New answers tagged rigid-body-dynamics
1
vote
How thick would a coin have to be s.t. the probability to land on its "side" is exactly 1/3?
I'll try to generalize by reducing the case of different bouncing behaviors to the one proposed in the question.
With the prescribed rule, the coin bounces once, and only once. Suppose the bouncing ...
- 461
0
votes
Accepted
Difference between torque about point and torque about axis
Difference between torque about point and torque about axis
One difference is the net torque about an axis produces angular acceleration about that axis. The net torque about a point produces angular ...
- 77.9k
0
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Non-Holonomic constraint in rigid body dynamics
You should still be able to write the equations of motion using a Lagrange multiplier coordinate (or constraint force.)
If the orientation of the rod is given by an azimuthal angle $\phi\in[0,2\pi)$, ...
- 3,749
-2
votes
Accepted
In rotation context, why is there a $r$ squared in the Moment of Inertia?
Answer
Ladies and Gents:
To understand why the moment of inertia has ( r^2 ), I believe that we need to consider two contexts involved in rotation:
The tangential application
The angular effect
The ...
- 47
0
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Rotational kinetic energy of rigid bar
The rotational energy $E_{rot}$ is defined as:
$$
E_{\text{rot}} =\frac{1}{2} \int_C (v_{\text{particle}}-v_{CM})^2 dm
$$
We know that the position of a particle in a rigid body can be expressed as ...
- 101
0
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Rigid body dynamics derivation from Newton's laws for higher dimensions
Too long for comments. I noticed that Chiral Anomaly's answer has the equation
$$\mathbf x_n=R\mathbf b_n+\mathbf x\tag1$$
as an assumption. But we can derive $(1)$ from the assumptions that, for all $...
- 111
0
votes
Why does this gyroscope derivation fail for non-punctual mass distributions?
Without going into the specific expressions:
Assuming the spin angular momentum is parallel to the gyroscope axis [...]
There is the following: during gyroscopic precession there is continuous ...
- 22.6k
2
votes
When transfoming an intertia tensor from one set of principle axes to another, why does it not change the tensor?
The inertia tensor of a cube with edge length a is
$$\mathbf I_1=\frac M6\,a^2\,\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{bmatrix}
$$
with arbitrary ...
- 12.9k
5
votes
In rotation context, why is there a $r$ squared in the Moment of Inertia?
You want an insight of Moment of Inertia without the concept of work - energy? That won't workout. Let me explain why.
How did those terms come?
After Newton brought a new era in Physics with his ...
- 1,131
0
votes
When transfoming an intertia tensor from one set of principle axes to another, why does it not change the tensor?
Not only a 45° rotation yields the same MMOI, but also any rotation $\varphi$.
Given a general rectangular prism of size $\Delta x$, $\Delta y$, $\Delta z$ has MMOI about the inertial frame after a ...
- 39.3k
0
votes
When transfoming an intertia tensor from one set of principle axes to another, why does it not change the tensor?
I also get the same result.
I took the formula for the moment of inertia for a solid cuboid:
$${\displaystyle I={\begin{bmatrix}{\frac {1}{12}}m(h^{2}+d^{2})&0&0\\0&{\frac {1}{12}}m(w^{2}+...
- 904
7
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In rotation context, why is there a $r$ squared in the Moment of Inertia?
Another way of looking at this is that angular momentum is $${\vec J} = {\vec r} \times {\vec p} \,.$$
For circular motion ${\vec p} = m {\vec v} = m {\vec \omega} \times {\vec r}$. Using $${\vec a} \...
- 26.6k
15
votes
In rotation context, why is there a $r$ squared in the Moment of Inertia?
Suppose we have a point mass $m$ and we apply a force $F$ to it then we get an acceleration $a$ given by Newton's second law:
$$ F = ma $$
To convert this to an angular calculation we simply pick a ...
- 363k
3
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In rotation context, why is there a $r$ squared in the Moment of Inertia?
The dimensions of moment of inertia can be found by dimensional analysis.
The moment of inertia of an object about a given axis is defined as the ratio of the object's angular momentum about that axis ...
- 60.5k
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