All Questions
19
questions
1
vote
0
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38
views
Weird sign in EOM: Centripetal vs. centrifugal term [duplicate]
Something goes wrong when I was deriving the equation of motion in Kepler's probelm, as below,
Angular momentum conservation $L = Mr^2\dot{\theta}^2$.
And Lagrangian is $L = \frac{1}{2}M(\dot{r}^2 + ...
- 425
0
votes
0
answers
80
views
Substituting the conservation of angular momentum into the Binet formula results in contradiction [duplicate]
Background Information
The lagrangian of a particle in a central force field $V(r)$ is
$$
L=\frac12m(\dot r^2+r^2\dot\theta^2+r^2\sin^2\theta\dot\varphi^2)-V(r).
$$
The particle must move in a plane, ...
- 675
1
vote
2
answers
101
views
How would a game of (American) football work on a space station generating artificial gravity by using spin? [closed]
Using rotation to generate artificial gravity is pretty common in sci-fi. I know the TV show "The Expanse" features it on the Mormon's vessel. I also remember a small-scale rotating space ...
- 1,666
3
votes
3
answers
1k
views
Why do we consider $L^2/(2mr^2)$ part of effective potential energy?
The energy of a particle under the action of a radial conservative force is given by
$$E = \frac{1}{2}m\left(\frac{dr}{dt}\right)^2+ \frac{L^2}{2mr^2} + U(r),$$
where the last two terms provide the ...
- 799
1
vote
2
answers
250
views
Centrifugal Force & Rotating Frames [duplicate]
In Thornton & Marion's Classical Dynamics, the following relation is given for the rate of change of an objects position in the two coordinate systems (according to the picture shown at the bottom)...
- 529
0
votes
1
answer
101
views
Why does the amplitude of a pendulum increases when continuously changing the rope length? [duplicate]
The simplistic answer is that I'm pumping energy into the system thus the velocity increases and so is the amplitude.
I'm more interested in understanding it from forces considerations.
- 11
1
vote
0
answers
376
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Rewriting the Lagrangian in terms of the constant(s) of motion doesn't work. Why? (spherical pendulum) [duplicate]
I am trying to solve for the equations of motion to simulate a spherical pendulum. I decided to use the spherical coordinates. The Lagrange equation is,
$$
L=T-V=\frac{1}{2}m\left(l\dot\theta\right)^2+...
- 317
3
votes
1
answer
417
views
If you have a conserved quantity, why can't you use it to eliminate a variable in the Lagrangian? [duplicate]
Suppose, for example, we take a particle in polar coordinates $(r, \theta)$ with a central force, so $U = U(r).$ The Lagrangian is $$\mathcal{L} = \dfrac12 m (\dot{r}^2 + (r\dot{\theta})^2) - U(r).$$
...
- 53.2k
1
vote
1
answer
1k
views
Lagrangian, central forces and conservation of angular momentum [duplicate]
When studying central forces it is possible to propose the Lagrangian:
$$ L = T-U=\frac{1}{2}m \dot{r}^2+\frac{1}{2}mr^2 \dot{\theta}^2 - U(r)$$
Then we can solve the equation of motions for $\...
- 980
3
votes
2
answers
565
views
Deriving effective potential energy from the Lagrangian of a two-body system [duplicate]
I'm having some issues understanding how the effective potential energy of a two-body system is derived from the Lagrangian of the system. Specifically my issue is with one step...
Suppose we are ...
2
votes
1
answer
20k
views
Acceleration in plane polar coordinates [duplicate]
When we express acceleration in plane polar coordinates, we can find that
$\vec{a}= \left(\ddot{r} - r \dot{\theta}^2\right)\hat{r} + \left(r \ddot{\theta}-2\dot{r}\dot{\theta}\right)\hat{\theta}$.
...
user174490
0
votes
1
answer
2k
views
Significance of centrifugal potential
While dealing with central forces (purely using newtonian mechanics) I've came across this result:
$$U_\text{eff}(r)=\frac{l^2}{2\mu r^2}+ U(r) \, .$$
I'm not at all fluent with the lagrangian ...
18
votes
6
answers
9k
views
How can the centripetal force lead to objects flying apart?
I don't understand how the centripetal force, which always points to the center of our circular motion can cause this scenario:
We have a big stone which spins very fast, so fast that a part breaks ...
- 396
15
votes
4
answers
24k
views
Shape of water in rotating bucket
I need to show that the surface of water in a bucket rotating with constant angular velocity will have parabolic shape. I'm quite confused by this problem, but here's what I did:
$$\vec{F}_{cf} + \...
- 568
17
votes
2
answers
7k
views
Lagrangian of an effective potential
If there is a system, described by an Lagrangian $\mathcal{L}$ of the form
$$\mathcal{L} = T-V = \frac{m}{2}\left(\dot{r}^2+r^2\dot{\phi}^2\right) + \frac{k}{r},\tag{1}$$
where $T$ is the kinetic ...
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