# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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