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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Why isn't the moment of inertia of a cylindrical tube the difference of those of two cylinders?
The moment of inertia of a solid cylinder is $\frac{1}{2}mr^2$, and the moment of inertia of a composite object is the sum of the moments of inertia of the objects it consists of minus the moment of ...
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Confusion about torque [duplicate]
Consider a free body, not hinged about any point. If a force is applied to one end of the body, the body has a net nonzero torque about many points in space. About which will it rotate? Am I wrong in ...
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Why does a free body rotate about centre of mass when torque is applied? [duplicate]
Suppose a body is kept in space and is not fixed about any point (i.e, not hinged). If an external force is applied, the body rotates about the centre of mass.
I saw one answer pointing out that ...
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Do magnetic fields cause rotational inertia proportional to their magnetic moment?
If I have a stick that has a spinning disc on the end of it, and I try to rotate the stick, I will feel more inertia the faster the disc is spinning because I have to transform its rotational energy ...
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Very Basic Question: Calculation of Moment of Inertia Tensor
I am studying from the book Classical Dynamics: A Contemporary Approach by Jose & Saletan, and at page 496, they work through an example about the moment of inertia tensor of a uniform cube
(sorry ...
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Principal Axis and Conservation of Angular Momentum
In the video, the rotation of the wooden block around the principal axis with intermediate moment of inertia is an unstable rotation, where the axis of rotation and therefore the angular velocity ...
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Why does MOI of rectangle depend on length of diagonal of the rectangle? [closed]
If we have a rectangle of length $a$ and width $b$
my physics textbook showed that the MOI about the axis of rotation in the picture is given by
\begin{align}\label{MOI rectangle}
\dfrac{1}{12}M(a^2+...
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Why does the centrifugal force act normal to the surface?
This questions comes from understanding the Tennis Raquet's Theorem, also known as Dzhanibekov's effect in this video: Veritasium's video
On the second 8:36, we can see that when they try to add some ...
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Inertia tensor in canonical and rotated basis
So say we can describe some angular velocity $\boldsymbol{\omega}$ in canonical basis, $\{\mathbf{e}_i\}$, and a rotated basis, $\{\mathbf{\tilde{e}}_i\}$, like
$$\boldsymbol{\omega}=\omega_1\mathbf{e}...
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Parallel Axis Theorem Derivation
My professor was deriving the parallel axis theorem wherein he took the Center of mass of an object as some point O and was calculating the moment of inertia about an axis through point P located at a ...
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A question regarding momentum of inertia of L-shaped bar [closed]
I have trouble deriving moment of inertia $\Theta_A$ of the bar rotating around the point A shown in the image.
The answer of the problem says:
$\Theta_A=\frac{m}{3}\frac{a^2}{3}+\left[\frac{(2a)^2}{...
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Second mass moment for a specific rotating object
The polarization amplitude for gravitational waves can be written as a function of second mass moment. For example for a propagating wave along the $z$-direction, we have
$$ h_+ = \dfrac{1}{r}\dfrac{G}...
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Inertia tensor for spinning top
I was trying to find the equations of motion of a symmetric, solid and fixed spinning top employing Euler's angles and lagrangian mechanics. As for Euler angles I used the following convention (the ...
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Question about product of inertia of rolling disk [closed]
If a uniformly distributed mass, as a perfectly symmetric circular disk, performs a rolling motion or trans-rotational motion on a stationary plane, with $\alpha$ angular acceleration about it's ...
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Please help me understand what the tipping point of this Pedestal Table will be? [closed]
I am brand new here. I am a furniture maker and designing an Oval Pedestal Table for a client. I presented a design that I felt comfortable with, however my client has asked that I shrink the ...
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What is the total kinetic energy of a pendulum?
I have a sphere of a mass $m$ and radius $r$, attached to a massless string with a length $l$. The pendulum moves with an angular velocity $\omega$.
I know that axes of rotation can be relative. If I ...
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Calculating moment of inertia for a hollow cylindrical shell of varying amounts of water within it for an experiment about rotational motion [closed]
I am doing an experiment with the overall research question of: To what extent does the amount of fluid within a hollow cylindrical can affect its dynamics while rolling down an inclined plane
I was ...
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Why distance in Moment of Inertia definition arises squared? [duplicate]
The moment of inertia is defined as: $$I=\int_{}^{}r^{2}dm=\int_{}^{}r^{2}\rho dV$$
While it is obvious from the experience that the father away the mass from the axis of rotation the more it adds to ...
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Energetics of a Discus Throw
When a discus thrower releases the discus, the discus will have a certain forward speed that will be the same as the thrower's rotational speed multiplied by the distance between the thrower's center ...
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Inertias of the drivetrain components of a 4x4 vehicle or any geared system for that matter
I’m making a vehicle simulator and I’m not entirely sure if my inertia calculations are right or are completely wrong. What I currently have now is something like this:
1. Engine:
engine_Out_Inertia =...
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Deduction of mass moments up to the fourth order
Could anyone show me the details on the deduction of a discrete object's zeroth, first, second, third and fourth mass moment? For example, for the hydrogen fluoride with symmetry $C_{\infty V}$? I ...
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Query regarding parallel axis theorem
Suppose there are two points at (4,0) and (-4,0) each of mass 1 kg and the origin is (0,0). The rotational inertia of the combined system is 32kg.m^2. But if we shift the axis of rotation to left by 2 ...
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What axis of rotation should be used for rotational kinetic energy?
I know the kinetic energy of a rigid object is
\begin{align}\tag{$1$}
KE = \frac{1}{2}mv^{2} + \frac{1}{2}I\omega^{2}
\end{align}
where $v$ is the velocity of the center of mass of the object, $\...
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Moment of inertia of hollow body and solid body
I read in my textbook about the various results of moment of inertia for different geometrical shapes like solid and hollow cylinder, sphere, disc and ring etc. Something general I noted is that $M.I$ ...
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Intuitive idea behind the moment of inertia, torque
I'm not sure what idea I have of the concept of twisting moment, moment of inertia. The moment of inertia of a particle is $mr^2$. The torque indicates how much force $F$ applied at a distance d from ...
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How can I make the equation of warped disk?
My teacher gave us the assignment to find the moment of inertia of any shape you want. So I decided to find the moment of inertia of our milky way galaxy.
I found out that our galaxy is shaped like a ...
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Apparent contradiction between moment of inertia and Archimedes's law of the lever
Suppose you have a (nearly massless) lever you are using to lift something up. On the other end there is a $2\,\text{kg}$ object at $1\,\text{m}$ away from the pivot. Let's say I input some force on ...
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Misunderstanding properties of principal axes for moment of inertia
My lecturer has stated that the principal axes of the moment of inertia (hereafter MOI) are a set of axes such that the off-diagonal deviation terms of the MOI tensor disappear. He then said that in ...
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Moment of inertia of solid cone [closed]
Can we compress a solid cone parallel to the axis of rotation, into a disk of same mass to find moment of inertia of solid cone (uniformly dense)?
What I think...
As we are compressing the cone ...
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Angular momentum of a spinning solid sphere about an axis other than the spinning axis [closed]
A solid sphere is spinning about z axis. I know that its angular momentum about the z axis will be L=Iw. Where I is the moment of inertia about its central axis and w is its angular speed about z axis....
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What the theory behind the diameter of spheres and the time taken for it to roll down a track?
I am working on an experiment to learn about rotation motion and find the correlation behind the between the spheres’ diameters and the time taken for them to make their way down an track at constant ...
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In rotational motion, can we assume the point masses system as continuous body systems? [closed]
I was solving a question of rotational motion given below:
Four point masses, each of mass $m$, are fixed at the corners of a square of side $L$. The square is rotating with angular frequency $ω$, ...
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How do I prove that a fast spinning disk requires more force to stop it than a slowly spinning disk?
I am trying to prove that with all things the same, a rapidly spinning disk requires more force to stop it than a slowly spinning one.
For example, when you first hit the brakes in a fast moving car, ...
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How to find Moment of Inertia at a different axis?
So I have a solid disk:
m = 2.98 kg
r = 0.2 m
and an axis in the - and + z direction (in unit vector form, the k dimension)
And I have to find the Moment of Inertia of the disk if the axis is at point ...
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Compound physical pendulum with spinning disk
I want to know in which situation the disk rotates about its center of mass? It seems it’s in the first situation when it’s fixed. But how does it rotates if it’s fixed and what does cause it to ...
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Rotating disk at the end of a pendulum [closed]
I don’t understand the answer to part b. Why the disk won’t rotate about its center of mass if it is mounted to the rod by a frictionless bearing. Why can the disk be a mass point at the end of the ...
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Translate inertia tensor & outer product of a vector with itself
I'm trying to understand a mathematical expression where it's like this:
$$
I' = I+ m \left( d^2 E - \mathbf{d}\otimes\mathbf{d}\right)
$$
where $I'$ is the new tensor flow, $I$ is the tensor flow, $...
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Intermediate axis theorem - why can't we have exponential decay? [duplicate]
I was reading about the intermediate axis theorem and its mathematical proof. Typically one starts with the torque-free Euler's equations
$$
\begin{align}
0&=I_1\dot\omega_1 + (I_3-I_2)\omega_3\...
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In case of an axis where moment of inertia changes with time which of the following equations is valid? [closed]
$$ T = I \alpha $$
$$ L = I \omega $$
$$ T d(\theta) = d(\tfrac12 I \omega^2) $$
If I differentiate the second and third one with respect to time… all three equations give a different expression for ...
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Force that an axle exerts on wheel [closed]
Consider a wheel attached to an axle.
Since the torque on both must be the same:
$$F_wR_w=F_aR_a$$
$$F_w < F_a$$
because the radius of the wheel is greater than radius of the axle.
My question is: ...
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Is there an analogous method of finding moment for inelastic bending like in the case of inelastic torsion?
When calculating for Torque in torsion or for Moment in the case of bending, the concept is usually first introduced with the simplification of the assumption that the stresses and strains are in the ...
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Center of Mass calculation in configuration of $3$ pennies inscribing equilateral triangle [closed]
I'm working on a problem that is asking me to solve the moment of inertia about the center of mass of a $3$ penny system where the edge of each penny is touching the edge of the others and the ...
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Formula for Bifilar Pendulum
What is the formula for a Bifilar Pendulum's Period? Assuming the Pendulum oscillates sideways. ( Spins to the left and then to the right etc. ). I can't seem to find the formula online.
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Moment of inertia of a rectangular plate
I am currently trying to solve some moment of inertia questions with the help of "simple" volume integrals.
We haven't really been taught at all the derivation of the moment of inertia and ...
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What type of cylinder would roll down a slope faster; A solid cylinder or an inverted hollow cylinder? [duplicate]
I am conducting a lab right now in which I have been given a cylinder/can, which I can modify on the inside as I wish. The goal is to create a can that can roll down the fastest.
After some testing I ...
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Angular momentum and velocity about a point about which a rigid body is not rotating
Suppose a rigid body of mass $m$ is rotating about its centre of mass with angular velocity $ω$, and the centre of mass is translating with linear velocity v, consider two cases:
(I) we want to ...
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Kinetic rotational energy at different rotational axis
Consider a homogeneous wheel with a moment of inertia of $\frac{1}{2}mr^2$ around its center of mass. Said wheel rolls along a horizontal area. I am wondering about some of the quantities of this ...
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Does moment of inertia about an axis depend on whether the object is rotating around it or not?
In this solved problem in my textbook:
A disc is freely rotating with an angular speed on a smooth horizontal plane. It is hooked at a rigid peg P and rotates about P without bouncing. What will be ...
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Moment of inertia of a pipe
the wall of the pipe is very thin then my ideas was:
$$ I = \int^L R^2dm $$ where L is th elenght of the pipe then I will a bunch of hoops of dm:
$$dm = p2\pi RldR$$
replacing in the first eq and ...
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Moment of inertia of a disk with a hole
I was trying to find the moment of inertia of the disk object the following way
$$I_o = I_d-I_h$$ where $$I_0$$ is the moment of inertia of the disk with a hole and $$I_d$$ is the moment of inertia of ...