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. Benjamin_harsh

    Moment of Inertia: Calculation Method

    Ratio of moment of inertia of a circle and that of a square having same area about their centroidal axis is: Sol: both area and square have same area: ##a^2 = \large\frac{π}{4}\normalsize d^{2}; \large\frac{d}{a} =\large \sqrt\frac{4}{p}## Ratio of moment of inertia of a circle and that of a...
  2. P

    Determining the Moment of Inertia about an angle θ to the x axis

    I thought about solving it using components of IAB but since it is a scalar quantity it doesn't seems to be correct . I don't think Perpendicular Axis theorem will work as required Inertia is in the same plane.
  3. Benjamin_harsh

    Exploring the Mystery of Moment of Inertia

    While I am reading this article, it says moment of inertia is same at radius of ring and at the center of it. How is that possible?
  4. Physics lover

    Moment of inertia of a square along an axis inclined at an angle

    My attempt-:I extended the axis and made a triangle by joining other adjacent vertex to the line such that its angles are 15°,75° and 90°.I found the distance between the centre of square and upper vertex of triangle by using law of sines.And then i found out inertia along upper vertex of...
  5. L

    Spinning top's moment of inertia I (disc on shaft )

    I hope you guys can help me with this problem.. A top in the form of a flat, circular disc spins on a shaft that is inclined at an angle alpha to the vertical. Now I have to find the moment of inertia I for the disc about its centre on the shaft. My attempt was building I with spherical...
  6. EEristavi

    Energy Conservation in Angular motion / Moment of Inertia

    I write Conservation of Energy: Potential Energy loss(change): U = m g ##\Delta##h = m g (R+r) (1-cos##\alpha##) kinetic Energy gain(change): K = (##\frac {m v^2} 2## + ##\frac {I \omega^2} 2##) + (##\frac {M v_2^2} 2## + ##\frac {I_2 \omega_2^2} 2##) U = K m g (R+r) (1-cos##\alpha##) =...
  7. I

    General Physics I Question: Moment of Inertia Problem Difficulty

    Hi everyone, I am honestly pretty confused as to where to start with this. Can anyone give any pointers as to where I could possibly go, for (a) and (b)? Thanks!
  8. EEristavi

    Calculating the moment of inertia of Cone

    I'm Summing the Inertia of "donuts" with width dr and radius - r. I'm also "flattering" the cone into 2D and considering that each donut has different mass - because of the different height - h so: dm = 3 m h / (pi R2 H) dr I = ∫ dm r2 = 3 m h / (pi R2 H) r2 dr from triangle similarities H/R...
  9. R

    How is moment of inertia calculated and translated between units?

    Calculating moment of inertia and translating it between units, I've become confused. The example is a mass of 1kg at 2000mm from the pivot. The force is applied at 1000mm from the pivot. Basics as far as I'm aware: Moment of inertia = mass * Distance to center of rotation^2 Torque = Moment...
  10. Romain Nzebele

    Is My Calculation of Moment of Inertia Using the Parallel Axis Theorem Correct?

    The picture of the problem and my attempt to solve it are below. Please let me know if my resolution is correct. Thank you in advance.
  11. D

    Determine the acceleration of the cylinder axis if there is no slip

    So I first wrote the moment of inertia of the cylinder, since it says that it is thin-walled, I think that its moment of inertia is ##I=\eta mR^2##. After that I wrote the sum of torques, I think that there are three forces that cause torque, the two forces of friction, the one caused by the...
  12. LeoChan

    What is the principal moment of inertia tensor for a lamina?

    The moment of inertia tensor I found out is (1/3) (-1/2) 0 (-1/2) (4/3) 0 0 0 (5/3) The principal moment of inertia tensor I found out is (1/3-I) . (-1/2) . 0 (-1/2) . (4/3-I) . 0 0 . 0 . (5/3-I) det of principal of...
  13. D

    Find the acceleration and the work done by a force pulling a spool?

    I already solved the first part, but according the my book, the answer isn't quite correct. This is what I did. Finally, I ended up with ##a=\frac{F(r-R\cos\alpha)}{Rm(\gamma+1)}##. But according to my book, the answer is ##a=\frac{F(\cos\alpha-\frac{r}{R})}{m(1+\gamma)}##, what am I doing...
  14. D

    How to calculate the moment of inertia of a bar with two masses on it?

    I first tried to calculate the moment of inertia of the bar. The problem is that I don't understand exactly how are the dimensions of the bar. The fact that it has a width ##a## means that its height is ##a##, or that it has an unknown hight and the width is ##a##, like a parallelepiped? After...
  15. jaewonjung

    Moment of Inertia of this combined system?

    Moment of inertia for a rod rotated around one end is 1/3ML^2 using the parallel axis theorem Moment of inertia for a sphere is 2/5 MR^2 Itotal=Isphere+Irod=2/5MR^2+1/3ML^2 However, the answer is 2/5MR^2+1/3ML^2+ML^2 Why is there an extra ML^2 added to the moment of inertia?
  16. J

    Rotation of a Submerged Disc

    1. A 20-lbf disc with diameter 18" and thickness of 3" is held static while completely submerged in water. Upon release of a lock, the disc experiences a torque from a torsional spring that causes rotation about its center of mass along the x/y axis (think coin toss, not wheel). If the spring is...
  17. M

    Torque required to tilt 1200 kg mass?

    Object to be tilted = 1200mm diameter cylinder Cylinder height = 2.5 meter weight of tilting assembly = 1200 kg Cylinder is on platform which is "L" shaped and tilting point is on that corner of "L" as in attached image. Shaft diameter on which it is tilted id 63mm Angular acceleration = 0.05236...
  18. brotherbobby

    Rotation of a Solid Cylinder

    Problem : A cylinder of mass ##M## and radius ##R## rotates with an angular velocity ##\omega_1## about an axis passing through its centre of symmetry. Two small masses each of mass ##m## (small in comparison to the radius of the cylinder) are glued to either of the two circular faces of the...
  19. D

    Moment of inertia of a branched cantilever beam

    Hey there! I am working on a project concerning the mathematical modeling of nano-swimmers in a viscous medium. Assuming the nano-swimmers to be cantilever beams, the project involves calculation of moment of inertia of the said nano-swimmers. While calculating the moment of inertia of a simple...
  20. IanAlpace

    Moment of inertia of two rods (T shape)

    Homework Statement Hey, just wanted to ask a quick question about how to find the moment of inertia of two rods (of equal mass and length) attached together in a T shape (the axis is at the bottom of the vertical rod). My calculations got the correct result but I just wanted to check whether my...
  21. A

    Moment of inertia of a thin uniform rod

    I was thinking that if a uniform rod of mass M and length L remains static ,then it's centre of mass will be at L/2 from one end (e.g total mass assumed to be concentrated at L/2 ) But if this rod is moving with uniform angular velocity ω about an axis passing through it's one end and...
  22. K

    Problems calculating the New Moment of Inertia

    Homework Statement Posted Photo with Work done so far, but my MOI is off Homework EquationsThe Attempt at a Solution
  23. sams

    Questions Regarding the Inertia Tensor

    In Chapter 11: Dynamics of Rigid Bodies, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, pages 415-418, Section 11.3 - Inertia Tensor, I have three questions regarding the Inertia Tensor: 1.The authors made the following statement: "neither V nor ω...
  24. H

    Confusion about moment of inertia

    When a question only asks for the moment of inertia (of say, a T-section), do I have to find the moment of inertia with respect to both the x and the y axis?
  25. S

    Moment of inertia of door rotation

    Homework Statement A 23 kg solid door is 220 cm tall, 95 cm wide. What is the door's moment of inertia for rotation about a vertical axis inside the door, 17 cm from one edge? Homework Equations I = I_cm + MR^2 The Attempt at a Solution I = I_cm + MR^2 I = (1/12)(23kg)(0.95m)^2 +...
  26. S

    Moment of inertia about a perpendicular axis through its center?

    Homework Statement A 15-cm-diameter CD has a mass of 21 g . What is the CD's moment of inertia for rotation about a perpendicular axis through its center?Homework Equations I = (1/2)MR^2 3. The Attempt at a Solution I = (1/2)(0.21g).((15/2)^2) = 5.9*10^-4 kg*m^2 cant see what I am...
  27. R

    Moment of Inertia of an Ammonium Molecule

    Homework Statement The ammonium ion NH4+ has the shape of a regular tetrahedron. The Nitrogen atom (blue sphere) is at the center of the tetrahedron and the 4 Hydrogen atoms are located at the vertices at equal distances L from the center (about 1 Å). Denote the mass of the hydrogen atoms by Mh...
  28. S

    Rotational moment of inertia for 3 balls on a pipe

    Homework Statement Homework EquationsThe Attempt at a Solution i just don't know from where to take the R which is colored with brown, in the 2(MR^2), what is the R here exactly, because i took the radius of the sphere , but i get different answer althought it's near.
  29. J

    Subtracting an Assembly's MOI -- Help please

    It's been ages since I have done Moments of Inertia of complex objects! :( Can anyone help?? Say you have an assembly like an airplane and you want to just get the moment of inertia(MOI) of the tail section and you are given the Mass of whole plane,mass of tail section, moment of inertia of...
  30. J

    Rotational motion: Conservation of energy doesn't work....

    http://www.animations.physics.unsw.edu.au/jw/rotation.htm#rolling I have set up an apparatus similar to what the above link says (the first bit about brass object with shaft). So basically, the shaft is in contact when the brass is first rolling, then it suddenly accelerates when the edge of...
  31. N

    Inertia tensor of cone around its apex

    Im trying to calculate the principals moments of inertia (Ixx Iyy Izz) for the inertia tensor by triple integration using cylindrical coordinates in MATLAB. % Symbolic variables syms r z theta R h M; % R (Radius) h(height) M(Mass) % Ixx unox = int((z^2+(r*sin(theta))^2)*r,z,r,h); % First...
  32. P

    What is the moment of inertia for a water bottle?

    Hi, I'm setting up an equation for initial gravitational potential energy, and I'm just wondering what the moment of inertia, I, would be for a water bottle rolling down an inclined ramp? Is it mr2 or ½mr2 Or would it be something else?
  33. Physicsterian

    Effect of density on downward rolling spheres

    Homework Statement If two balls, being identical in volume, but different in density (one ball is made of iron, the other of aluminum) roll down from an inclined plane, which will reach the bottom first and which will cover a larger distance after having reached the bottom? IMPORTANT NOTE...
  34. kasoll

    Inertia tensor v.s pincipal axes moment of inertia

    Is there a method to calculate inertia tensor form principal axes moment of inertia? Like now we have moment of inertia: (Ix,Iy,Iz)=(20,18,25), and hot to calculate the inertia tensor like (Ixx,Ixy,Ixz Iyx,Iyy,Iyz, Izx,Izy,Izz)? I have read about this page several times, but still have no idea.
  35. M

    Thickness from Moment of Inertia

    Hi guys... I am creating a parametric spreadsheet to predict the behaviour of a sign post when subject to a specified wind speed for a uni assignment . Currently half way through creating the spreadsheet and one of the questions requires me to calculate a wall thickness of the pipe using the...
  36. S

    Rod w/ Pivot and Attached Mass Pendulum

    Homework Statement A solid rod of length L and mass M has a pivot through its center and is originally horizontal. Another mass 2M is then attached firmly to one end of the rod, and released. What is the maximum speed of the mass 2M attained thereafter? (Cornell 2009) Homework Equations Not...
  37. Krushnaraj Pandya

    Moment of inertia of a leftover square

    Homework Statement A square plate is of mass M and length of edge 2a. Its M.I about its centre of mass axis, perpendicular to its plane is equal to I(1). Four identical disks of diameter a are cut from the plane. The MI of leftover square about the same axis? Homework Equations 1) MI of...
  38. Math_Maniac

    Moment of Inertia of Solid Sphere - Proof

    So I have been having a bit of trouble trying to derive the moment of inertia of a solid sphere through its center of mass. Here is my working as shown in the attached file. The problem is, I end up getting a solution of I = (3/5)MR^2, whereas, in any textbook, it says that the inertia should...
  39. Krushnaraj Pandya

    Moment of inertia of a metal plate about three points

    Homework Statement Moment of inertia of a metal plate (center at origin) which was a square before 1/4th of it was cut off (4th quadrant), about three points a)- top leftmost corner in 2nd quadrant b)- origin and c) (where the right-lowest corner used to be in 4th quadrant) ranked in decreasing...
  40. symbolic

    Moment of inertia with system of rods

    Homework Statement Three slim rods with the same length L are placed in a inverted U shape, as the figure shows. The masses of the vertical rods are the same, while the third bar has the mass 3 times bigger than the first or the second rod. The rods' thickness should be ignored. Find the moment...
  41. R

    Where is the hinge point? & what is the moment of inertia of a square?

    Homework Statement :[/B] A uniform wire of linear mass density λ having three sides each of length 2a is kept on a smooth horizontal surface. An impulse J is applied at one end as shown in the figure. P is the midpoint of AB. Now answer the following questions. 1) The angular velocity of system...
  42. Raghavendar Balaji

    Moment of Inertia of an inclined disk

    Homework Statement 1. Homework Statement An uniform disk of mass m, thickness L and radius r is suspended from the ceiling by a wire. It is attached to the disk at a distance, d from the center of mass, so that it inclines at an angle A to the vertical. What is the moment of inertia of the...
  43. P

    Find the moment of inertia of a washer

    Homework Statement Find moment of inertia of a washer with a hole of radius r/2. Homework Equations Moment of inertia = integral of x^2 dm γ = m / L The Attempt at a Solution I think I can find the moment of inertia of the washer using integral of x^2 dm. So, I know my γ = m / L = m /...
  44. P

    Moment of inertia of a system of masses

    Homework Statement The moment of inertia about an axis through the center of mass of a system consisting of two masses 3.0 and 5.0 kg connected by a rod of negligible mass 0.8m long is Homework Equations [/B] I = I1 + I2 The Attempt at a Solution I added all the moments of inertia of the...
  45. P

    Moment of inertia of a meter stick

    Homework Statement A meter stick has a mass of 0.2kg. A small hole is bored in it at the 10cm mark so the meter stick can be hung from a horizontal nail. The moment of inertia of the meter stick around an axis at the 10cm mark is Homework Equations m = 0.2kg I = mr^2 The Attempt at a...
  46. V

    Moment of inertia of a half disk about an axis

    Homework Statement Consider a half disk (of uniform density) with the flat end lying on the x-axis, symmetric about the y-axis (i.e. being cut into two quarters by the y-axis). Calculate the moments of inertia about each of the axes. Homework Equations $$I_{rr}=\sum_{i}m_ir_i^2$$ The Attempt...
  47. jybe

    Angular momentum changing as mass moves to center

    Homework Statement A 25kg child is spinning on a merry-go-round of mass 150kg and radius 2m at a constant angular velocity of 1rev/s. The child slowly walks to the center of the merry-go-round. Treat the child as a point mass and the merry-go-round as a uniform solid disk, and neglect friction...
  48. yecko

    Moment of inertia formula help

    Homework Statement Homework Equations I=1/12*bh^2+Md^2 The Attempt at a Solution for the highlighted 600mm^2 (and the 300mm^2 below) in the middle of the page, when finding the moment of inertia, what the solution used is an area, yet for related formula required the use as for mass in the...
  49. B

    Parallel Axis Theorem Experiment

    Homework Statement I am currently working on a physics experiment to confirm the parallel axis theorem. To do this, I have the following setup: In this experiment I change the distance between the centre of the rotating disc and the central axis. I record the time for 5 complete rotations...
  50. B

    Finding the coefficient of kinetic friction on an incline

    << Mentor Note -- Two threads on the same subject have been merged >> I am a junior enrolled in IB Physics at the standard level at my high school. As a part of the curriculum we must perform an Internal Assessment (IA) which involves performing an experiment and performing calculations and it...
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