All Questions
Tagged with newtonian-gravity integration
45
questions
12
votes
2
answers
22k
views
Gravitational potential energy of any spherical distribution
The general formula to get the potential energy of any spherical distribution is this :
\begin{equation}\tag{1}
U = - \int_0^R \frac{GM(r)}{r} \, \rho(r) \, 4 \pi r^2 \, dr,
\end{equation}
where $M(r)$...
- 7,592
8
votes
3
answers
6k
views
Integrating radial free fall in Newtonian gravity [duplicate]
I thought this would be a simple question, but I'm having trouble figuring it out. Not a homework assignment btw. I am a physics student and am just genuinely interested in physics problems involving ...
- 93
4
votes
2
answers
326
views
How do you know which way to choose the limits of an integral?
I am reading http://www.feynmanlectures.caltech.edu/I_13.html#Ch13-S4
In the beginning of equation 13.18, in which Mr. Feynman calculates the potential energy of an object outside a spherical shell, ...
- 527
4
votes
1
answer
3k
views
Newton's original proof of gravitation for non-point-mass objects
Suppose we have two bodies, one very large (Earth), and one very small (a cannon ball). If the cannon ball is some distance away from the Earth, to find out the force produced on the cannot ball, we ...
- 143
3
votes
1
answer
491
views
Better derivation for the gravitational potential energy
I was shown this derivation for the gravitational potential energy, and I'm not very happy about it assuming that $\frac{1}{\infty} = 0$. Is there a better derivation, either using a completely ...
- 73
3
votes
3
answers
125
views
What is wrong with this calculation of work done by an agent bringing a unit mass from infinity into a gravitational field? [duplicate]
Let us assume that a gravitational field is created by a mass $M$. An agent is bringing a unit mass from $\infty$ to distance $r < \infty$, both measured from mass $M$.
The agent is always forcing ...
- 49
3
votes
1
answer
204
views
How to calculate the gravitational binding energy of a uniform cube of length $L$ and mass $M$?
The functional form is known already (as attached). But what is the solution for this integral?
- 425
3
votes
1
answer
112
views
Confusions on The Gravitational Energy of a Point P in a Cube
I have been working, quite tirelessly, to try and find an answer to a question that has been bothering me for some time now.
I have been working over some proofs, in the Newtonian Mechanics world, to ...
- 152
2
votes
3
answers
832
views
What's my $\mathrm dM$? Gravitational Potential inside a circle of mass
I'm trying to find the gravitational potential for an arbitrary point within a ring of uniform mass density. The point is constrained to be in the same plane as the ring.
So we start with:
$$\Phi=\...
- 53
2
votes
1
answer
11k
views
Gravitational force exerted by a rod on a point mass
I have doubts with the solution of a certain problem. I will give the entire solution below and will lay out my doubts as well.
A point mass $m_1$ is separated by a distance $r$ from a long rod of ...
- 383
2
votes
0
answers
123
views
What is the correct way of integrating in astronomy simulations? [closed]
I'm creating a simple astronomy simulator that should use Newtonian physics to simulate movement of plants in a system (or any objects, for that matter). All the bodies are circles in an Euclidean ...
- 2,119
1
vote
1
answer
1k
views
From Paris to ... London [closed]
(Excuse the pun in the title, couldn't resist)
Paris and London are connected by a straight underground tunnel, as shown in the diagram below. A train travels between the two cities powered only by ...
- 3,569
1
vote
2
answers
416
views
Gravitational force of point mass on a rigid body - Integral proof
Assume a point mass $m$ located at $\vec{x}$. Assume also a solid body whose coordinates $\vec{x}'$ belong to a connected subdomain $\vec{x}' \in \Omega$. The solid body has a non-uniform mass density ...
- 201
1
vote
1
answer
130
views
Finding velocity $v$ and position $r$, given a time $t$ under the acceleration of a gravitational force [closed]
I was messing with the maths, when I tried to find the velocity as a function of time, $v(t)$, and the position, also, as a function of time, $r(t)$ under the gravity force.
$$ m \ddot{r} = -G \frac{...
- 1,525
1
vote
1
answer
51
views
Problem with Deriving work done by gravitational force and gravitational potential energy from the first principles
Suppose we have a system with Two point masses of mass $M$ and mass $m$. And we want to derive Work done. Lets say M is fixed or $M>>m$. Initially assume mass m is at rest at a distance of $a$ ...
- 45