All Questions
Tagged with newtonian-gravity integration
45
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Integrating acceleration + escape velocity over distance [closed]
I am not sure how to title this question so apologies if it's inaccurate.
If I throw an object at thrice the escape velocity of earth, what would be its velocity very far away from earth, (at a ...
- 129
3
votes
1
answer
112
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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
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0
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14
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Describing force accumulation trend of an infinite volume with evenly distributed radiative sources
I am looking for confirmation if I've built my equation properly.
My goal is to describe the change in force over time at a given point if evenly distributed radiators (in-phase or cumulative energy/...
- 25
1
vote
1
answer
51
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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 $a$ ...
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5
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(Not a flat-earther) The mathematics of an infinite flat earth using gauss' law for gravity
On the flat earth website, they prove that the gravitational pull of an infinite flat earth is finite. Is their proof correct?. I'm not that good at physics and can't determine if they're correct ...
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1
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How to calculate the potential energy of a large object as an integral?
Usually when calculating the potential energy of a body it is sufficient to take its center of gravity’s distance from the ground in order to get a result according to the formula $E_p=g*h*M$. But the ...
1
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1
answer
71
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Question on Gravity (Ring and Sphere) [closed]
Find the gravitational force of attraction between the ring and sphere as shown in the diagram, where the plane of the ring is perpendicular to the line joining the centres. If $R\sqrt8$ is the ...
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2
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What is the gravitational field intensity of a uniformly distributed mass content in Newtonian gravity?
In an infinite universe composed of single point masses which can be simplified as a uniformly distributed mass density, what is the equation for the gravitational field intensity in Newtonian gravity?...
- 476
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1
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98
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Triple integral gravitational potential between point and sphere [closed]
Problem for self-study: Gravitational potential between a point $\mathbf a$ and a uniform sphere, leaving out gravitational and density constants
$$
V(\mathbf{a})=\int\int\int\dfrac{1}{\vert\mathbf a -...
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1
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63
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How do you integrate a gravitational or electric field from $r=0$ to $r=\infty$?
I'm interested in determining the total gravitational and electric field of a charged particle. At reasonable distances the value of each field at a point is given by:
$$g = G\frac{m}{r^2}$$
$$E = \...
- 57
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2
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239
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Direct calculation of the gravitational potential inside a hollow sphere
I calculated the gravitational potential inside a massive sphere with constant density and got the result:
$$\Phi = -2\pi G\rho R^2 + \frac{2}{3}\pi G\rho R_p^2$$
Where $R$ is the radius of the sphere ...
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1
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521
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How do you integrate by parts when you have a triple integral?
I'm studying how particles of equal mass behave in a spherical cluster held intact by gravity. I will assume that the mass density $\rho(R)$ of the cluster is a function of the magnitude of the ...
- 1,222
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86
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How to derive gravitational potential from Navier-Stokes equation?
Starting from the Navier-Stokes equation I want to be able to derive the gravitational potential using the Poisson equation but am unsure how to do it in spherical polar coordinates.
This is what I ...
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1
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438
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Gravitational potential of a disc [closed]
The question says
Find the potential at the center of a disc whose surface area density varies as $$σ = σ_0(1+\cosθ)r $$ where theta is the angle made by the radius with the horizontal and $r$ is the ...
user318937
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2
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166
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Calculating the gravitational field on a point mass at central axis of a uniform ring [closed]
Consider a uniform ring of mass $M$ and radius $r$ and centre $O$. Let $P$ be a point on the central axis of the given ring at a distance $a$ from the centre of the ring (line passing through the ...
user297948
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2
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What is the meaning of Gravitational Potential when multiple point masses are involved?
According to Wikipedia "The gravitational potential $V$ at a distance $x$ from a point mass of mass $M$ can be defined as the work $W$ that needs to be done by an external agent to bring a unit ...
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1
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Better understanding of the definition of Gravitational Potential as the improper integral $\frac{1}{m}\int^x _{\infty}G\frac{Mm}{x^2}dx$
According to Wikipedia "The gravitational potential $V$ at a distance $x$ from a point mass of mass $M$ can be defined as the work $W$ that needs to be done by an external agent to bring a unit ...
- 379
1
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1
answer
130
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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.
$$ m \ddot{r} = -G \frac{...
- 1,525
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2
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729
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What is the gravitational potential of a homogeneous sphere? [closed]
I am studying gravitational potentials from the book Galactic Dynamics by James Binney and Scott Tremaine. They provide the equation from where the potential of a spherical system is to be derived as:
...
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1
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Doubt regarding solving an integration for radial flow of matter around a star in Newtonian gravity
The spherically symmetric flow of matter around a star in Newtonian gravity is governed by the equation
$$v\frac{dv}{dr}+\frac{1}{P+\rho}\frac{dp}{dr}+\frac{1}{r^2}=0$$
The equation of state is chosen ...
- 2,015
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3
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147
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The force of gravity between a shperical shell and a particle
I am trying to understand the proof of why the force acting on a spherical shell and a particle is
$$\frac{GMm}{r^2}$$
Where M is the mass of the sphere and m is the mass of the particle.
I am looking ...
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1
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84
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Why does this volume integral vanish?
I am stuck on this problem concerning the gravitational potential of a body. The body has a mass density $\rho(\mathbf x)$ and I have to calculate a contribution to the total gravitational potential ...
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1
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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 ...
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2
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787
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How to calculate center of mass of a hollow hemi-sphere with some thickness?
When we calculate Center of mass (COM) of a hollow sphere, we assume that it's thickness is
infinitesimally small, but in real world, we do not have any object with zero thickness, so how can we ...
3
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3
answers
125
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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 ...
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1
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Why do I have an extra factor of 3 for self-gravity?
So, I'm trying to calculate the "acceleration" (force / mass) on a spherical object of mass $M$ and radius $R$ due to its own gravity that holds it together. So, here is what I figured. The "...
-1
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1
answer
356
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Integration and average in physics? [closed]
Many applications of physics theory involve computations of integrals. Examples are voltage, force due to liquid pressure, surfaces...
In some cases, when there is linear dependence between two ...
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3
answers
1k
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Derive gravitational potential energy for this system [closed]
This is on a study guide for my Physics 221 final. I feel like I almost got it but I am off by a sign error. Here is the question:
Here is what I got so far:
Known:
$$F_g = \frac{GMm}{r^2}$$
$$U_g =...
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1
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Finding suitable element to perform integration upon [closed]
Is there any precise (proper) method or technique to specify the element on which integration will be performed. Is it the same for all properties like moment of inertia, gravitational potential, ...
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2
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480
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Feynman's proof for Newton's shell theorem [closed]
I have two questions concerning this proof:
Firstly, what is the difference between the increments ds and dx? Are they not just the same thickness of the strip?
Secondly, why can the integral ...
- 406
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1
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204
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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
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Nonlinear GPE of a solid block seems wrong
I am trying to calculate the gravitational potential of a solid block, and I have a nonlinear answer which strikes me as wrong.
A block with horizontal surface area $A [m^{-2}]$ and uniform density $\...
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1
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How to find gravity field of a solid square body?
I was programming gravity simulation and stumbled upon a problem, that Newton's formula for point masses is not enough for me, I need gravity field formula of a solid square body (2D).
To simplify ...
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2
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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 ...
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2
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708
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Integrating a dot product gives wrong sign for work done [duplicate]
Consider a point mass which creates a gravitational field. The gravitational force pulls a 'test mass' towards the point mass. Since the displacement and gravitational force are in the same direction, ...
12
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2
answers
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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 $M(r)$...
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1
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899
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Period of a pendulum [closed]
In the book 'Calculus the Early Transcendetals' at page 776 (7th edition) they give that the period of a pendulum with length $\text{L}$ that makes a maximum angle $\theta_0$ with the vertical is:
$$\...
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3
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832
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What's my $\mathrm dM$? Gravitational Potential inside a circle of mass
I'm trying to find the gravitational potential for an arbitrary point within a ring of uniform mass density. The point is constrained to be in the same plane as the ring.
So we start with:
$$\Phi=\...
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1
answer
91
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How to find an equation for $x$ in terms of $t$ for a particle falling under gravity with resistance given by $mkv^2$? [closed]
Okay so I have determine the velocity $v$ and displacement $x$ as functions of $t$ for a particle falling under gravity with resistance given by $mkv^2$.
I have set up the equation of motion divided ...
- 35
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1
answer
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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
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1
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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 ...
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2
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326
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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, ...
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1
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11k
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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 ...
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3
answers
6k
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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
2
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0
answers
123
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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 ...
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