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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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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How exactly does (the weak form of) Newton's third law hold in the case of a force applied onto a lever?
Suppose we have an ideal massless lever with a fixed pivot point and a mass $m_{1}$ (at radius $L_{1}$) on one end and mass $m_{2}$ (at radius $L_{2}$) on the opposite end, and assume gravity plays no ...
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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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How to numerically simulate a rolling object? [closed]
I am a maths person and definitely not a physics person, so maybe this is taught somewhere and I haven't learned it. I am trying to understand and figure out how to simulate rolling an object on a ...
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Is Energy conserved in a spinning top?
When we spin a top (give it kinetic energy $K_0$) and place it on a table, it starts precessing around the vertical axis.
Is the total energy of the spinning and precessing top equal to the initial ...
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Moment of Inertia of a Rectangular Parallelepiped
Moment of inertia calculated about an edge for a rectangular parallelepiped is given by
$$I = (m/3) (a^2 + b^2), $$
my question is:
when m(a^2+b^2) is added to I, the new value obtained is Moment of ...
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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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Angular Momentum and Kinetic Energy of a Rigid Body
The inertial tensor of a homogeneous rectangular sheet of mass m with sides of respective length a and b and negligible thickness is:
$$I = \frac{m}{12}\begin{bmatrix}b^2 & 0 &0 \\ 0 & a^2&...
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Does rigid body rotation contradict Newton's first law?
Per Newton’s law, an object will move along a straight line with a constant speed if no force is acting upon it.
No portion of a rotating ball is moving in a straight line (except those on the axis ...
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Angular Momentum of a rigid, extended object: When we see a rotating object, is the state of rotation totally relative?
Angular momentum of an object is a physical quantity that depends on the chosen point about which to calculate the angular momentum.
It is often said that an object that has been thrown up in the air ...
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The motion of an ideal dumbbell [closed]
Good day, everybody.
I program a simple computer game and there is a "dumbbel" inside of it, which should using some kind of rocket engine. The engine is fixed on one side of the dumbbel and ...
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Gyroscope precession and Euler equations
I've been trying for so long to solve this problem, but the solution I have found isn't the one I expected. Basically, I have to solve Euler's equations for a gyroscope with a weigth at a distance d. ...
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Mathematically proving that it is always possible for a rigid body to maintain its rigidity
Consider a rigid body $\mathcal{B}$ modeled by a system of $n$ point masses $B_1,B_2,\dots, B_n$ constrained to keep constant distance from each other. I wonder how it is possible to mathematically ...
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Force not applied on center of mass
I have a question about rigid body motion when a force acts on off-center of mass.
I read the answer to the post force applied not on the center of mass but I'm still confused.
I understood that ...
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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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What happens when you try to rotate a system around not it's center of mass?
What happens when you try to rotate a system around not it's center of mass?
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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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Weird rotational motion of a heavy thick disc
I and my friend noticed a strange behaviour of a rotating disc (or a cylinder) and we don't know how is that happening.
This is the video.
Actually what is happening in the video is that the disc ...
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What is a detailed explanation for why friction does zero work on an object rolling on a surface without slipping? [duplicate]
I'm working through Taylor's Mechanics and he seems to assume this implicitly at a few points. Intuitively, I agree that this should be the case. For while at any time during the rolling there ought ...
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Question about Euler's equation for rigid bodies
I wished to understand a particular case of Euler's equation applied in the following cylindrical body:
where $I_{1,2,3}$ are the moments of inertia. By symmetry, $I_1=I_2=I_T$.
Here I consider that ...
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How to find angular velocity vector of a three dimensional rigid body given velocity of three non-collinear points on the same?
The position of a three-dimensional rigid body is completely defined by specifying position vectors of three non-collinear points on the body.
Similarly, one can define the motion of the rigid body by ...
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Why does a smooth rolling ball roll indefinitely despite there being static friction? [duplicate]
For a rigid smooth rolling ball rolling down a ramp (as seen above), the acceleration of the center of mass is given by:
$$a_{\text{com}, x} = - \frac{g \sin θ}{1+\frac{I_\text{com}}{MR^2}}.$$
However,...
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How to calculate/use horizontal coefficient of restitution value for basketball?
I'm trying to run an experiment in which a basketball is dropped straight down with a certain angular speed and then undergoes a bounce. My goal is to create a relationship between the initial angular ...
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From where does the perpendicular component of force arise in this lever? Does it conflict with the strong form of Newton's third law?
Newton's third law states that for every force applied by object $A$ on object $B$, there is an equal an opposite force by object $B$ on object $A$. The strong form of Newton's third law states that ...
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How does a spaceship respond to a sling?
If you are in a spaceship and you spin a sling with a rock in it, does the spaceship rotate in the opposite direction as the sling with the same angular, kinetic energy as the sling?
If the sling is ...
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Precession of Angular momentum of Symmetric Top
For a torque free symmetric top, is the angular momentum in body fixed coordinates in same direction as instantaneous axis of rotation?
I know that instantaneous axis of rotation precesses about ...
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Conservation of Angular Momentum about the Instantaneous centre of rotation
While solving some problems on rotational kinematics involving ,say, a cylinder rolling without slipping against a rough surface, I used conservation of angular momentum about the instantaneous centre ...
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Collisions with Spheres (with Different radii) on a plane
How do you calculate the new velocity of the spheres after a collision when the spheres have different radii?
I thought you could just decompose the horizontal velocities and use the standard ...
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Computing angular velocity on a different point of body
Suppose you have a rigid body and you have two local coordinate systems on the body at $P_1$ and $P_2$ and we can write a vector in the $P_2$ coordinate system using $P_1$ via the transformation $v_{...
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Astronaut with Jet Engines: Applied Forces and Angular Momentum over Time
Let's say that we have an astronaut in space of length $L$ and mass $m$:
Because of some truly irresponsible space agency, they've attached jet engines to this fellow's arms as depicted in the ...
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The tennis racket theorem with degenerate eigenvalues $I_1, I_2 , I_3$: Are the rotations around the principal axes stable?
If a rigid body has a symmetry such that two of the principal moments of inertia are equals, i.e. $$I_1=I_2> I_3 \qquad{\rm or}\qquad I_1>I_2=I_3.$$
Are the rotations around the principal axes ...
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Off center, perpendicular force applied to rigid body with pivot/ fulcrum at center of mass?
I feel like this question has been asked in so many ways, but I am still unable to explain my personal experience based on previous answers. From other discussions:
You can always replace an off-...
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Translate inertia tensor & outer product of a vector with itself
I'm trying to understand a mathematical expression where it's like this:
$$
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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Angular velocity of a cylinder rolling in another cylinder
I'm having troubles with the following problem:
Consider a hollow cylinder of radius $2R$ which has a point of its rim fixed in an inertial frame and can rotate about this fixed point. Inside it there'...
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Apparent contradictions in the mathematical model of a rigid body and Newton's Second Law in Newtonian mechanics. What gives?
In all books and texts I've seen so far, Newton's Second Law is used to prove that the net torque acting upon a system of particles with respect to its center of mass is equal to the rate of change of ...
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Addition of forces on a rigid body instead of a point
When two forces act on a point mass,we add the forces like we usually do and i have no problem understanding that. When the same forces are applied on a rigid body,how are we able to add them the same ...
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Work done by internal forces of a rigid body
I am reading Goldstein's Classical Mechanics book, and I came across that:
In a rigid body the internal forces do no work
Is this statement based on the assumption that the internal forces are ...
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Angular Velocity in the Plane of a Lamina
A rigid body (i.e., a 2-dimensional object) has principal moments of inertia about the centre of mass of $I_1 = (\mu^2 -1), I_2 = \mu^2 + 1, I_3 = 2\mu^2.$ I wish to show, using the Euler equations, ...
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Is the free 3D rigid body/Euler top an integrable system?
Or is there chaos?
The base of the configuration space is 3D. There are 4 constants of motion, namely, the energy, and the 3 components of the angular momentum.
So there are still 2 degrees of freedom....
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Why do we use $\Delta m$ in the derivation of angular momentum of a rigid body?
while I was watching a derivation of angular momentum of a rigid body on youtube, it came to my attention that the person who was doing the derivation, used $\Delta m$. my question is, why did he use ...
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In a rigid body rotating in 3D around the origin, is the norm of the linear velocity of each particle constant?
I was trying to prove that kinetic energy is constant in a rigid body undergoing some arbitrary rotation around the origin in a 3D space, and I ended up proving a stronger statement: that the norm of ...
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Inertia tensor formula for point masses in rigid assembly?
Suppose I have $N$ 1kg point masses in a massless rigid assembly such that the center of mass of the assembly is at the origin and point mass i is at $(x_i, y_i, z_i)$.
The inertia tensor of the ...
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Is a free rigid body in 3D space an integrable system? [duplicate]
I am trying to find three integrable systems with 6 degrees of freedom using the Liouville–Arnold theorem. That means that a set of integrals of motion that correspond to a conserved quantity for ...
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Euler's equation of motion for rigid bodies rotating with one rotation axis not through the body's center of mass [closed]
this is my first question in this forum. Thanks for all the knowledge and support shared throughout the whole website!
I have a body with rotations around 3 axes. I am looking for the external torques ...
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Nutation frequency for a weighted gyroscope
So, I've been led to believe that the frequency of nutation of a gyroscope can be calculated using the formula
In which the I's are the moments of inertia around the principal axes and omega-3 is the ...
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Ball rolling in a cylindrical trough
I am trying to understand an interesting effect I observed while playing with my kids' toys (video). The energy in this system seems to slosh back and forth between the trough and the ball, with the ...
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Moment of Inertia of a screw rotating on a horizontal plane [closed]
Since a screw is kinematically equivalent to a cone, you can consider it like one, but I wanted to mathematically model the motion of a screw/cone as it is rotating on a plane. However, the moment of ...
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How to transform Arbitrary Rotation Matrix $A$ to a coordinate system where the $z$ axis lies along the axis of rotation by Similarity Transformation?
In Chapter 4 of the book Classical Mechanics by Goldstein, it was written that
"By means of some similarity transformation, it is always possible to transform the matrix A to a system of ...
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