 100kg & Radius 1m

    A solid sphere w/ mass 100kg and radius 1m is spinning on axle. A brake pad is used to slow it down to a stop. While braking a force of 1N is applied on the pad. The coeff. of friction b/t pad and sphere is 0.5. The sphere is initially spinning at 100 rev/s, how long will it take to stop...
  41. R

    Understanding the Physical Significance of Polar Moment of Inertia

    Hi guys Can anyone explain what is physical significance polar moment of inertia. Well i know it's formula e.g in case of shafts but not it's physical meaning.
  42. J

    Moment of Inertia about the line x=y

    Homework Statement Let \Omega represent a two dimensional material region who density is given by \rho(x,y). Establish an integral formula for the moment of inertia of the material region about the line y=x. Homework Equations The moment of inertia about the x axis...
  43. V

    Calculate Moment of Inertia - Stick and Ball

    Homework Statement What is the moment of inertia about the y-axis for the system as shown. The stick has a length of 2L and a mass of M. The solid sphere has a diameter of L and a mass of 2M. Homework Equations I=\intr2dm \lambda=\frac{m}{l} The Attempt at a Solution Should...
  44. C

    What is the moment of inertia of the particle at the end of the meter stick?

    Moment of Inertia A particle of mass 0.400 kg is attached to the 100-cm mark of a meter stick of mass 0.100 kg. The meter stick rotates on a horizontal, frictionless table with an angular speed of 4.00 rad/s. Calculate the angular momentum of the system when the stick is pivoted about an axis...
  45. W

    What Is Moment of Inertia in Rotational Motion?

    please help me to understand moment of inertia
  46. C

    Question on total moment of inertia

    Homework Statement A lawn sprinkler has three spouts that spray water, each r = 15.5 cm long. As the water is sprayed, the sprinkler turns around in a circle. The sprinkler has a total moment of inertia of 9.10 10^-2 kg · m2. If the sprinkler starts from rest and takes 2.60 s to reach its...
  47. M

    Moment of inertia and friction

    I'm looking at some wheel/tire combinations for auto racing, and trying to decide which combination would out-accelerate or have more grip cornering. combo 1: 225/45r15 tire on 15"x8" wheel; 34.2 lbs total weight; diameter 23.0" combo 2: 195/50r15 tire on 15"x7" wheel; 28.4 lbs total weight...
  48. A

    Moment of Inertia problem with multiple masses

    Homework Statement A string is wrapped around a uniform disk of mass M = 2.5 kg and radius R = 0.04 m. Attached to the disk are four low-mass rods of radius b = 0.09 m, each with a small mass m = 0.6 kg at the end in an "x" configuration through the center of the disc. What is the moment of...
  49. E

    Product moment of inertia of an inclined section of a beam

    Hi guys, I would like to derive the Ixy equation from simple integration and I can't seem to get the right answer (third equation down the picture). I seem to be able to derive Ixx and Iyy easily but product moment of area requires first moment of area to be calculated and I just don't know...
  50. Q

    Moment of inertia of solid sphere

    Hi, i am trying to find the moment of inertia of a uniform density solid sphere about z-axis I = integrate => x^2 dm x = perpendicular distance from z-axis to anywhere in sphere so by pythagorus theorem, r^2 - z^2 = x^2 since dm = p dV and V = 4/3 (//pi)r^3 dV = 4(//pi)r^2 dr so I =...
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