Questions tagged [moment-of-inertia]

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Dependence of moment of inertia on radius [duplicate]

How do we derive that moment of inertia I=mr^2 depends on square of radius or it is proved experimentally.
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$I=m r^2$, moment of inertia

Why does moment of inertia dependent on square of radius and can this be derived by some means? If so then how? or only can be proved experimentally?
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How to find the moment of inertia for a cylindrical segment

I want to find the moment of inertia for a cylindrical segment, show below: On Wolfram MathWorld, I found a formula for the volume of a cylindrical segment. Let $$h(r,\theta) = h_1 +\frac{1}{2}\left(...
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Why can I use different formulas for moment of inertia to model this simple rod system?

Suppose I have a rod of uniform mass as follows. enter image description here I'm confused how to pick the moment of inertia for this rod. One possibility would be to use the formula corresponding to ...
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1answer
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Moment of inertia change

Does the moment of inertia of an object change with the presence/absence of large masses in the vicinity of that object?
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to compute the mass moment of inertia tensor for a trapezoidal prism, can I break up the prism into 3 regions and then sum their inertias?

I want to find the mass moment of inertia tensor for a trapezoidal prism, like this: My first step was to find the COM. First, I found the geometric centroid of a trapezoid. $$\begin{align} \bar{x} &...
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2answers
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Could anyone provide me with derivation for planar moment of inertia? I also have a couple of aditional doubts [closed]

I've been searching for the derivation of planar moment of inertial but I can't find it anywhere $(I = \iint y^2dA)$. I don't understand why there's a $y^2$ term in there but I do understand that you ...
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2answers
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Rod's Kinetic Energy in the pendulum problem

On the MIT OCW for engineer dynamics the cart pendulum problem is solved by using the Lagrange Method here. This is a 2D problem, so rotation only occurs on the Z axis. While obtaining the rotational ...
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Is moment of inertia additive? If so, why doesn't adding two halves of solid box work?

I'm doing a project where I want to calculate the moment of inertia for some objects. I've broken the objects down into simple objects like cubes, spheres, cylinders, etc -- things I know the moment ...
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What is the moment of inertia of a free electron?

Well, elementary particles have no moment of inertia $I$. But what is the nearest possible answer to the question for a free, spinning electron? In general, one has a relation with spin $J$ given by $$...
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Applying conservation of angular momentum when string of a pulley suddenly gets taut

A block of mass $M$ is attached to one end of a light string which is wrapped on a disc of mass $2M$ and radius $R$. The total length of the slack portion of the slack portion of the string is. The ...
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1answer
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How does the amount of liquid in a cylindrical can affect its motion when rolling down an inclined plane?

I am currently working on my physics extended essay for IB and I have chosen to investigate how the amount of water in a cylinder affects its motion down an inclined plane. I am planning to carry out ...
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Objects rolling down inclined plane at different speeds

Consider the case of objects rolling down an inclined plane at different speeds depending on their moment of inertia. Some basic equations are shown in this screenshot from Michel van Biezen. He is ...
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How to find the moment of inertia of an object when we know the moment of inertia of its symetric part?

I am getting a little bit confused with calculating moment of inertia of symetric objects. For example if the task was to find $I$ of a thin equilateral triangular plate of mass $m$ and side $a$ ...
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1answer
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How to interpret the values of the inertia matrix?

I've a foldable quadrotor with rotating arms. So, the quadrotor can take different morphologies, like the classical one "X", an "H" morphology, "Y", etc. I've calculated the inertia matrix for each ...
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3answers
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Why will the maximum value of static friction act? [closed]

I viewed a similar question but that didn't solve my query.In the shown problem, while solving the equations my teacher just assumes that the static friction will have its maximum value and solves the ...
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1answer
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Gyroscopic forces due to counter rotating electric motors [closed]

I am currently working on a tilt rotor aircraft, for that I need to size a motor the applies the torque to swivel the propellers from a down facing position into a rear facing position. We are using ...
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How is ground speed simulated for an accelerating vehicle?

For vehicles on "treadmills", how is the ground simulated by the drum (as depicted) to accurately represent the energy loss to drag. I have been conceptualizing the process below, but am skeptical ...
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How does moment of inertia transfer through a series of levers? (Specifically, a Piano Action)

Ultimate Goal: Calculate the mass moment of inertia that a finger experiences as it depresses a piano key. Background on the question is at the end if you need, but I'll keep this high-level so that ...
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How to calculate the distance travelled by a particle using the radius of gyration of its trajectory (represented like a body)?

According to the Wikipedia page on Radius of Gyration: One can represent a trajectory of a moving point as a body. Then radius of gyration can be used to characterize the typical distance travelled ...
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How much torque needs to be applied?

I'm using a physics programming library named Bullet and the documentation is really lacking so I have to figure much of it out as I go. I have a box with side lengths of 1 and mass 1. The "local ...
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1answer
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Moment of inertia and centre of masses of continuous bodies [duplicate]

I'm quite accustomed with integration and all those calculus involved in finding moment of inertia as well as center of mass. But a random thought is wiggling in my mind. Why do we take $dm$ instead ...
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1answer
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Moment of Inertia of a rod rotated around a point [closed]

The rod rotates about an axis perpendicular to the rod and at a point $d$ distance from the starting end. How would I go about finding the rotational inertia?
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4answers
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About which axis we should take moment of inertia?

In the formula of rotational kinetic energy $\frac{1}{2} I \omega^2$; which axis should we take the moment of inertia (about the centre of mass (COM) or about the axis of rotation)? For a pure rolling ...
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1answer
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Rotation calculation of an imbalanced object

Let's assume I have a scale like object hanging from a point. Now if I put an object inside it stays as it is, as we have replaced the spring with some sort of fabric, that doesn't expand itself. ...
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2answers
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Two values of moment of inertia ($I$) at $x=l/4$ in rod, when calculated differently

I'm using parallel-axes theorem in both the following methods to calculate $I$ of a rod about a point (say $P$) at $x=l/4$ from C, $\perp$ to the rod. Method 1: We know, $I$ about the centre C, $\...
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Why does my book consider moment of inertia as a scalar when it is a tensor?

I found in the internet that the moment of inertia of a rotating body is a tensor quantity. But in my book it is considered as a scalar quantity. Won't doing this give wrong results? So how does it ...
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1answer
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Are the linear velocities of rotating disc and hanging mass equal?

In that picture there is a system that shows a disc that is rotating due to the work done by gravity on the mass hanging on the pulley. There is no friction or any other non-conservative force acting ...
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Momenta of objects when combining/separating

Let's say I have two solid objects, $A$ and $B$, with inertia tensors $I_A$ and $I_B$. Imagine they are suddenly combined into one object, i.e they go from travelling in free space to being connected ...
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2answers
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Integrating moment of Inertia elements/Choosing the infinitesimal element for integration for a disc [closed]

Moment of inertia is defined by this integral $ I =\int r^2 dm$ where $I$ is the moment of inertia $r$ is the distance from axis of rotation of a mass element 'dm' r Now, If I have a disc I could ...
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2answers
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Moment of Inertia of a Solid Sphere

I just seen the following derivation of I for a solid sphere about an axis through the center of the sphere: $$I= \int V\hat{r}^2dm =\int_0^\pi\int^{2\pi}_0\int^{R}_0\rho(r\sin\phi)^2r^2\sin\phi\,...
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Moment of inertia of a Cardan/Hooke joint?

I'm working on a model that approximates a rotating body as a double Cardan joint: one of these guys. In my scenario, torque is applied to the center shaft and both end shafts are free. The angle ...
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1answer
60 views

Ping Pong Ball and Spinning Billard Ball [closed]

A student sent me a Tik Tok with saying it "Breaks Physics". A person hits a ping pong ball to a billard ball. The ball does not move. The person spins the billard ball, then hits the ping pong ...
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Moment of inertia tensor and symmetry of the object

What information does the moment of inertia tensor give on the structure of an item. I was told that its eigenvectors give the principal axes of the object. Do you know more about this?
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If masses are stuck on a rod, how it's moment of inertia changes? [closed]

Supposedly I keep a rod of mass 3 kg and length 2 meters on a smooth desk, such that masses of 1 kg and 2 kg are stuck firmly to top and bottom of the rod respectively. Now, the center of mass of the ...
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Is there any possibility to use the inertia tensor to express the Kinetic energy? [closed]

$$T=\frac{1}{2} \mathscr M \omega^2$$ Well in this situation the particle is in rotation. Can we express $T$ using the inertia tensor $I_{ij}$
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Understanding and Expressing the Definition of Inertia Tensor in the Language of Differential Geometry

I am confused about translating the definition of the inertia tensor I know into the language of differential geometry. Part of this confusion arises because in every physics textbook I read, the term ...
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1answer
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Two expressions of kinetic energy of rotation [closed]

The moment of inertia matrix for rigid body in general case is $I= \begin{bmatrix} I_{xx} & I_{xy} & I_{xz}\\ I_{xy} & I_{yy} & I_{yz}\\ I_{xz} & I_{yz} & I_{zz} \end{bmatrix} $...
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Conceptual question about calculation of moment of inertia of a rolling wheel

In the problem above shouldn't the moment of inertia be $\frac{1}{2}mr^2+mr^2$ by the parallel axis theorem rather than $\frac{1}{2}mr^2$ since the instantaneous centre of rotation is the contact ...
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2answers
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How to explain visually and in a mathematically simple manner why the moment of inertia of a system is minimum at its center of mass? [duplicate]

I've been solving different problems related with finding the moment of inertia in a set of different particles, and ojects of known rotational inertias. Let's say spheres, cylinders, rings, rods and ...
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1answer
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Moment of inertia of a body free to rotate

A square plate $ABCD$ of mass $m$ and side $a$ is suspended from point $A$ in vertical plane. A disc of the same mass and $\sqrt2a$ diameter is attached at point $C$ as shown in the figure. The disc ...
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Angular velocity of a rod [closed]

Hello, so I've been struggling with part b of this problem. The answer I got online for part b is that the angular velocity of the rod is still the same even after the rings have left the rod. But ...
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How do you find the Triangle Inequality from an Inertia Matrix?

If you have an inertia matrix of the form $$\begin{pmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{yx} & I_{yy} & I_{yz} \\ I_{zx} & I_{zy} & I_{zz} \end{pmatrix}=I$$ If the matrix ...
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Rotational dynamics perpendicular axis theorem

I was taught in class that in case of perpendicular axis theorem $I_z=I_x+I_y$ (therefore,$I_x=I_z-I_y$, according to me) but in the next class my teacher told me that $I_x=I_z+I_y$.
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Derivation for moment of inertia of the shell

I'm trying to find the moment of inertia of a shell(basically a sphere where all the mass is concentrated on the surface of the sphere) using triple integration. The equation of the sphere $x^2 +y^2 ...
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What do moments of inertia do in the potential terms of Lagrangians?

I am struggling to understand the Lagrangian computed in this paper. In particular, a binary spacecraft-debris system is assumed as below. The analysis goes as follows. 1- I am in trouble to ...
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Way to reduce lateral movement of a rod when trying to calculate the mass moment of inertia using bifilar pendulum?

Basically I'm trying to measure the moment of inertia of a hollow rod using bifilar pendulum. I will rotate the rod, and measure the period, and use the written formula to calculate the moment of ...
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1answer
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Rotational Kinetic Energy of Rigid Bar

Consider a rigid bar (infinitely thin and with uniform mass density) of length $L$ with $x_1(t), x_2(t) \in \mathbb{R}^3$ each describing the positions of an endpoint of the bar in some fixed inertial ...
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What is the interplay between the “spinning tennis racket” effect on a planet with three principle axis of rotation and its moon?

Suppose I have a tennis racket spinning on its second axis of rotation such that it is subject to periodically flipping over on its spinning axis, as per the Dzhanibekov effect (explained in ...
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1answer
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Moment of Inertia using centre of mass knowing distance and weight of rod [closed]

I know the mass, I know the center of gravity position. Can i use this to work out the moment of inertia of a rod. So my ruler is 30 cm long and I know its mass to be 11.8g I have some blu tac which ...

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