# Questions tagged [rigid-body-dynamics]

The study of the movements of a collection of connected bodies subject to external forces in the absence of deformation. This tag should be used for questions on the analysis of 2D/3D dynamics of rigid bodies, do NOT use this tag because your question contains a rigid structure.

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### Why does this object periodically turn itself?

See below gif image taken from here. Or see this Youtube video about 30 sec in. Is this a real effect? Why does it seem to turn periodically? Can it be explained by classical mechanics alone? Is ...
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### Why don't helicopters use reaction wheels to counter the main rotor?

As the main title says. I'm finding myself wondering about helicopters. The tail rotor is a vulnerable and key piece of equipment, especially on military helicopters. I know some helicopters instead ...
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### What do people actually mean by "rolling without slipping"?

I have never understood what's the meaning of the sentence "rolling without slipping". Let me explain. I'll give an example. Yesterday my mechanics professor introduced some concepts of rotational ...
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### Stability of rotation of a rectangular prism

I've noticed something curious about the rotation of a rectangular prism. If I take a box with height $\neq$ width $\neq$ depth and flip it into the air around different axes of rotation, some motions ...
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### How can the contact point of a body rolling without slipping have zero velocity?

They say that for a rolling body, the velocity of the contact point is zero. I'm not getting this. How can it be zero when it's in continuous motion?
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### Is it correct to say that it is theoretically impossible for perfect rigid bodies to exist?

If perfect rigid bodies were to exist, then consider a scenario in which two rigid bodies of equal masses moving with velocities of equal magnitude but opposite in direction colliding against one ...
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### How to combine these equations of constraint?

I want to model a nonholonomic system of an arbitrary rotating disk in 3D, which rolls without slipping, and doesn't have to stay vertical. (think spinning a penny on the table) I want to use the ... 4k views

### Why certain rotations are unstable? (Euler Equations)

We have the Euler equations for a rotating body as follows $$I_1\dot\omega_1+\omega_2\omega_3(I_3-I_2)=0\\ I_2\dot\omega_2+\omega_1\omega_3(I_1-I_3)=0\\ I_3\dot\omega_3+\omega_2\omega_1(I_2-I_1)=0$$ ...
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### Why does a Yo-Yo sleep, and then awaken?

What are the mathematics / mechanics principles behind a sleeping Yo-Yo, and in particular, what changes with a wrist-snap flick that causes it to "awaken" and return to your hand?     &...
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### Is there a quantum analogue of the "Tennis Racket" theorem?

A non-trivial result from studying the classical mechanics of an extended object shows that rotation about an axis whose moment of inertia is between the largest and smallest moment-of-inertia axes is ...
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### Will a pure rolling cylinder stop on a rough surface? [closed]

Will a disc or cylinder (rigid body) executing pure rolling on a rough surface stop, neglecting air drag and other heat losses and rolling friction but not static and kinetic friction? If yes, due to ...
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### How can friction do no work in case of pure rolling?

I have read various answers, on PSE and elsewhere, and most of them explain that the point of contact of the rolling object undergoes no instantaneous displacement in the direction of friction, I ...
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### What is the proof that a force applied on a rigid body will cause it to rotate around its center of mass?

Say I have a rigid body in space. I've read that if I during some short time interval apply a force on the body at some point which is not in line with the center of mass, it would start rotating ...
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### Why does a cuboid spin stably around two axes but not the third?

Let $C$ be a cuboid (rectangular parallelepiped) with edges of lengths $a < b < c$. Consider an axis that passes through the centers of two opposite faces of $C$. There are three such axes, ...
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### Degree of freedom paradox for a rigid body

Suppose we consider a rigid body, which has $N$ particles. Then the number of degrees of freedom is $3N - (\mbox{# of constraints})$. As the distance between any two points in a rigid body is fixed, ...
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### How is Chasles' Theorem, that any rigid displacement can be produced by translating along a line and then rotating about the same line, true?

Chasles' Theorem in its strong form says: The most general rigid body displacement can be produced by a translation along a line (called its screw axis) followed (or preceded) by a rotation about ...
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### How to prove that any rotation can be represented by 3 Euler angles

How can one prove that any rotation of a rigid object in 3-dimensional (3D) space can be represented by a sequence of three rotations around pre-fixed axes by 3 Euler angles? I see this statement in ...
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### Does a rotating rod have both translational and rotational kinetic energy?

I've recently learned about rotational kinetic energy and how an object can have both translational kinetic energy and rotational kinetic energy at the same time. However, I get confused when I try to ...
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### Intuitive Explanation of Tippe Top Effect?

A friend showed me a tippe top (a special kind of spinning top) lately and asked me about the physics behind it. I thought about it for a while but cannot quite figure it out. So I will throw the ...
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### What is the physics of a spinning coin?

When we spin a coin on a table, we observe 2 things: It slows down and stops after sometime. It does not stay at just one point on the table but its point of contact with table changes with time. I ...
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### Angular Velocity expressed via Euler Angles

On the top of the fourth page from here, the author trivially derives the components of angular velocity, expressed via Euler angles of the system. I fail to understand how the components of angular ...
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### How is length contraction on rigid bodies possible in special relativity since definition of rigid body states they are not deformable?

This is more like a conceptual question. We define rigid bodies as solid bodies with zero or almost zero deformation (meaning the deformation should be negligible). So no distance between two points ...
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### How long would it take for an upright rigid body to fall to the ground?

Let's suppose there is a straight rigid bar with height $h$ and center of mass at the middle of height $h/2$. Now if the bar is vertically upright from ground, how long will it take to fall on the ...
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### Which is the axis of rotation?

This should be simple, but it keeps bothering me. If a rigid body has no fixed axis, and a torque (defined relative to a point $A$) is applied, it will rotate around $A$. But often I can also ...
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### General motion of a cone on an inclined surface

Suppose that a solid cone is placed horizontally on an inclined surface and is initially at rest. How will the cone move when it starts motion due to its weight? I know that its motion depends on the ...
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### How can I approximate Jacobian Elliptic Functions in terms of basic integrable functions for the $\operatorname{SO}(3)$ rotation of a rigid body?

So, the rotation of a 3d body can be described with Euler's equations of motion giving the rotational velocity in components along the principal axes of inertia. As shown in f.ex. this paper, Euler ...
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### How to determine the probabilities for a cuboid die?

Imagine we take a cuboid with sides $a, b$ and $c$ and throw it like a usual die. Is there a way to determine the probabilities of the different outcomes $P_{ab}, P_{bc}$ and $P_{ac}$? With $ab$, $bc$...
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### Can a ball be struck to give it a spiral spin?

I play table tennis and we can hit balls to make them spin in pretty much every way possible except to spiral about its direction of travel like an American football throw. Of course it could be shot ...
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### What is the physical meaning of the principal axes of inertia?

What is the physical meaning of the principal axes of inertia? I used to think that the axes of inertia are, in some sense, the only axes about which the body can rotate without the angular momentum "...
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### Proving that the angular velocity vector is equal to a limit involving the rotation vector

The angular velocity vector of a rigid body is defined as $\vec{\omega}=\frac{\vec{r}\times\vec{v}}{|\vec{r}|^2}$. But I'd like to show that that's equivalent to how most people intuitively think of ...
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### How can different points on a rigid body move with different speeds but also be relatively at rest?

For a rigid body rotating with a constant angular speed, the points near the axis must have lower linear velocity than the points farther away. If they have different linear velocities, they must have ...
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### Paradox in the two block problem [closed]

UPDATE (regarding duplicate) : This question is not a duplicate of another question. Sure, the situation in both the questions is same and, yes, both questions ultimately provide a methodology to ...
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### Why does a rigid body rotate and not simply translate when pushed with an instantaneous force?

Let's say we have a metal rod of consistent density sitting flat on a frictionless surface. I intuitively understand that if I push one of its ends away from me, (at a right angle to the length of the ...
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### How fast does force propagate through matter? [duplicate]

Possible Duplicate: Is it possible for information to be transmitted faster than light? Consider the following thought experiment. You have a long perfectly rigid beam (for the sake of simplicity,...
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### Motivation for the definition of angular momentum?

As the title asks, what is the motivation for the definition of angular momentum and by extension torque? In all the books, be it undergrad or grad, the definition of the above-mentioned is just put ...
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### What is the relationship between rolling resistance and velocity?

I'm a games programmer, trying to write a simple car physics simulation. I'm aware that a car travelling in a straight line will exert a traction force that drives it forwards (by turning the wheels ...
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### What is the problem of having an inertia tensor not satisfying the triangle inequality?

It is well known that rigid body inertia tensors are 3 by 3 positive semidefinite matrices, which is the same as saying that their eigenvalues are all non-negative. A little less known is the fact ...
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### elliptical ring rolling on a horizontal surface

Consider a ring rolling without slipping along a horizontal surface. Regardless of the speed of the ring, it is continuously in contact with the surface. Let's deform the ring slightly so that it ...
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### What makes a forever spin top special?

What is the difference between a regular spin top and a forever spin top (which spins for far longer than a regular spin top).? The forever spin top is made out of stainless steel. I would like an ...
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### Defy gravity torques with gyroscopes?

Context On the following drawing, a platform is hung from the ceiling not exactly from its centre of gravity. Because of this it can't sustain an arbitrary orientation for long; I want to increase ...
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### The angular velocity

Is the angular velocity of a rigid body about any point the same as that about the axis of rotation. Also, can we even define angular terms (Angular Velocity, Angular Acceleration, etc) about any axis,...
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### Rolling without slipping, general analysis [closed]

Consider first, the simple case of a disc rolling horizontally on the x-axis, having angular velocity $\mathbf{\omega}$ Let p be any point on the edge of the disc. Let $\mathbf{r_{c}}$ be the ...
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### Rotationally invariant body and principal axis

Suppose a rigid body is invariant under a rotation around an axis $\mathsf{A}$ by a given angle $0 \leq \alpha_0 < 2\pi$ (and also every multiple of $\alpha_0$). Is it true that in this case the ...
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### -Rewritten question- Why do drop spindles spin faster when they have a top whorl?

New version of this question So, I've tried putting two bounties on this question in hopes that somebody understands what I am asking, in hopes for a satisfactory answer. I'm also going to include ...
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### Conservation of linear and angular momentum

Suppose I have two rigid bodies A and B and they are connected by a spring which is attached off-center (thus possibly causing torques). Due to the spring a force $f$ acts on A and a force $-f$ acts ...
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### Is there an equivalent of a scalar potential for torques?

For a given scalar potential $V$, it is known that the corresponding force field $\mathbf{F}$ can be computed from $$\mathbf{F} = -\nabla V$$ Suppose a rigid body is placed inside this potential....
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### How do I treat the Lagrangian in the case of a rigid body?

Here's Exercise 1.11 from Goldstein's Classical Mechanics 3rd edition (the first one after having derived the Lagrangian basically): Exercise 1.11: Consider a uniform thin disk that rolls without ...
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### Why don't we use the concept of axis of mass in place of center of mass?

Being a high school student, I read the concept of center of mass and it was written in my book that When a spinning ball is projected with some velocity , then all the points on the ball have ...
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According to wikipedia, the inertia tensor of an ellipsoid with semi-axes $a,b,c$ and mass $m$ is \left[\begin{array}{ccc} \frac{m}{5}(b^2+c^2)&0&0\\ 0&\frac{m}{5}(a^2+c^2)&0\\ 0&...