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. I

    Angular Momentum & Moment Of Inertia

    Homework Statement Two children on opposite ends of a merry-go-round of radius 1.6 m throw baseballs at the same speed of 30 m/s but in opposite directions as shown. The mass of each baseball is 0.14 kg, and the moment of inertia of the merry-go-round and children combined is 180 kg-m^2. If...
  2. H

    Moment of Inertia vs. Inertia Constant

    The following equations are found in the following reference (Page 119): http://www.eeh.ee.ethz.ch/fileadmin/user_upload/eeh/studies/courses/power_system_dynamics_and_control/Documents/DynamicsPartI_lecture_notes_2012.pdf By definition, the inertia constant for a synchronous machine is...
  3. J

    Moment of inertia of a suspended cylinder

    Homework Statement A uniform cylinder 20 cm long, suspended by a steel wire attached to its mid-point so that its long axis is horizontal, is found to oscillate with a period of 2 seconds when the wire is twisted and released. When a small disc, of mass 10 g, is attached to each end the...
  4. G

    Moment of Inertia of spherical masses

    Homework Statement Three small spherical masses are located in a plane at the positions shown below. The masses are Q=0.700 kg, R=0.400 kg, and S=0.800 kg. Calculate the moment of inertia (of the 3 masses) with respect to an axis perpendicular to the xy plane and passing through x=0 and y=-3...
  5. T

    Finding rate of change of moment of inertia tensor

    Homework Statement The Wikipedia article on spatial rigid body dynamics derives the equation of motion \boldsymbol\tau = [I]\boldsymbol\alpha + \boldsymbol\omega\times[I]\boldsymbol\omega from \sum_{i=1}^n \boldsymbol\Delta\mathbf{r}_i\times (m_i\mathbf{a}_i). But, there is another way to...
  6. B

    Rotating parts of a motor have a moment of inertia

    Homework Statement The rotating parts of a motor have a moment of inertia of 15 kgm^2 and an optimum running speed of 1400 rev/min. When operating the motor is connected at optimum speed , by means of a clutch, to a shaft which has a counter rotation of 600 rev/min. The shaft has a mass of...
  7. B

    MHB Moment of Inertia: Motor, Shaft Speed & Torque Calculation

    The rotating parts of a motor have a moment of inertia of 15 kgm^2 and an optimum running speed of 1400 rev/min. When operating the motor is connected at optimum speed , by means of a clutch, to a shaft which has a counter rotation of 600 rev/min. The shaft has a mass of 80 kg and a solid...
  8. F

    Expression for the moment of inertia

    Homework Statement The thin, homogeneous bent rod has the mass m and the total length of 4b. It rotates with the angular speed of ω = ω0(24i + 12j - 6.0k) (only rotation) Determine the expression for the moment of inertia with consideration of the center of mass of the rod The figure...
  9. A

    Calculate the moment of inertia

    Homework Statement A mass m is tied with a light string,which it's another end is winded at a axle fixed at wall,in which it's radius is r. Assume there is no friction.The mass is released from rest and falls a distance S after time t. Find the moment of inertia of the axle.(represents I in...
  10. Z

    Relative difference between moment of inertia for the earth

    Homework Statement The Earth is slightly thicker around the equator and hence $I_{0}\neq I_{\zeta}$ I am curious in finding the angular velocity for the **precession** between using the fact that the distance between the spin axis and the precession axis is 10 meters on the surface of earth...
  11. kontejnjer

    Moment of inertia around displaced and rotated axis

    Homework Statement A homogenous disk with radius R and mass m lies in the xy plane so its center matches the origin O. Point O' is on the z axis at a distance s from point O. Axis y' passes through point O' at an angle \theta with the z axis. Find the moment of inertia around axis y'...
  12. J

    Small mass element for laminar (moment of inertia)

    Hi, I was just going over the moment of inertia for a 2D lamina, I've been happy with writing the small mass element dM as dM = ρdxdy where ρ is the area density, but for some reason decided on doing it like this, M(x,y) = ρxy so dM = \frac{∂M}{∂x}dx + \frac{∂M}{∂y}dy = ρ(ydx +...
  13. S

    Problem on moment of inertia

    Find moment of inertia of a uniform quarter disc of radius R and mass M about an axis through its centre of mass and perpendicular to its plane ... I tried in the following way: I considered the relation. I= Icm + Md2 Where d is the distance between required axis and centre of...
  14. K

    Moment of inertia of an ellipse

    Homework Statement To calculate I, the moment of inertia of an ellipse of mass m. The radius are a and b, according to the drawing. Homework Equations I=mr^2 Ellipse: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \Rightarrow y=b\sqrt{1-\frac{x^2}{a^2}} Area of an ellipse: \piab The Attempt at a...
  15. M

    Question on finding moment of inertia about the center of mass

    So I have a question. I (moment of inertia) is basically mr^2 right? And r is supposed to be the distance from the axis of rotation. When the axis of rotation is directly through the center of mass, how is there Icm (moment of inertia about the center of mass). It's confusing to me, because so...
  16. Dethrone

    Deriving the moment of inertia of a thin rod.

    Homework Statement I need to derive the moment of inertia of a thin rod with its axis of rotation at the end of the rod. http://en.wikipedia.org/wiki/List_of_moments_of_inertia The third one. Homework Equations I = mr^2 The Attempt at a Solution I completely understand how the...
  17. L

    Moment of Inertia of a Wheel

    Twelve uniform, thin rods of mass and length are welded together to form a “wheel” as shown in the figure. What is the moment of inertia of this wheel for rotation around an axis through its center and perpendicular to the plane of the wheel? The welds contribute no mass to the wheel. I...
  18. Feodalherren

    Moment of inertia, double integral

    Homework Statement Homework Equations The Attempt at a Solution For part B, why is he using the formula for the moment of inertia about the y-axis? Why isn't he using the formula for the moment of inertia about the origin...
  19. Rococo

    Moment of Inertia about axis through body diagonal of a Cuboid

    Homework Statement Consider a cuboid of lengths a, b and c along the x, y and z axes respectively, centred at the origin. The task is to show that the moment of inertia of the cuboid of mass M and mass density ρ about an axis along the body diagonal, from (-a/2, -b/2, -c/2) to (a/2...
  20. P

    Why is Ix/Iy used instead of Iz for the mass moment of inertia?

    Homework Statement for the mass moment of inertia, why did they use Ix/Iy and not Iz? Homework Equations Ix=Iy= 1/12 m (3(r^2) + (h^2)) Iz= 1/2 m (r^2) The Attempt at a Solution
  21. B

    What is the moment of inertia of the flywheel?

    Homework Statement An energy storage system based on a flywheel (a rotating disk) can store a maximum of 4.4 MJ when the flywheel is rotating at 21,300 revolutions per minute. What is the moment of inertia of the flywheel? Homework Equations K= Ktranslational + Krotational Krot=...
  22. chongkuan123

    Rotational Energy, Moment of Inertia Problem

    Homework Statement Energy is released by the Crab Nebula at a rate of about 5×10^31W, about 105 times the rate at which the sun radiates energy. The Crab Nebula obtains its energy from the rotational kinetic energy of a rapidly spinning neutron star at its center. This object rotates once...
  23. G

    Find Moment of Inertia Around CoM: Summation Formula & Point Mass

    How do I find the moment of inertia around the CoM of an object when the axis of rotation is not through the CoM? When Are summation formula used in equations and what exactly constitutes a point mass? regarding moments of inertia?
  24. A

    Determination of moment of inertia

    Homework Statement I've been trying to find out what is the period os this kind of pendulum decribed here: http://www.eng.uah.edu/~wallace/mae364/doc/Labs/mominert.pdf The thing is, I've came to the same result shown in equation (11) but my reasoning it's different. I would even say that...
  25. N

    Moment of inertia and angular velocity

    Homework Statement (a) Calculate the moment of inertia I of the disc when it rotates about the pivot as shown in the figure. (b) If the disc is released from rest, determine the angular speed, ω, of the disc at its lowest point. Homework Equations a) Id = Icm + md^2 Icm = 1/2*M*R^2...
  26. G

    Hollow Sphere Moment of Inertia

    I need to find the moment of inertia of a sphere of radius ##r## and mass ##m## about an axis through it's centre. I've already done it and got the correct answer of ##\frac{2}{3}mr^2## however I have tried doing it using a different method to see if I get the same answer, but I don't, and I...
  27. P

    Calculating Mass Moment of Inertia in a Two Stage Planetary Gearbox

    hi everyone, help me, To calculate the mass moment of inertia at output shaft with respect to input shaft in the two stage planetary gearbox. Torque at input = 15 Nm input speed = 1440rpm stage I zs=14 zp=23 zR=61 fixed stage II zs=21 zp=40 zR=102 fixed
  28. G

    Moment of Inertia: Kinetic Energy, Momentum & Conservation

    I read that for a rotating body the kinetic energy ##E_k = \sum \frac{1}{2}mv^2 = \frac{1}{2}{\omega}^2∑mr^2 = \frac{1}{2}I{\omega}^2## where ##I## is the moment of inertia. If we did the same thing for momentum then ##P = ∑mv = \omega\sum mr## So why is angular momentum ##I\omega=\omega\sum...
  29. B

    What is the Relationship Between Moment of Inertia and Relativity?

    In classical mechanics you want to calculate the moment of inertia for hollow & solid: lines, triangles, squares/rectangles, polygons, planes, pyramids, cubes/parallelepiped's, circles, ellipses, parabola's, hyperbola's, sphere's, ellipsoid's, paraboloid's, hyperboloid's, cones & cylinder's...
  30. carllacan

    Moment of inertia of a cube along the diagonal.

    Homework Statement Calculate the moment of inertia of a cube which rotates along an axis along its diagonal. Homework Equations Moment of inertia definition: I = \int \rho (\vec{r}) \vec{r} ^2 dV Angular velocity vector; \vec{\omega}=\omega (\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}...
  31. H

    Moment of Inertia of Hollow Sphere about Center Axis x-y-z method

    Homework Statement Find the moment of inertia of a hollow sphere about a vertical axis through its center in terms of its mass M and radius R. Homework Equations I=\int r^{2} dm The Attempt at a Solution I've been curious about different methods for finding moments of inertia...
  32. D

    Change in orbit due to moment of inertia change?

    Hi, So recently I read about the massive 3 gorges dam changing the mass moment of inertia of the Earth to such an extent that the days will now be 60ns longer. Then I thought, how will this effect the orbit of the Earth about the Sun? Any thoughts?
  33. K

    Answer check please - Moment of Inertia Calculations

    Answer check please -- Moment of Inertia Calculations For question 1 I got T=mnut * r pulley * gravity For question 2 I got Isystem = 1/2 M(R23 + R24 + m (R21 + R22) First day of physics lab and I just wanted to double check that these are correct. Thanks.
  34. H

    Moment of Inertia question (dead simple)

    Homework Statement A thin disk is 100g. It's diameter is 20cm. It's thickness is 2cm. Rotation is about the central axis (ie. perpendicular to the symmetrical plane). Answer in kg*m2 Homework Equations I=(1/2)MR2 M is the mass R is the radius I is the moment of inertia The...
  35. binbagsss

    Moment of Inertia Tensor Cylinder.

    I am computing the \hat{I} - moment of inertia tensor - of a cylinder with height 2h and radius R, about its axis of symmetry at the point of its centre of mass. I am working in cartesian coordinaes and am not sure where I am going wrong. (I can see the cylindirical coordiates would be the...
  36. J

    Moment of inertia tensor for a laminar

    Hi, Consider a 2D laminar only rotating about the z axis, with the axis origin at the bottom left hand corner and adjacent sides coinciding with the z and x axes. so ω = (0,0,ωz) y = 0 I don't understand how the IXZ component is 0 to just leave the IZZ component?
  37. B

    Center of mass and moment of inertia of catenary

    Homework Statement A homogeneous catenary ##z=acosh(x/a)##, ##y=0## and ##x\in \left [ -a,a \right ]## is given. Calculate the center of mass and moment of inertia Homework Equations The Attempt at a Solution I started with ##x=at##, for##t\in \left [ -1,1 \right ]##, therefore...
  38. binbagsss

    Moment of Inertia tensor - displaced axes theorem:

    Ok, so the system consists of two massive spheres, m1 and m2, of radii a and b respectively, connected by a massless rod of length R, as seen in the diagram attached. The question is to calculate the moment of inertia tensor. Sol: Set the origin at the centre of mass . So that we are in...
  39. M

    Buckling in various planes, finding moment of inertia

    Homework Statement Homework Equations The Attempt at a Solution With this problem and in general, I am having difficulties knowing what should be the cubic and what shouldn't be from visual inspection, so in this case I can't tell why I_x is 1/12ba^3, as opposed to 1/12ab^3. How can I tell...
  40. KiNGGeexD

    What is the Moment of Inertia of a Flywheel?

    Ok me and my friend have clashed with solutions to calculate the moment of inertia of a flywheel and I think I have it after a while but I'm not so sure about my maths! The solution he ended up with was a lot larger and more complex :) really enjoying this forum:)
  41. M

    Moment of inertia of triangle about centroidal axis

    Homework Statement There is the moment of inertia about an x and a y-axis named, I_{x}, I_{y}. Then there is the moment of inertia about the centroidal x and y-axis named, \overline{I}_{x}, \overline{I}_{y}. Often we can look up these values in a table (like the figure included) and apply...
  42. W

    Understanding Moment of Inertia for a Beam Bending about the Y-Axis

    Homework Statement I'm supposed to solve for the maximum moment assuming the beam bends about the y-axis (not the z axis as shown in the image. Same image for different questions). I don't understand how to find the moment of inertia in this case. The solution gives the moment of inertia for...
  43. KiNGGeexD

    Moment of inertia of circular flywheel

    Hi I'm new to this forum so still getting used to it! I have to find the moment of inertia of a circular flywheel which has radius, a and mass m! But also had area density (mass per unit area) of ρ0, Where ρ0=ρa(1+r/a) I know moment of inertia is the integral of a^2 dm and in order to...
  44. S

    Moment of inertia of a cylinder about perpendicular axes

    Homework Statement Calculate the moment of inertia of a uniform, solid cylinder about it's perpendicular axes. The cylinder has length L, radius R, and total mass M. It is centered on the origin with the z-axis running through the center of it's circular faces. Homework Equations I =...
  45. A

    Moment of inertia lab graph reading assistance

    The problem I am currently trying to right a lab report on a experiment trying to find a flywheels moment of inertia. In the lab we had a fly wheel connected to a mass on a string, and measured the position of a point on the wheel as the mass was accelerating downwards. The data was analysed...
  46. N

    Moment of inertia of a spherical shell

    Hey. There's one thing I've always been wondering about when it comes to deriving the expression for the moment of inertia of a spherical shell. Namely, why is the length of the infinitesimal cylinder used in the derivations (like here ) equal to ##R d \theta##, instead of ##R d \theta...
  47. P

    How Does Mass Distribution Affect Rotational Dynamics of a Rod?

    Homework Statement A rod of mass M and length l rotates in a vertical plane about its centre which is on a frictionless, horizontal pivot. On the ends of the rod are point-like masses m1 and m2, where m1 != m2. a)moment of inerta about the center of the rod b)Determine the angular momentum...
  48. G

    Center of gravity & moment of inertia

    Dear forum, while standing, spreading your legs helps your stability because you have a wider base. but doesn't spreading your legs lowers your center of gravity, thus shortening the distance (r) from your center of gravity to the ground, and therefore lowering your moment of inertia = making...
  49. P

    Calculate the moment of inertia of an object

    So we just learned about moment of inertia in my first year physics class, and how to calculate it. Though I do know how to calculate the moment of inertia of an object, I don't really know what it is. I tried looking on wikipedia and the explanations just seem to be equations. What exactly is...
  50. D

    Moment of Inertia (Triangular Prism)

    Homework Statement A triangular prism (like a box of toblerone) of mass M, whose ends are equilateral triangles parallel to the xy plane with side 2a, is centered on the origin with its axis along the z axis. Find its moment of inertia for rotation about the z axis. Without doing any...
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