Moment of inertia Definition and 161 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. Aaron Mac

    Torque needed to rotate a rack

    The weight of the rack is supported on an axial bearing as seen in the attached pdf below. I have made an attempt to calculate the torque by taking a look at the chain traction force and the required shaft power to make the plates rotate. For the moment of inertia case i don't know how to treat...
  2. AF Fardin

    Moment of inertia of a double physical pendulum

    I am having trouble to find the moment of inertia of the second rod! Is it related to the first rod?? At the beginning I thought It's not! But when took those as constant,the equation had become way much simpler and there is nothing about chaos! My approach is given below
  3. L

    Moment of inertia of a uniform square plate

    I placed my Oxy coordinate system at the center of the square, the ##x##-axis pointing rightwards and the ##y##-axis pointing upwards. I divided the square into thin vertical strips, each of height ##h=2(\frac{L}{\sqrt{2}}-x)##, base ##dx## and mass ##dm=\sigma h...
  4. V

    A problem on rotational motion and angular momentum (small disc rotating on top of a larger disc)

    I think the the time given doesn't matter since no torque is acting on the system, but not sure. Therefore, all we need is to determine the angular momentum about the axis passing through O and perpendicular to the plane of disk. This will involve finding the moment of inertia of smaller disk...
  5. V

    What is the "r" in the moment of inertia?

    We solved this problem in class as follows: Net torque about the center of the pulley taking counterclockwise rotation to be positive = m1gR - m2gR = I_tot α, where I_tot is the moment of inertia of the full system. My professor said that I_tot = I + m1R^2 + m2R^2, where m1R^2 is the moment...
  6. George Keeling

    I Time derivative of the moment of inertia tensor

    I am completely stuck on problem 2.45 of Blennow's book Mathematical Models for Physics and Engineering. @Orodruin It says "We just stated that the moment of inertia tensor ##I_{ij}## satisfies the relation$${\dot{I}}_{ij}\omega_j=\varepsilon_{ijk}\omega_jI_{kl}\omega_l$$Show that this relation...
  7. L

    Disk with rod attached rotating about the center of the disk

    1) Since the rod is uniform, with mass m and length l, it has a linear mass density of ##\lambda=\frac{m}{l}##, so ##I_{rod_O}=\int_{x=r}^{x=r+l}x^2 \lambda dx=\frac{\lambda}{3}[(r+l)^3-r^3]=\frac{\lambda r^3}{3}[(1+\frac{l}{r})^3-1]=\frac{1}{3}mr^2[3+\frac{3l}{r}+\frac{l^2}{r^2}].##...
  8. Tapias5000

    How to determine the same moment of inertia in two different ways?

    My solution is now I am asked for the same result but in this form but I don't know where to start.
  9. L

    Disk hit by two masses

    1) By conservation of linear momentum: ##m_1 v_1-m_2v_2=(m+m_1+m_2)v_{cm}\Rightarrow v_{cm}=\frac{m_1}{m+m_1+m_2}v_1-\frac{m_2}{m+m_1+m_2}v_2=\frac{3}{8}\frac{m}{s}##; 2) By conservation of angular momentum: ##-Rm_1v_1-Rm_2v_2=I_{total}\omega=(I_{disk}+m_1R^2+m_2R^2)\omega## so...
  10. L

    Tangential velocity of rotating rod

    1) ##LT\sin(\frac{\pi}{2}-\theta)-\frac{L}{2}mg\sin\theta=0\Rightarrow T=\frac{mg}{2}\tan\theta##. ##N_{x}-T=0, N_{y}-mg=0\Rightarrow N=\sqrt{N_x ^2+N_y ^2}=mg\sqrt{(\frac{\tan\theta}{2})^2 +1}## 2) ##E_{k_{fin}}=mg\frac{L}{2}[1+\cos\theta]## 3)...
  11. Ang09

    Moment of Inertia with varying distance from Centre of Mass

    h = d1 + 0.08 d1 = h - 0.08 d2 = h + 0.08 I of the vertical portion = 1/12 m (l^2 + b^2) + md1^2 = 1/12 m (0.28^2 + 0.04^2) + m(h - 0.08)^2 I of the horizontal portion = 1/12 m (l^2 + b^2) + md2^2 = 1/12 m (0.28^2 + 0.04^2) + m(h + 0.08)^2 The moment of inertia for the whole T-shape about...
  12. P

    Lagrangian mechanics - rotating rod

    Hello, It might sound silly, but when I try to calculate the kinetic energy of a rotating rod to form the Langrangian (and in general), why it has both translational and rotational kinetic energy? Is it because when I consider the moment of Inertia about the centre I need to include the...
  13. PiEpsilon

    Ball collides with the two rotating discs on a bar -- What is the resulting motion?

    What we know: The ball is dropped at the tip A with some speed ##v_0## and rebounds with speed ##v##. This collision produces an angular impulse, changing the angular momentum of the bar with the flywheels. Solution inspired by an answer provided by @TSny in the similar question. Angular...
  14. TheBigDig

    Determine the moment of inertia of a bar and disk assembly

    I have been given an answer for this but I am struggling to get to that point $$ANS = 0.430\, kg \cdot m^2$$ So I thought using the moment of inertia of a compound pendulum might work where ##I_{rod} = \frac{ml^2}{12}## and ##I_{disc} = \frac{mR^2}{2}## (##l## is the length of the rod and ##R##...
  15. P

    Moment of Inertia of a sphere about an axis

    I = 2/5M R^2 + Md^2 This is analagous to Earth's movement about the Sun. Is the moment of inertia of Earth about the centre of mass of the Earth Sun system = 2/5MR^2 + Md^2, where: M = Mass of earth, R = Radius of Earth, d = distance from Earth to centre of mass of earth-sun system.
  16. PiEpsilon

    Freely hinged rods on a table - Linear velocities of CM after the impulse

    We know that impulse is $$\vec J = \vec F \Delta t = \Delta \vec p$$ Let ##l, m## be the length of single rod and its mass respectively. Analyzing torques and forces on each rod separately we have: Rod ##AC##: $$F\Delta t +N_x\Delta t = mV_{ac,x} \space\space\text{ eq. }(1)$$ $$F\Delta t\cdot...
  17. Who_w

    Moment of inertia of a regular triangle

    Please, I need help! I need to calculate the moment of inertia of a triangle relatively OY. I have an idea to split my triangle into rods and use Huygens-Steiner theorem, but after discussed this exercise with my friend, I have a question: which of these splits are right (picture 1 and 2)? Or...
  18. wcjy

    Moment of inertia of a uniform 2D triangular plate

    Answer is 37.8 g cm^2 new to latex
  19. G

    What is a "Torsional Constant"?

    The question was: I will also include the solution: So, what is the justification of the first formula [ω=√(C/I)]? I know how to derive simple harmonic equations, this one as I guess is probably similar? But I cannot connect as to how C is used exactly. And the second formula [ω'=ωβ], I...
  20. N

    T-Shaped Pendulum

    My answer to part A is correct but for Part B I got an incorrect answer of 0.204J. My working out is sent as an attachment. Part A: Part B: Part C:
  21. P

    Minimum force required to rotate a lamina

    When the lamina rotates about A, FA must act on B (because it is the farthest away) perpendicular to AB (so that all of FA contributes to rotation). Same argument is valid for rotation of lamina about B as well. Having noted that, I tried two approaches: Approach 1- If I assume that the...
  22. Y

    Moment of inertia where mass and torque are at a different positions

    The formula for moment of inertia is: I=mr^2 A common derivation for this is: 1. F=ma 2. τ=rma 3. τ=rmrα = r^2 mα This is a rotational version of Newton’s second law, where torque replaces force, moment of inertia replaces mass, and angular acceleration replaces tangential acceleration...
  23. C

    How can I calculate the moment of inertia of any object?

    Apologies if I make anyone frustrated. To start, I've only had up to Calculus II so far but I was curious how to use and evaluate integrals used for moment of inertia. I know that the moment of inertia is basically an object's resistance to rotation, and is the rotational analog of mass. I know...
  24. A

    Stabilizer Leg Linear Actuator Force to Jack up a Truck's rear tyres

    Hi, I previously posted about the statically indeterminate truck problem. Thank you to everyone who helped me. However, I now realized that isn't the problem I need to solve. I need to know the force of the linear actuators to lift the rear tyres off the ground. Since the tyres will be...
  25. C

    About the Moment of Inertia

    Hello, I am a computer science major and Ex-Biology grad student, my knowledge in physics is humble, but I got a little curious when my professor derived the expressions of moment of inertia for different objects. The moment of Inertia of a thin disk is 1/2MR2, but it is the same as the moment...
  26. O

    Analytical mathematical modeling of the torque caused by the laundry

    Hello my dear physicists, I'm trying to model the varied generated (needed)Torque to rotate a washing machine Drum during a Washing Process so i assumed that the Model has as Input the target vilocity and as an Output the new needed torque to rotate the Drum(to be as a input for the motor...
  27. L

    Calculate the moment of inertia of this body

    I use the moment of inertia I = 1/12ml2 for an axis perpendicular and passing through the center of mass of a rod. In a cube built out of 12 rods I have 8 rods at a perpendicular distance l/2 from the axis through the midpoint of a cube. These 8 rods contribute the moment of inertia I1 =...
  28. CanFan

    B What sort of can would roll the fastest?

    Hello, I'm new here so apologies if this is in the wrong place! I was wondering, hypothetically, what sort of canned food would roll the fastest (assuming the ramp remained constant and the can couldn't be altered)? I've been looking this up and I know a solid can would roll faster than an...
  29. A

    Torque on a circular disk

    moment of inertia= [(1/2)(1.1kg)(0.96)^2+ (1.1kg) (0.75*0.96)^2]= 1.08 kg*m^2 θ=9.8 degrees= 0.17 rad torque= (mass*gravity) * radius * sin(theta) radius= 0.17rad * 0.96m = 0.16m torque= (1.1kg*9.8m/s^2) * 0.16m * sin(0.17rad) = 0.29 N*m torque = inertia * angular acceleration 0.29N*m=...
  30. dlxgains

    Moment of inertia project help- experiment 1: Bifiler Rod

    Homework Statement: So i need to find equations to help me with a bifiler suspension experiment in which i will use a rectangular drop bar as the oscillating object, also any help with the method of this experiment would be greatly appreciated. The end goal is to find the moment of inertia...
  31. A

    Moment of inertia of a tire

    I=(1/2)(6kg)(0.4^2 + 0.52^2) = 1.29 kg*m^2 initial: 34 mph= 15.2m/s 15.2m/s = (ω) (0.52m) ω= 29.2 rad/s after: 19 mph = 8.5m/s 8.5m/s = ω(0.52m) ω= 16.3 rad/s acceleration = (16.3 rad/s - 29.2 rad/s) / 5s= -2.58 rad/s^2 torque= |-2.58 rad/s^2 |(1.29 kg*m^2 ) = 3.3 N*m I am confused...
  32. A

    Torque and inertia of a space object

    I= 524kg * 6m^2 = 18.86E3kg*m^2 KE=(1/2) (18.86E3kg*m^2) (0.16 rad/s )^2= 241.5 J torque= 0.25* 57N* (6m)=85.5 N*m
  33. Celso

    Rigid body motion - thin disc

    Why is the gravitational potential energy of the chain's center of mass equal to the total kinetic energy of the disc after it was fully wrapped? My first thought was to write ##E_{0}=(M/2+M)g∗2πR=E_{f}= Ep## (from the chain) ##+Ec## (from the disc). Instead he wrote ## mg \frac{l}{2} ## = ##...
  34. The Bizarre Behavior of Rotating Bodies, Explained

    The Bizarre Behavior of Rotating Bodies, Explained

    Spinning objects have strange instabilities known as The Dzhanibekov Effect or Tennis Racket Theorem - this video offers an intuitive explanation. Part of th...
  35. Kaushik

    Moment of Inertia of a hollow sphere with Mass 'M' and radius 'R'.

    Homework Statement: Derive the formula for moment of inertia of a hollow sphere. Homework Equations: Required answer ##\frac{2MR^2}{3}## Consider a Hollow sphere. At an angle ##Θ## with the vertical, consider a circular ring whose moment of inertia is given by ##MR^2##. The most basic...
  36. Prabs3257

    Rotating pendulum

    is this eqn correct
  37. G

    I The time for a tilted brick to hit the ground

    I am interested in climate change and thereby interested in tipping points. So for the last few months, I have been investigating the dynamics of a toppling brick. I derived a differential equation of for the motion and wrote a computer program to solve this from initial conditions. I have more...
  38. 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...
  39. 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...
  40. 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##) =...
  41. R

    Moment of Inertia and 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...
  42. 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...
  43. 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...
  44. 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...
  45. 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...
  46. 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...
  47. 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...
  48. 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...
  49. 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?
  50. 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...