Moment of inertia Definition and 1000 Threads

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...

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...

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...

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...

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...

Homework Statement I need to know how to derive the equation for the moment of inertia of a rectangle rotated about an axis through its center. The rectangle has sides a and b. I know the equation to be (1/12)M(a2+b2), but I am having trouble deriving it. I have searched all over the...

I know that the moment of inertia of a disk about the axis perpendicular to the plane of the disk is I=1/2 MR^2 and by the perpendicular axis theorem the moment of inertia about the other axis is I=1/4 MR^2 I want to get the 2nd result with calculus. I can't get the right answer...

Homework Statement Landau&Lifshitz Vol. Mechanics, p101 Q1Find the moment of inertia of a molecule of collinear atoms Homework Equations The Attempt at a Solution I defined the origin alone the orientation of the molecule. I_3=0 obviously. For I_2 I wrote I_2=Ʃm_b[x_b-\frac{Ʃm_a x_a}{μ}]^2...

Hi, I have a question regarding a dynamics question that be found here:http://by.genie.uottawa.ca/~mess/files/MCG_2107/MCG2107_M2005_p2.pdf, It is the first problem on that link (labeled number 2.) I did ƩFx, ƩFy, ƩMG which allowed me to find angular acceleration, frictional force, and...

I have been trying to do the moment of inertia of a rectangle and I have it figured out when we have the center of the rectangle as the center of the rotation. The equation is ∫∫ρ(x^2+y^2)dy dx where the first integral is from -b/2 to b/2 (if b is the height) and the second integral is -a/2 to...

Homework Statement The Crab Nebula is a cloud of glowing gas about 10 light-years across, located about 6500 light years from the Earth (the figure (Figure 1) ). It is the remnant of a star that underwent a supernova explosion, seen on Earth in 1054 a.d. Energy is released by the Crab Nebula...

Homework Statement Find an expression for the moment of inertia (I) of the platform in terms of acceleration of the mass (a), the value of the hanging mass (m), the radius of the spindle (r) and the constant (g). Diagram: http://i.imgur.com/rv3zFYG.jpg Homework Equations F=ma t=Iα...

Greetings! I enjoyed the definition of moment of inertia for a volume and for an area in the form of matrix. It's very enlightening! I = \int \begin{bmatrix} y^2+z^2 & -xy & -xz\\ -yx & x^2+z^2 & -yz\\ -zx & -zy & x^2+y^2 \end{bmatrix}dxdydz '->...

Homework Statement A black ring is placed concentrically on a turntable and is being pulled by a constant force of 12.5 N and the force is applied at a distance of .01m from the spindle/pulley. It's known that the ring accounts for about 8.50% of the total moment of inertia. Find moment of...

Hi friends the problem is as: https://fbcdn-sphotos-d-a.akamaihd.net/hphotos-ak-ash4/1379445_1440214886205617_1276077735_n.jpg While finding moment of inertia of the system about the point from where the rod is fixed, the book says that M.I. of the system will be, I = Irod + Idisc I =...

Homework Statement Small blocks, each with mass m, are clamped at the ends and at the center of a rod of length L and negligible mass Compute the moment of inertia of the system about an axis perpendicular to the rod and passing through a point one-fourth of the length from one end...

Homework Statement Q is the edge length of the cube. The moment of inertia of the cube about an axis passing through its center and the center of two opposing faces is (1/6)mQ^2 (PCM in the diagram). Use the parallel axis theorem to find the moment of inertia if the axis is along one face of...

Generally, when we talk about moment of inertia, we talk about rotation and inherently, we talk about moment of inertia about an axis. But when we talk about inertia tensor, we calculate about a point. Is there a reason for this difference? Am I missing something? I am new to tensors.

I know some of you may have given up on me understanding moment of inertia/second moment of area, but here is another problem. I am using the this table to get the equations for the polar moment of area of a hemicircle around the x-axis, then I am applying the parallel axis theorem to find the...

]1. Homework Statement [/b] Homework Equations The Attempt at a Solution On paper it looks like not much of an attempt has been made, but I have tried a few different ways to find I_y. I thought to take the entire rectangle and subtract the parts on the upper left and right...

Hey, sorry for not using the exact template; I just have a general question about how to calculate the moment of inertia that I will have to apply to a number of instances. In this particular case, I need to calculate the moment of inertia for a rod pendulum. Of course, I could just use...

Hello, This is a very basic question. Wikipedia says moment of inertia I of a point mass m is calculated as: I = m * r^2 Where, r is distance from the axis of rotation. What this actually mean? I mean what is an r?

Hello, If I have a rigid cube rotated about an axis with dimensions L x W x H centimeters, how can we find the moment of inertia? Thank you.

Hi, I'm trying to find out what effect an increase in moment of inertia of a cylinder would have on slipping when the cylinder rolls down an incline. Starting with: ma=mgsin(theta)-f, where f is the friction force, which is less than or equal to µmgcos(theta), normal force times µ. We already...

Homework Statement A solid wood door 1m wide and 2m high is hinged along one side and has a total mass of 40kg. Initially open and at rest, the door is struck at its center by a handful of sticky mud with mass .5kg. What is the original and final moment of inertia for the door? What is the...

Homework Statement Find the moment of inertia of a rectangular plate of mass m, sides 2h and 2k , rotates about its diagonal. Homework Equations I= Ʃmr^2 The Attempt at a Solution I=Ʃ m(x^2 +y^2) , let 2h parallel to x-axis, 2k parallel to y-axis. I= m(x cosθ)^2 + m (y cos ∅)^2 where...

Homework Statement Four T-shaped objects are constructed by connecting two identical rods of equal mass and equal length. Rank in order, from LARGEST to SMALLEST, the moments of inertia for rotation about the axis indicated by the dashed line. Homework Equations The Attempt at a...

Homework Statement A cylinder with radius R and mass M has density that increases linearly with distance r from the cylinder axis, ρ = αr, where α is a positive constant. Calculate the moment of inertia of the cylinder about a longitudinal axis through its center in terms of M and R. This is...

Homework Statement If we multiply all of the design dimensions by a scaling factor f, it's volume and mass will be multiplied by f^3. (a) By what factor will its moment of inertia be multiplied? (b) And if a 1/48 scale model has a rotational kinetic energy of 2.5J, what will be the kinetic...

Hi everyone. I'm presently improving a 2D physics engine i wrote, with angular dynamics. Moment of inertia is becoming a big sticking point for me. However, given that my applications are fairly simple, I'm hoping someone can help me come up with a simple way of calculating the moment of...

Homework Statement There are four masses (particles) with a mass of 30gr each and they are at the vertices of a square that has each side with a length of 90cm. What is the moment of inertia of this system through a perpendicular axis that is at the center of the square?The Attempt at a...

Homework Statement A dumbbell has two spheres of radius 0.10m and mass 10kg (each). The two spheres are connected using a bar with length 1.0m and mass 12kg. What is the moment of inertia of the dumbbell about an axis perpendicular to an axis at the bar's center of mass? Moment of inertia...

Hello, I am trying to find moment of inertia of various random 3D shapes and I have few general questions. I take very thin 'plates' of the shape, calculate moment of inertia of that area, multiply it by thickness of one 'plate unit', also I add a square distance it is from the origin...

Hello everybody! This is the derivation (for a single particle). \tau = F_{\perp }r \ = ma_{\perp}r \ = \alpha mr^2 \\ \text{if }\ \tau = I\alpha \text{ where } I \text{ is resistance to accleration then } \\ I = mr^2 I'm curious what the problem with this is because I haven't...

Hi there, I am trying to calculate both volume and moment of inertia of shapes which surface coordinates are depicted by following equation: |xn| + |yn| + |zn| = Rn When n is 1 it is simple octahedron, when it is 2, it is sphere and then choosing any n, it becomes something else. I...

Minimizing the Moment of Inertia while keeping the Moment constant Hi there. I am dealing with a mathematical problem which seems to be much harder than I initially expected: Minimize the functional J(\Omega) = \frac{1}{\rho} I_{z} = \int \!\! \int \!\! \int_\Omega \left( x^{2} + y^{2}...

Hi, I read in a textbook that area Moment of Inertia (M.I.) of any shape would be Ʃ of (y2 χ A)1,2...n where, A represents the Area of strip no. 1, 2... n y is the distance between globlal centre of gravity (C.G.) of the shape to th local C.G. of strip no. 1, 2... n (pl. look at the image)...

Homework Statement Square lamina (of side a) of uniform density. Find I about a diagonal. Homework Equations I = ∫ dm*l^2 The Attempt at a Solution So I drew a square and its diagonal and I imagine a differential mass drawn somewhere on the lamina. The distance squared to that...

Homework Statement A rod is 10 ft long and has a density that goes from 4 to 24 lb/ft. a) Find the mass. b) Find the center of mass. c) Find the moment of inertia through the center of mass. Homework Equations I = Ml^2 I = I(cm) + Md^2 (parallel axis theorem) The Attempt at...

Homework Statement An oxygen molecule consists of two oxygen atoms whose total mass is 5.3x10^-26 kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is 1.9 x 10^-46 kg*m^2. From this data, estimate the effective distance between...

Hi everyone, Is there a general method for finding out the moment of inertia of an irregular convex 2D polygon given the coordinates of its vertices? I have thought of one possible method: Split the polygon into multiple triangles and find the moment of inertia of each triangle around the...

Why can't moment of Inertia be never greater than MR2 for uniform bodies with simple geometrical shapes?

Hi, I've attached a word document with my problem since I've used the Mathtype program (sorry, I didn't quite know how to use the tools on this forum), hope you don't mind :)

Homework Statement A hoop with mass m= 10 kg, radius r=0.6 m, and moment of inertia I=mr^2 is rolling without slipping with a linear velocity of 8.5 m/s. What is the total kinetic energy (in J). Homework Equations K=1/2 m v^2 Kr=1/2 I ω^2 The Attempt at a Solution The kinetic...

Regarding the moment of inertia constant, k, as in I = kmr^2, k is often a fraction less than one, and sometimes is equal to one. http://en.wikipedia.org/wiki/List_of_moments_of_inertia Is there an example of k being greater than one?

Homework Statement How do you calculate the moment of inertia of a regular hexagonal plate of side a and mass M along an axis passing through its opposite vertices?Homework Equations Moment of inertia for a right triangle with an axis running along its base would be I = m h2 /6 where h and m...

hello! I have to verify the formula of the moment of inertia of an ellipse about its' centroidal axis, is it M*(a^2 + b^2)/4.This is the one I got by myself. But in a webpage it was given as Pi*a*(b^3)/4. NOTE: don't ask for the proof of what I did.Its' a bit longer.I just want to know...

Homework Statement What must the ratio of height to radius of a cylinder be so that every axis is a principal axis (with the CM as the origin)? Homework Equations Moment of inertia tensor. I need I_yy = \sum m *(x^2 + z^2) The Attempt at a Solution I calculated I_zz = MR^2 /12...

Homework Statement A 4.00-m length of light nylon cord is wound around a uniform cylindrical spool of radius 0.500 m and mass 1.00 kg. The spool is mounted on a frictionless axle and is initially at rest. The cord is pulled from the spool with a constant acceleration of magnitude 2.82 m/s2...

Homework Statement The two small spheres of mass m each are connected by the light rigid rod which lies in the x-z plane. Determine the mass moments of inertia of the assembly about the x, y, z axes. I have attached an image of the question Homework Equations The Attempt at a...