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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A paradox of the precession of a stick
I can see that a heavy flywheel around an axis experiences precession. But when I look at the equation:
$$ \dot{\omega} = (I \omega) \times \omega + \tau ,$$
where $\omega$ is angular velocity in the ...
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What are the Euler-Lagrange equations of an approximated rigid body and a point planet interacting via gravity?
The potential due to a rigid body will be approximated using a Lagrange polynomial expansion as
$$
U(x,y,z)=-\frac{\mathbb{G}Mm_p}{r} - \frac{\mathbb{G}m_p(I_1 + I_2 + I_3)}{2r^3} + \frac{3\mathbb{G}...
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Principal axis of circular disc with removed circular portion [closed]
From a circular disc, a circular portion of radius R/2 has been removed....it is asked if the axis YY' is principal axis or not...so my first intuition was since the axis is not symmetric this should ...
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Is there a meaningful way to define an inertia tensor for a wave function?
In classical mechanics the inertia tensor for a density field $\rho(x,y,z)$ inside volume $V$, is defined as:
$$
\iiint_V \rho(x,y,z)(||{\mathbf r} ||^2 {\mathbf I}_3 - {\mathbf r}\otimes{\mathbf r})...
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Does the moment of inertia of a body change with angular velocity?
In the wikipedia of Moment of Inertia wiki-page:-
Let 𝑅 be the matrix that represents a body's rotation. The inertia tensor of the rotated body is given by:
$$\textbf{I}=\textbf{RI}_0\textbf{R}^T$$
(...
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Period of a Oscillating Disk [closed]
What is the period of this solid disk as it rocks on a flat surface? In terms of the disk's mass (M), gravity (g), diameter (D), and moment of inertia (I)? Please show how to derive and calculate I ...
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Correcting the angular momentum in GR for non-point-like bodies?
Background
Consider I'm studying the orbit of planets using the Schwarzschild metric is given by:
$$
ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right) c^2 dt^2 + \left(1 - \frac{2GM}{c^2 r}\right)^{-1} dr^2 +...
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Help understanding moment of inertia [closed]
I'm learning about moment of inertia for the first time in the prep work set by my university before term starts (I'm going into first year in a few weeks). And as I understand it, it's basically the ...
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Minimum Static Friction Under a Driven Wheel
Say you have an idealized wheel in the form of a narrow cylinder of mass $m$, radius $r$, and uniform density balanced stationary on its curved edge on a level surface.
The wheel suddenly experiences ...
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Angular acceleration about different axes
Say you have a wheel floating in space with forces of equal magnitude acting upward at the COM and downward at the right edge. I have always been under the impression that no matter the inertial frame ...
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Non-ideal pulley
In a non-ideal pulley (with friction at axle to be negligible) the tension on both side of rope is not equal because friction is present between rope and pulley, but when we write torque on the pulley ...
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Periodic orbits for the motion of a body with a fixed point (without forces)
It seems that non-obvious periodic orbits for the Euler-Poinsot problem do exist (inertial tensor with 3 different eigenvalues). But I could not find any precise reference for the validity of this ...
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Why are there two different tension forces on a continuous string in the pulley system shown?
In an ideal pulley system with massless and frictionless strings and pulleys, the tension is typically uniform throughout a continuous string, even when the system is accelerating. However, when ...
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Derivation of moment of inertia formula
According to Physics Libre texts:
We defined the moment of inertia $I$ of an object to be:
$$
I = \sum_i m_i r_i^2
$$
for all the point masses that make up the object.
Now my question is why is the ...
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Moment of inertia in rolling of a disc on a rough surface
A disc resting on a rough surface is given an impulse. The disc eventually starts pure rolling. In such a case when we apply conservation of angular momentum at the point of contact of the disc with ...
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