Questions tagged [moment-of-inertia]

The moment of inertia, or rotational inertia, determines the torque needed for a desired angular acceleration about a rotational axis. Like inertial mass is the resistance to being linearly accelerated, the moment of inertial is the resistance to being rotationally accelerated.

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Moment of inertia of a cylinder with variable density [closed]

I have a cylinder with radius $R$ and a density that depends on the radius as follows: $ρ = ar$, where $a$ is just some constant. I tried to find its moment of inertia this way: $$ I=\int_0^R{r^2dm} \\...
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How to calculate the moment of inertia of convex polygon? (two-dimensions)

I've read some equations for a 2D polygon's moment of inertia using Green's Theorem (constant density $$I_y=\frac{\rho}{12}\sum_{i=0}^{i=N-1}(x_i^2+x_ix_{i+1}+x_{i+1}^2)(x_iy_{i+1}-x_{i+1}y_i)$$ $$I_x=...
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How does attaching a point mass to the CM of an object affect its moment of inertia about the CM?

How does attaching a point mass to the CM of an object affect its moment of inertia about the CM? Intuitively, it seems to me that this would not affect the mass distribution about the center of mass, ...
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Why can't we use integral of $x$, $y$ and $z$ in calculating moment of inertia

I've got no problems with calculating the moment of inertia/tensor of inertia of a cube using an integral over the lamina of a cube. However, I must be missing something obvious or making some sort of ...
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Are there physics theorems that can prove maths theorems? Eg Pythagoras' Theorem

There's this recent post on maths overflow Which theorems have Pythagoras' Theorem as a special case? that has an answer by dxiv that appears to use a physics theorem to a prove a maths theorem, ...
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How do the inertia tensor varies when a rigid body rotates in space?

The inertia tensor is clearly constant the in a frame moving with the rigid body. But what is the simplest way to see why its columns can be considered rotating vectors in space with the angular ...
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Moment of Inertia of a solid hemisphere. What am I doing wrong? [closed]

I want to calculate the MOI of a uniform solid hemisphere about Axis passing through its centre of mass (COM) and perpendicular to the circular base. Axis coinciding with any diameter at the ...
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Question on the proof of Moment of Inertia for a solid sphere

I was trying to prove myself the moment of inertia of a solid sphere. Here's it, Let us consider a sphere in $3$-dimensions with radius $R$ and mass $M$. Moment of inertia is the product of mass with ...
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Inertia tensor for rotors

For vectors we can use Inertia tensor. But if I want to use bivectors (Rotors), what should I use for the inertia tensor? I want to make a 2d game and progressively to 4d.
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Inequivalence of 2nd Moment of Mass and Moment of Inertia

My question is concerned with the difference between two ways of defining the moment of inertia, and how to interpret the difference. Let me begin by saying I understand moments of inertia are ...
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Moment of inertia when velocity is zero

The most basic form of the moment of inertia comes from fix axis rotation, that is $$L=\int rv\, dm=\int(r\omega)(r\,dm)=\omega \int r^2\, dm\tag{1}$$ Here $r$ is the perpendicular distance from the ...
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Doubt on the application of conservation of angular momentum

The following is the picture from the you tube channel physics desmos. In this the man is demonstrating the conservation of angular momentum. Now the explanation of it is also given in my book as ...
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What's exactly is moment of inertia?

I know that angular momentum can be expressed in terms of moment of inertia tensor as follows, $$\vec{L}= I_{\text{tensor}}\vec{w}$$ Where $I_{\text{tensor}}$ is tensor for moment of inertia. It can ...
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Does the angular velocity of a rotating rod change when its axis of rotation changes?

If a rigid rod of length $l$ rotates on one end around the origin with an angular velocity of $\omega$ and suddenly the end fixed to the origin is released allowing the rod to move freely without any ...
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Will an object undergoing translation move faster than if it underwent both translation and rotation?

So I'm revisiting my physics, as I'm looking to solidify my foundations. I was going over the chapters concerning rotation of rigid bodies and I just thought of this problem: If an impulse is applied ...
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Moment of Inertia of solid cylinder about the exist through it centre and perpendicular to its axis of cylindrical symmetry

I have seen the derivation of it first we find the moment of inertia of the disc about its diameter: I =(MR²/4l)dx and then we apply the parallel axis theorem <...
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Can anyone guide me on torque equation to rotate a part?

hope you all are doing good. I am a bit confused on which equation to use to calculate the torque required to rotate a part. I want to use a hydraulic motor at the 6th axis (end of robot arm) of ...
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Why is mass changed for moment of inertia in the formula for rotational kinetic energy?

The linear kinetic energy is $$\frac{1}{2} \cdot m \cdot v^2$$ and the rotational kinetic energy is $$\frac{1}{2} \cdot I \cdot \omega^2$$ Why is mass changed for moment of inertia in the equation for ...
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Moment of Inertia of Triangular loop made of rods only about an axis passing through one of the vertex and in the plane of loop parallel to the base

This question is from electrodynamics although I am stuck in the mechanics part here. $AB$,$BC$,$CA$ are wires/rods, we needed to calculate Moment of Inertia(MOI) about an axis $xx'$ parallel to BC ...
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Regarding simple pendulum

When I was going through the wikipedia article I have read clock pendulums are usually made of a weight or bob attached to the bottom end of a rod, with the top attached to a pivot so it can swing. ...
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Moment of Inertia of a Rectangular Parallelepiped

Moment of inertia calculated about an edge for a rectangular parallelepiped is given by $$I = (m/3) (a^2 + b^2), $$ my question is: when m(a^2+b^2) is added to I, the new value obtained is Moment of ...
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Question regarding $ΣM= Iα$ and an example problem given by my professor

In this example problem, an applied moment and two hanging masses of different values and in different positions caused a double pulley to rotate and angularly accelerate. As you can see, on the ...
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Why do objects with lower rotational inertia have more translational kinetic energy?

In my physics class, we have seen experimentally that objects with lower $I$ values (like spheres) will reach the bottom of a ramp sooner and with a higher final velocity than objects with higher $I$ ...
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Can passing an alternate current through an inductor be compared with speeding up and then slowing down a very massive object?

Can passing an alternate current through an inductor be compared with speeding up and then slowing down a very massive object? So, for example, when a capacitor discharges through an inductor the ...
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Moments of inertia

I know the concept of moment of inertia from $L=I \omega$, $\tau =I \alpha$, $I=mr^2$, and so on. I also know the that, in general, $I=\int r^2dm$ I would like to know how we derived the moments of ...
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Does moment of inertia tensor keeps changing if object is rotating about multiple axes?

Consider a plane circular disc kept in X-Y plane with Z axis passing through its centre. It is rotated about all threes axes with some angular velocities. In such a case, to find the inertia tensor, ...
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Why am i getting the wrong inertia moment tensor?

Let the x axis be parallel to one side of the rectangle, so as the z xis. Let the y axis be parallel to the normal. The x side of the rectangle is a and is 2a for the z axis. I should calculate $Ixx, ...
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Moment of inertia of a disc with rounded sides

I'm trying to formulate a dimensional equation for the MOI of a disc with rounded chamfer, where the chamfer is circular in shape and has radius $b$, and the thickness of the disc is the same as the ...
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Intuitive explanation for parallel axis theorem

The parallel axis theorem states that you can relate the moments of inertia defined with the center of mass as the origin to the moments of inertia defined with respect to some other origin. It is ...
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Does moment of inertia has to be with respect to a rotational axis?

Say we have a disc with its center at the origin. The disc has a mass M and a radius R, the density distribution of the disc is constant and is spinning as shown. The moment of inertia with respect to ...
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Why is radius of gyration for an ellipsoid derived from a tensor different than the formula?

The first method one can use to solve for radius of gyration, $R_g$, of an ellipsoid with semi-axes a, b, and c involves using the formula: $$ R_g = \sqrt{\frac{a^2+b^2+c^2}{5}}$$ However, it appears ...
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Inertia tensor as two times contravariant pseudotensor

I need to prove that the inertia tensor is a (2,0) pseudotensors, how can i do it? I keep getting that is a two times contravariant tensor, and that is what every source i checked says.
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Confusion regarding angular momentum of a tilted rod

I have a question regarding angular momentum and torque in the following example. This is not a homework problem, I just believe understanding some parts of this particular problem would help me ...
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Ball hits a rod

So let's imagine we have a homogeneous rod of length L and mass M and a tiny ball of mass m, all lying on a smooth horizontal table. Rod is stationary and ball is projected toward the rod at the right ...
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Is torque dependent on the moment of inertia?

I think this is why more force is required for a shorter distance; the inertia causes the object to stop, and greater force is required to cover the angular displacement.
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If you hang a mass with a rope on a wheel, how do you calculate the resulting moment of inertia?

How do you calculate the moment of inertia of the left object with axis of rotation in M. Is it the moment of inertia of the wheel $I$ and the $m *r^2$, where $r$ is the position vector of $m$ w.r.t. ...
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Guidelines to calculate moment of inertia

The moment of inertia is defined as $$I = \int r^2 dm$$ but I am not sure how to proceed with solving the above integral. All examples I have seen seem to be done with different strategies. They ...
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Moment of inertia of tetratomic molecule

I am reading Landau's Mechanics. About the problem 1(c) on page 101, Ch.32, with all masses equal and the molecule being a regular tetrahedron, the solution gives that all principal moments of inertia ...
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Confusion in using angular momentum conservation to solve this problem

Suppose I have spherical shell kept on a rough horizontal surface. The radius of this shell is given as $R$. Let us call the center of the sphere $O$. At some height $h$ above the center of the sphere,...
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How do you calculate the moment of inertia of a 3 quarters full cylinder? [closed]

If the cylinder's cross-section is as followed: Assuming that the bottom part of the cylinder is filled because it is filled with water. What would be the moment of inertia about the center of the ...
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Compute angular acceleration from torque in 3D

I am coding a simulation in which a force is applied to the corner of a cube Here is a picture to understand the problem better, the force is represented by the segment IF I first developed the ...
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Torque and moment of inertia with bikes

Can someboby please explain to me why it is better for a bicycle/ motor cycle to have lower center of mass when it rounds a turn. What I could gather from the rule I = mr2 Was that the arm is shorter ...
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Valence of the Moment of Inertia tensor

In "euclidean" classical mechanics the distinction between vectors and covectors is rather blurry due to the "trivial" nature of the isomorphism between tangent and cotangent ...
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Moment of inertia of the Cavendish balance

In her paper, Henry Cavendish: The man and the measurement, Isobel Falconer uses, Newton's force equation as, $$\mu \theta = 4 G M m a / d^2$$ for the force that turns the arm 1 radian. Where does she ...
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Derivation of Moment of Inertia of a solid sphere [closed]

I was deriving the moment of inertia of a solid sphere taking a solid disc as an element opposed to a hollow sphere. During derivation I found a problem that the integrand was wrong as it should've ...
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How many types of inertia are there?

I was looking for types of inertia, but I am confused. My book says there are three types of inertia, namely inertia of rest, inertia of motion, and inertia of direction. But when I searched for these ...
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Can I represent an Inertia Tensor with a Quaternion?

We can store three-dimensional rotations in 3x3 matrices, and in quaternions. since we usually store angular Momentum within a 3x3 matrix, would it be possible to use a quaternion to store/repersent ...
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Why can we calculate moment of inertia, but not inertia?

I'm learning about rotational motion and the moment of inertia. Unlike inertia that I learned before, there is a formula to calculate rotational inertia. I'm having trouble understanding why it's ...
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What is the distance travelled by a rolling object which is released from an inclined plane?

Suppose that a solid sphere with moment of inertia $I=\frac{2}{5}mr^2$ is rolling down an inclined plane. It's velocity at the end can be calculated using $mgh=\frac{1}{2}mv^2 + \frac{1}{2} I w^2$. ...
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Why does the angular velocity use $r$ instead of $R$?

I am calculating final velocity of a Maxwell wheel rotating and translating down due to gravity with energy. I split the kinetic energy into translational and rotational kinetic energy. I am little ...
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