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. S

    Tubular Column Moment of Inertia

    Can anyone explain why the moment of inertia for a tubular column in that textbook is like so? (take a look at the attachments). It should be (I = MR^2), as far as I know.
  2. C

    Moment of Inertia of a Solid Cylinder With a Wedge Removed

  3. Ugnius

    Rotating body moment of inertia

    I have done some lab work , and now i have to answer some theoretical questions , but i can not find any data about this on the web or atleast i don't know where to search , i will add some pictures of experiment for you to better understand it. I was wondering can someone share their knowledge...
  4. MattGeo

    Torque and Moment of Inertia of a Lever Arm

    I never really considered this back when I was taking physics in college but imagine for the sake of thought experiment that you have an extremely and impractically long wrench and it is fixed to the bolt you wish to tighten. Now the longer the lever arm the greater the torque so if you double...
  5. C

    Confused by Unexpected Results: Acceleration & Moment of Inertia

    Like I said, objects with the higher acceleration are giving me the lowest values. For a hoop, I got I=0.1*MR^2 For a cylinder, I got I=0.7*MR^2 this seems backwards, no?
  6. T

    Moment of Inertia with pulley and two masses on a string (iWTSE.org)

    Note: the working (taken from iWTSE website) refers to inertia as the symbol ‘J’ (in-case there was any confusion).I found equations of motion for mass m and 2m which were ‘T1 = ma + mg’ and ‘T2 = 2mg – 2ma’, respectively. I know they are connected particles with the same acceleration ‘a’.I have...
  7. wcjy

    Moment of inertia of a uniform 2D triangular plate

    Answer is 37.8 g cm^2 new to latex
  8. andyonassis

    Moment of inertia (Perpendicular axis theorem)

    So i derived the moment of inertia about the axis of symmetry (with height h) and I am confused about the perpendicular axis theorem. The problem ask to find the moment of inertia perpendicular to axis of symmetry So the axis about h, i labelled as z, the two axis that are perpendicular to z, i...
  9. M

    Mass moment of inertia of a composite shape

    My thought process was to get the mass moment of inertia of the rectangle and then subtract the mass moment of inertia of the quartercircle from it. The MMoI of the rectangle is: (1/3)(0.005*7850*.6*.3)(.3^2)= 0.212 meters The MMoI of the quartercircle is: (1/4)(0.005*7850*¼π 0.3^2)(.3^2) + ? My...
  10. Nexus99

    Moment of inertia of an isosceles triangle

    I did in this way: ## I = \int dm \rho^2 ## Dividing the triangle in small rectangles with ##dA = dy x(y) ## where ##x(y) = 2 ctg( \alpha ) (h - y) ## we have : ## dm = \sigma 2 ctg( \alpha ) (h - y) ## Now i have ## \rho^2 = x^2 + (h-y)^2 ## Now I don't know what I can do because it would be...
  11. Nexus99

    Moment of inertia, center of mass and vincular reaction of a rod

    I'm struggling doing point 5, i have no idea how to solve that question. In point 1 i obtained the following result: ## I=\frac{ML^2}{2}## calculating the integral of dI, the infinitesimal moment of inertia of a small section of the rod of length dr. 2) Through the conservation of angular...
  12. Nexus99

    Moment of inertia of two coaxial disks

    The total moment of inertia is: ##I_{tot} = 2 M_1 R^2 + \frac{1}{2} M_2 R^2## We have ## M_1 = (4 \pi R^2) \sigma ## and ##M_2 = (\pi R^2) \sigma ## , where ## \sigma ## is the density of the disks. We also know that: ## \sigma = \frac{m}{ \pi 5 R^2} ## this leads us to say that: ##I_{tot} =...
  13. 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...
  14. N

    T-Shaped Pendulum: Find the Total Mechanical Energy

    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:
  15. 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...
  16. B

    Lead screw's moment of inertia

    In order to choose a DC motor according to the article titled: APPYING MOTORS IN LINEAR MOTION APPLICATION by PITTMAN - step 4 : "determine the total reflected inertia (Jt) back from the load to lead screw shaft " . The formula is: Jt = Jscrew + Jload. This calculation relies on the fact...
  17. 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...
  18. M

    Exploring Moment of Inertia & Angular Acceleration for Drone Propellers

    Hi, I am building a drone for a school project and I am also looking into how it flies. Recently I have been looking into angular momentum, torque, moment of inertia and angular acceleration. However I am struggling to understand moment of inertia and angular acceleration. If possible please...
  19. LCSphysicist

    How do calculate this moment of inertia?

    Inertia moment of a thin square side 2b about the center of mass... I put the coordinates in the center of the square and came to: Integral of (x²+y²)dm = Integral of (x²+y²)*(dxdy)M But, the interval of the integral is [0,b] to x and y And, since this consider just the integral of one...
  20. J

    Calculating Moment of Inertia from Ball Accelerations

    Hey guys. Im trying to figure out how to calculate the moment of inertia from a homogeneous ball based on a series of accelerations. The ball is released from the top of an incline plane (3.33 deg) and with a motion sensor, 5 values of acceleration where captured . Together with the radius and...
  21. cwill53

    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...
  22. LCSphysicist

    Moment of inertia of a cone -- Why do we divide the equation dI = dm*r^2 by two?

    Why we divide the equation dI = dm*r^2 by two when we calcule the moment of inertia of a cone relative to its symmetry axis?
  23. 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...
  24. Rongeet Banerjee

    Moment of Inertia for a Triangle with Masses at the Vertices

    If I take the three masses individually and try to calculate the moment of inertia of the system separately then I=(m*0²)+(m*(l/2)²)+(m*l²) =ml²/4 +ml²=(5/4)ml² But If I try to calculate Moment of Inertia of the system using its Centre of mass then As centre of mass is located at the the...
  25. Iqish

    Calculating Moment of Inertia for a Curved Rod with Respect to a Specific Axis

    Given:Thin, homogeneous, curved rod with radius of curvature 𝑅 See figure to the down. Find: The moment of inertia 𝐼𝑥′𝑥 ′ with respect to 𝑥′- the axis passing through the center of mass (point 𝐺). Can someone who can help me ?
  26. VVS2000

    Moment of inertia (experiment)

    So in the above image, I intend to find the moment of inertia of that black rotating object which rotates due to torque which is provided by placing mass on the pulley. But the thing is that this rotating object is kind of like a ball bearing kind of system and even for a small torque it starts...
  27. Eggue

    Moment of inertia of a rod bent into a square

    I tried to find the moment of inertia of 2 rods connected at the corners by adding up their moments of inertia: \frac{1}{3}(\frac{M}{4})a^2 + \frac{1}{3}(\frac{M}{4})a^2 = \frac{1}{6}Ma^2 I then tried to solve for the moment of inertia at the center of mass of the 2 rods using the parallel...
  28. Andrea Vironda

    I Polar moment of inertia

    Hi, A well-known part of the formula for calculating the deflection stress is ##I_z=\int \int r^2 dA## Usually a moment of inertia is something related to how difficult is to move an object. In this case is understandable but i don't understand the meaning of the double integral. Using ##r^4##...
  29. C

    Why Do Different Objects Share Similar Moments 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...
  30. 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...
  31. V

    Moment of inertia, conservation of angular momentum and energy

    When I solved the problem using the conservation of angular momentum, I have got the correct result (ω = 0.006 rad/s). However, when I tried to find the answer using the conservation of energy the result was incorrect and I do not understand why.
  32. B

    Instantaneous moment of inertia of an object

    Can someone guide me on how to approach this question? I tried to draw a quick diagram of what I think is happening here Does the question imply that this object is undergoing horizontal circular motion in the shape of a conical pendulum? Thanks for any help!
  33. B

    Rotational motion and finding the moment of inertia

    Here is the problem that I am finding difficult to answer I had tried using conservation of energy to do this question Where I know that the gravitational potential energy at the top of the slope equals to the sum of both the linear and rotational kinetic energy at the bottom of the slope...
  34. 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 =...
  35. H

    Engineering How to find the first area moment of inertia (Q) for an I-beam with flanges

  36. 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...
  37. A

    How Does Torque Affect Angular Acceleration in 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=...
  38. J

    Moment of Inertia: Thin Spherical Shell

    Homework Statement: Derive the formula for the moment of inertia of a thin spherical shell using spherical coordinates and multiple integrals. Homework Equations: Moment of Intertia is (2MR^2)/3 I = (2MR^2)/3
  39. 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...
  40. A

    What is the direction of the torque in this scenario?

    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...
  41. Heyxyz

    Moment of Inertia: Car vs Sphere

    Hello, I tried to put it in an equation, but it didn't really work out. In this situation, the car was about the size of a model, and, while not exact, the radius of each wheel couldn't have been more than like a centimeter. Conversely, the ball was like twice the size of the car and had a...
  42. 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
  43. jisbon

    Moment of inertia of 2 uniform thin rods

    So to start off, what I will do find the center of mass of each of the rods. So for the top rod, COM is at where y= 0.5 L and COM of the rod at the bottom is at x = 0.5 L. From there, how do I proceed in finding the moment of inertia using parallel axis theorem? Do I simply treat: ##I...
  44. kepherax

    Moment of Inertia of a Rectangular Picture Frame

    https://www.physicsforums.com/attachments/250905 I know the answer, but am not certain how they got Lsin(angle) for R?
  45. 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} ## = ##...
  46. 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...
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  47. jisbon

    Moment of inertia of this shape about the x-axis

    I've attempted this question, but the answer seems to be incorrect. Here's my workings: ##I=\int y^2 dm## - standard equation ##dM = \mu * dy * x## - take small slice and find mass of it ##x = 4y-16## - convert equation in terms of x to sub in later ##dM = \mu * dy * 4y-16## ##I=\int y^2 \mu *...
  48. 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...
  49. Prabs3257

    Rotating Pendulum: Is This Eqn Correct?

    is this eqn correct
  50. 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...
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