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 for door hinges

    How do we find out that the moment of inertia about the hinges of a door with a width w is I=M w2/3. I find it like this in a book and also in table for the moments of inertia of homogeneous rigid objects with different geometries (about a long thin rod with rotation axis trough end). Thank you.
  2. S

    Acceleration of objects related to moment of inertia

    Homework Statement A system is given below. Pulley A has radius R and mass 2M and pulley B has radius 1/2 R and mass M. If the pulley is assumed as solid cylinder, find the acceleration of each object Homework Equations τ = I.α F = m.a a = α.R The Attempt at a Solution Equation of...
  3. S

    Moment of inertia of solid slab

    Homework Statement Solid slab with moment of inertia 2 kg m2 is given the forces as shown. Find the angular acceleration if θ = 37°, R1 = 10 cm, R2 = 25 cm Homework Equations τ = I . α The Attempt at a Solution This is what I've tried although I don't know it is correct or wrong; I tried to...
  4. S

    Moment of Inertia of a Hemispherical Shell

    Prompt: Find the moment of inertia about the z-axis of the hemispherical shell of problem II-6. Additional Info.: -Problem II-6 states: the distribution of mass on the hemispherical shell z=(R^2-x^2-y^2)^1/2 is given by o(x,y,z) = (o/R^2)(x^2+y^2) where σ0 is a constant. Find an expression in...
  5. R

    Moment of Inertia of an L shaped beam

    Homework Statement I'm trying to work out the moment of inertia for this L shaped beam: http://dl.dropbox.com/u/630750/Screen%20Shot%202011-12-31%20at%2017.54.34.png Homework Equations (Included on my sheets below( The Attempt at a Solution I've made two attempts, the first...
  6. E

    Moment of Inertia of a non-uniform density paraboloid

    Homework Statement Find the moment of inertia of a paraboloid f(x,y)=x^2+y^2 whose density function is ρ(r)=cr=dm/dv. use mass M and height H to express your answer 2. The attempt at a solution I took the double integral ∫∫r^2 ρ r dr dθ to find the I of a single disk as a function of r...
  7. C

    Moment of inertia of triangle rotated about y axis

    Similar to my other thread https://www.physicsforums.com/showthread.php?t=557750 I think in this case I need to use triple integrals But what is the integrand and limits? Can someone please help me set it up? Here's all I know ∫∫∫dydx Please help I have my final next wednesday.
  8. A

    Calculating the Moment of Inertia of the Sun for Determining Rotational Energy

    I want to find the rotational energy of the sun. Erot=1/2*I*ω2 m=1.9891*10^30 kg r=6.955*10^5 km When I am using the formula for a sphere's moment of inertia: I=2/5*m*r2 I'm getting 3.848671797*1047 km*m2 Can i find the moment of inertia in another way? the moment of inertia needs...
  9. J

    Moment of inertia wheel and bug

    Homework Statement There is a wheel with spokes rotating on a horizontal axis about its center. the Wheel has a w(omega) of 6∏ and a radius of .04meters there is a bug on one of the spokes at exactly 1/2 radius or .02meters. the bug has a mass of .005kg. Homework Equations not really sure...
  10. P

    Moment of inertia of a solid sphere derivation.

    Hello! I'm trying to derive the formula for the moment of inertia of a solid sphere, and I keep running into a strange solution. I set up the infinitesimally mass of an infinitesimally thin "shell" of the sphere: dm = 4\rho\pir2 dr And then solved for the moment of inertia: I = \intr2dm =...
  11. C

    Moment of Inertia of a Small Sphere Revolving on a String

    Homework Statement A small sphere of mass 2.0kg revolves at the end of a 1.2 m long string in a horizontal plane around a vertical axis. Determine its moment of inertia with respect to that axis. Homework Equations Moment of inertia for sphere= (2/5)mr^2 The Attempt at a Solution...
  12. T

    Moment of inertia in n dimensions.

    I've just been thinking about moments of inertia in n dimensions and I just want to establish if this makes any sense: I'm considering doing a Monte Carlo evaluation of the moment of inertia of any n-ball - a solid sphere in n dimensions. Now I think you can say that the moment of inertia of a...
  13. D

    Finding moment of inertia of a flywheel using rotations, with friction.

    ive recently done a experiment which involved a flywheel which was rotated by the force of gravity on a mass which was wrapped around the flywheel axis. I have tried to calculate functions which will predict the behaviour as well as get the moment on inertia of the flywheel via torques of...
  14. S

    Moment of inertia tensor of three spheres

    Hi everyone Homework Statement I want to find out the moment of intertia tensor of the graphic below. Homework Equations parallel axis theorem The Attempt at a Solution We know the moment of inertia for one sphere, that's given, so I don't have to calculate it...
  15. Y

    Calculating Moment of Inertia of a cube

    Homework Statement Find the moment of Inertia of a cube (mass M, length L) around an axis going through one of the edges. Homework Equations I=Ʃmr2 The Attempt at a Solution Well I imagined the cube is formed of infinitely many square plates. The width of the square plate is dL...
  16. W

    Lab: Moment of Inertia - Find Torques & Slope for Masses 50-110g

    Homework Statement Mass m will be varied between 50-110 grams, in 20 gram increments. Homework Equations I need to find the torques of each run (50, 70, 90, then 110 grams). I already have the electronically-collected data for the linear accelerations of the masses: run 1: .050...
  17. sophiecentaur

    Steel Joists: Moment of Inertia?

    Steel Joists: Moment of Inertia? I thought it might well be worth while asking here, as the Forum is awash with knowledge about most things. I am planning to do a loft conversion and I need to know about the strength of steel beams (Universal Joists, in fact). I keep seeing the expression...
  18. B

    Calculating moment of inertia about a door hinge.à

    Homework Statement A solid door of mass 39.30 kg is 2.34 m high, 1.68 m wide, and 3.23 cm thick. What is the moment of inertia of the door about the axis through its hinges? If the edge of the door has a tangential speed of 76.5 cm/s, what is the rotational kinetic energy of the door...
  19. S

    Determine the moment of inertia of the wheel

    Homework Statement A cord 3.0 m long is coiled around the axle of a wheel. The cord is pulled with a constant force of 40N. When the cord leaves the axle, the wheel is rotating at 2.0 rev/s. Determine the moment of inertia of the wheel and axle. Neglect friction. (Hint: The easiest solution is...
  20. T

    What is the moment of inertia for a rod with an attached mass at the end?

    Homework Statement Consider a rod of length L and mass m which is pivoted at one end. The moment of inertia of the rod about an end is Irod = 1/3 mL2. An object with mass m is attached to the free end of the rod. Find the moment of inertia of the system with respect to the pivot point...
  21. G

    Moment of Inertia of a Conical shell

    Homework Statement Given that the conical shell has uniform density and thickness is made of one sheet, has mass M, height h, and base radius R, derive the moment of inertia about its axis of symmetry. Homework Equations I = MR^2 for a ring about its central axis. I = ∫dI - my approach The...
  22. M

    Moment of inertia for a thin rod

    A small 750 g ball on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. (a) Calculate the moment of inertia of the ball about the center of the circle. (b) Calculate the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts...
  23. C

    Find kinetic energy from potential energy and moment of inertia?

    Homework Statement A solid cylinder of mass 0.600kg is released from rest and allowed to roll down a board that has an incline of 30degrees to the horizontal floor. If the solid cylinder is released 0.60m above the floor, what is the cylinder's linear speed when it reaches the lowest end of...
  24. M

    Moment of Inertia Help: Calculating Away from Axis

    Moment of inertia help! I have a question about moment of inertia. How do you calculate an object that is away from the axis of rotation? I am trying to figure out how to find the total angular momentum of two disks. One disk is rotating around the origin. Disk 2 is connected to disk 1 and...
  25. B

    Moment of Inertia of Cabinet Door - 49.1x76.5cm, 550kg/m3

    Homework Statement A sheet of plywood 1.81 cm thick is used to make a cabinet door 49.1 cm wide by 76.5 cm tall, with hinges mounted on the vertical edge. A small 181-g handle is mounted 45 cm from the lower hinge at the same height as that hinge. If the density of the plywood is 550 kg/m3...
  26. R

    Proof for moment of Inertia of thin disc

    Homework Statement Its actually not a homework , i am just curious about this I have disc with a hole in middle , R1 is inner radius , R2 is outer radius , Mass M is the disc's mass . Let density of disc uniform . I did it this way . - λ.dA=dm λ.pie.r.dr=dm as Area=pie.r^2...
  27. G

    Calculating the moment of inertia

    I know that for for solid disks, the inertia is equal to (1/2)mr^2 and for hoops is just mr^2. This only works for if the mass is evenly distributed around though. So what about when the mass isn't equally distributed? How would you solve for the moment of inertia of a hollow disk if a small...
  28. E

    Finding Moment of Inertia of axle

    Homework Statement Not really a HW problem (no diagram), I just need to figure out this number for a design. I have an axle with two crank arms attached to either end, pointing in opposite directions. The entire thing will spin. I need to find the equivalent moment of inertia of the axle...
  29. T

    Moment of Inertia and velocity

    Homework Statement I was doing a experiment in class where I pretty much had to roll cylinders down a surface with a certain angle and then calculate the moment of inertia. After that, the teacher asked me to prove why moment of inertiea doesn't depend on the cylinder's velocity and I just...
  30. J

    What is the Moment of Inertia for a Door Rotating Around a Vertical Axis?

    Homework Statement A 32kg door that is 107 cm wide. What is the moment of inertia 10 cm inside the door with rotation around a vertical axis? Homework Equations L=Iω The Attempt at a Solution I=1/3mL^2 I= Icm+ md^2 1/3(32)(107)^2 Im not sure what to do after I find the I...
  31. P

    What is the derivation of the moment of inertia of a solid sphere?

    So here I just need help understanding a concept. I was reading this website's derivation of the moment of inertia of a solid sphere here: http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html#sph3 For the life of me I can't see where the (1/2)y2 term comes from. I would understand y2 because...
  32. X

    I with finding moment of inertia for pendulum

    Homework Statement A grandfather clock uses a physical pendulum to keep time. The pendulum consists of a uniform thin rod of mass M and length L that is pivoted freely about one end, with a solid sphere of the same mass, M, and a radius of L/2 centered about the free end of the rod. Obtain...
  33. M

    On the moment of inertia of a system of biparticles

    The moment of inertia of a system of particles, is given by: I_i^{\vphantom{^{\:2}}} = \sum_i \; m_i^{\vphantom{^{\:2}}} \, \vec{r}_i^{\:2} The moment of inertia of a system of biparticles, is given by: I_{ij}^{\vphantom{^{\:2}}} = \sum_i \; \sum_{j>i} \; \, m_i^{\vphantom{^{\:2}}}...
  34. C

    Angular acceleration and moment of inertia on a pulley

    Homework Statement Two forces are applied to a pulley with a moment of inertia I = .15kg*m^2. The pulley is mounted so that its frictionless axle is fixed in place. 1) What is the angular acceleration of the pulley? 2) If the pulley starts from rest, how long does it take to undergo...
  35. D

    How Do You Calculate the Moment of Inertia for a Disk Rotating About Its Edge?

    Homework Statement What is the moment of inertia of an 8 kg, 40 cm diameter disk for rotation through the edge of the disk? Homework Equations I = \frac{1}{2}MR^2 The Attempt at a Solution Inertia at the center of the disk would be I = \frac{1}{2}MR^2, right? I'm not sure what...
  36. V

    KE of Rotation & Moment of Inertia

    Homework Statement A light string is wrapped around the outer rim of a solid uniform cylinder of diameter 75.0 cm that can rotate without friction about an axle through its center. A 3.00 kg stone is tied to the free end of the string, as shown in the figure . When the system is released from...
  37. L

    Moment of Inertia for Ellipsoid

    Homework Statement a)Evaluate ∫∫∫E dV, where E is the solid enclosed by the ellipsoid x^2/a^2+y^2/b^2+z^2/c^2 =1. Use the transformation x=au, y=bv, z=cw. b)If the solid in the above has density k find the moment of inertia about the z-axis. Homework Equations ∅=phi The Attempt...
  38. M

    Moment of inertia of a stiff wire bent into a wheel with spokes

    Homework Statement A stiff uniform wire of mass M0 and length L0 is cut, bent, and the parts soldered together so that it forms a circular wheel having four identical spokes coming out from the center. None of the wire is wasted, and you can neglect the mass of the solder. What is the...
  39. T

    Mass Confined to Rotating Hoop with Moment of Inertia I3

    [b]1. Homework Statement A particle of mass can slide without friction on the inside of a small tube bent in a circle of radius r. The tube can rotate freely about the vertical axis, x3, and has a moment of inertia I3 about this axis. Homework Equations a) Derive the Hamiltonian for...
  40. J

    Vehicle Moment of Inertia? How?

    I have been playing with several vehicle dynamics simulators for a while now, and many of them require vehicle inertial properties to create realistic simulations. There is very limited published data out there so I was wondering how could one calculate these properties experimentally...
  41. mishima

    Moment of inertia of hemispherical shell

    Homework Statement The distribution of mass on the hemispherical shell: z=(R2-x2-y2)1/2 is given by σ(x,y,z)=(σ0/R2)(x2+y2) where σ0 is a constant. Find the moment of inertia about the z-axis of the hemispherical shell. Homework Equations I=∫r2dm The Attempt at a Solution...
  42. J

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

    Homework Statement Find the moment of inertia of a right circular cone of radius r and height h and mass m Homework Equations I = ∫r2 dm V = 1/3*π*r2*h The Attempt at a Solution Assume density is p dm = p dv divide both sides by dr dm/dr = p dv/dr dm/dr = p (d/dr * 1/3*π...
  43. K

    Moment of inertia of curved cuboid

    Hi Physics Forums! Moment of inertia question for you: I have a cuboid, like the first one in this link http://en.wikipedia.org/wiki/List_of_moments_of_inertia only it is curved around the y (w in pic) axis such that the x (h in pic) axis touches the top and bottom ends of the cuboid...
  44. T

    Moment of Inertia for Combination of Rod and Ball

    Homework Statement [PLAIN]http://www.smartphysics.com/images/content/mechanics/ch15/momentofinertia2new.png What is the moment of inertia of the object about an axis at the center of mass of the object? (Note: the center of mass can be calculated to be located at a point halfway between the...
  45. K

    Moment of Inertia for Curved Cuboid

    Hi Physics Forums! Moment of inertia question for you: I have a cuboid, like the first one in this link http://en.wikipedia.org/wiki/List_of_moments_of_inertia only it is curved around the y (w in pic) axis such that the x (h in pic) axis touches the top and bottom ends of the cuboid...
  46. G

    Moment of Inertia of a Square?

    Homework Statement See below for the diagram. Use this result to calculate the moment of inertia of the shape below about the axis shown by the dashed line. The object is a square with a side of length L, total mass of 4M, and a uniform density. Homework Equations The Attempt...
  47. K

    Moment of inertia, x and y coordinates

    Homework Statement Consider a rigid body that is a thin, plane sheet of arbitrary shape. Take the body to lie in the xy-plane and let the origin O of coordinates be located at any point within or outside the body. Let I_{x} and I_{y} be the moments of inertia about the x- and y-axes, and let...
  48. K

    Moment of inertia of a semicircle

    Homework Statement A uniform disk of radius R is cut in half so that the remaining half has mass M. (Figure (a)) 1. What is the moment of inertia of this half about an axis perpendicular to its plane through point A? Express your answer in terms of the given quantities. 2. Why did...
  49. G

    Moment of Inertia of a rectangle?

    Homework Statement I was trying to find the moment of inertia of a rectangle with width a and height b were axis of rotation is through it's center of mass Homework Equations The Attempt at a Solution I = integral r^2 dm rho = dm/dA dm = rho dA If I take an infinitely small...
  50. K

    Moment of inertia of the body Question

    Homework Statement A hole of radius R/2 is cut from a solid sphere of radius R. If the mass of the remaining plate is M, then moment of inertia of the body about an axis through the centre is____. Answer:57/140*M*R^2 Homework Equations Moment of inertia of solid sphere is...
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