Questions relating to Newton's Laws of Motion

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Attitude dependence on orbital elements

How would errors in the argument of perigee affect attitude? Is there a general approach documentation? Thanks
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2answers
506 views

Proof that the following function is a polynomial

I've been trying to get my head around this problem for a long time, yet I have not been able to make much progress. Let $\ell_0(j) = \left\lfloor \frac{1}{2}\left( \sqrt{8j^2 - 8j + 1} + 2j - 1 \...
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1answer
59 views

When do differential operators commute?

Given that the equation of motion of a particle placed on the apex of Norton's Dome is $$\frac{d^2 r}{dt^2}=r^{1/2}\qquad\longleftarrow\text{as proved in this previous question}\tag{1}$$ Solve this ...
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1answer
54 views

On the Liouville-Arnold theorem

A system is completely integrable (in the Liouville sense) if there exist $n$ Poisson commuting first integrals. The Liouville-Arnold theorem, anyway, requires additional topological conditions to ...
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2answers
49 views

Show that the equation of motion for a particle on Norton's Dome is $\frac{d^2 r}{dt^2}=r^{1/2}$

A particle sits at the top of a dome, whose height drops away from the centre, with a drop given by $$h=\frac{2r^{3/2}}{3g}$$ where $g$ is the acceleration due to gravity, and $r$ is a coordinate ...
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1answer
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projectile motion find the angle

I would please like help on the following question related to projectile motion. An object is projected in a $60$ m/s velocity with a $\theta$ angle to the ground. If the object has $30^\circ$ ...
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0answers
46 views

Possibly new solution to equal-mass three-body problem; refinement required

(Since I didn't know which authorities to contact, I thought I'd post this here.) While messing around in this Wolfram Demonstrations applet, I found a suspicious pattern, in which I could see ...
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0answers
47 views

How does one calculate a re-bounce sling shot trajectory such a Juno?

looking at this http://i.imgur.com/EHTN00d.gif What text book would allow one to calculate such things? DE books in university certainly had nothing that could be of use with such calculations. how ...
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1answer
79 views

Analytical solution of $\ddot r=ar^{-2}+b(\dot{r})^2$

I have this physics problem I'm trying to solve and its been a while since I've done differential equautions. The problem I'm trying to solve is: $$ -GMm/r^2 -k(\dot r)^2 = m\ddot r.$$ I know that I ...
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2answers
76 views

How much velocity can a canister of fuel give a spaceship?

I've recently considered the issue of how much velocity a canister of fuel can provide a 'spaceship'. I assumed we could approximate a basic solution If we know the mass of the fuel $m$, the mass of ...
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1answer
31 views

Calculate velocity $\nu$. [closed]

A lorry of mass $3.5\times10^4\text{ kg}$ attains a steady speed $\nu$ while climbing an incline of $1$ in $10$ with its engine operating at $175$ kW. Find $\nu$. $g=10ms^{-1}$. Neglect friction. The ...
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1answer
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Kinematics Mechanics Find the length of the belt and the speed of the rack. [closed]

For A. I know that the formula for belt is simply $L=\frac{Pi(D_a+D_b)}{2}+2C+\frac{(D_b-D_a)^2}{4C}$ Which gives me $L= 116.99"$ since C is equal to $50$ For B. However I'm stuck and can't get the ...
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1answer
25 views

Understanding elastic collision between two rocks with unknown masses

I have this problem here that goes like this: Two curling rocks of equal mass, one red and the other yellow, are involved in a perfectly elastic, glancing collision. The yellow rock is initially at ...
3
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0answers
107 views

generating functions for catastrophe theory

I am studying Thom's theorem in catastrophe theory and am having a hard time understanding what the "generating functions" actually do. How exactly are they used to classify generic caustics? The ...
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1answer
29 views

Comparison of Cartesian and Scalar Notation in Mechanics

In his book on Engineering Mechanics - Statics, R C Hibbeler provides many force problem solutions in both scalar and Cartesian notation (e.g Example 2.5 Chapter 2). It feels like he is trying to ...
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1answer
38 views

Two blocks and a frictionless pulley problem

Block B ($m_{B}$=0.36 kg) is connected to a lightweight rope that passes over a lightweight, low-friction pulley.The other end of the rope is connected to Block A ($m_{A}$=0.72 kg), which is on a low-...
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0answers
27 views

Question about G-Forces and Circular Motion on a Human Centrifuge

I don't really understand what a G-force is and how it can be used to solve problems using the formula: $$T=mv^2/r$$ T is tension, m is mass (in kg), v is velocity (m/s) and r is radius of the circle....
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1answer
46 views

The radius of the track.

A racing car completes $5$ rounds of circle in $2$ mins . It has uniform centripetal acceleration $\pi^2 t^{-2}$ then the radius of circle is?. I asked it on physics $SE$ but I dont know how to ask ...
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1answer
30 views

Accelerate to Max velocity, then decelerate to known velocity

I have an object traveling at a known velocity (Vi). It then accelerates (known A) to a known maximum velocity (Vmax), then decelerates (-A) to another known velocity (Vf). The total distance ...
0
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0answers
8 views

Function to represent discrete forces

I am trying to describe the force exerted by the two wheels of a 2d model of a car (one wheel in front of another) each with a magnitude of $F$. Is there any function I could use besides a piecewise ...
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1answer
25 views

Poisson brackets of angular momentum

So I'm trying to simplify this Poisson bracket of angular momentum vectors: {$L_1,L_2$} Where $L=r \times p$ I know that $L_1=r_2p_3-r_3p_2$ and $L_2=r_3p_1-r_1p_3$ (I can easily derive this from ...
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1answer
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Mathematical prediction of synchronizing multiple cams.

I am not a mathematician so be gentle with me but what kind of math equations (or what field of mathematics) would be needed for the following: I have designs for an old-style mechanical device I am ...
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1answer
48 views

Simple rocket model

I have a problem creating a model for a horizontal rocket flight. I want to model a rocket with constant force, drag constant and gravity. I also have to account for a changing mass and drag. I know ...
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0answers
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Mixed convection - Matlab

So I am trying to solve the following mixed convection problem in Matlab: for $x=0: u=v=0, T=Th$ (heated wall); for $x=L: u=v=0, \frac{\partial T}{\partial x}=0$; for $y=0: u=v=0, \frac{partial T}{\...
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2answers
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The position of a point mass that moves in a straight line .. / Determine the units

The position of a point mass that moves in a straight line is given by $x(t) = At^2 + Bt + C$, where $t$ is time. Determine the units of $A$, $B$ and $C$. The answer to the question is [A] = M/S^...
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0answers
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Elastic Body Simple Deformation

In continuum mechanics we can consider a reference frame $B = [0,1]$ along with a homogeneous deformation $F$ where $x = Fp$ for $x \in \mathbb{R}$ and $p \in [0,1]$ and $F = 2$ so $F[B] = [0,2]$. ...
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0answers
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Bead on rotating hoop with mass, determine the lagrangian

"Consider the system consisting of a bead of mass m sliding on a smooth circular wire hoop of mass 2m and radius R in a vertical plane, and the vertical plane containing the hoop free to rotate about ...
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5answers
71 views

Why don't we differentiate velocity wrt position in the Lagrangian?

In Analytic Mechanics, the Lagrangian is taken to be a function of $x$ and $\dot{x}$, where $x$ stands for position and is a function of time and $\dot{x}$ is its derivative wrt time. To set my ...
4
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1answer
22 views

A criterion for area preserving dynamical system

In my investigation of dynamical systems I was met with this seemingly easy question I could not find an answer to: If we have a two dimensional system of autonomous ODEs viewed as a 2D dynamical ...
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2answers
33 views

Difficult projectiles question

A particle $P$ of mass $m$ lies on a plane inclined at an angle $\alpha$ to the direction vector $\mathbf{i}$. At $t=0$ the particle is projected from the origin of the coordinate system with speed $U$...
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Taking Moments About a Point

This is a rod attached to a wall by a light inextensible string: And here is a diagram showing the forces acting: I want to find the normal reaction $R$ between the rod and wall using moments ...
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3answers
53 views

I did a Maths question on mechanics using calculus and physics students disagreed, why are the solutions different? [duplicate]

So, I did the following question and got 2 different answers. Question: A lighthouse located 300 metres from a straight shoreline sweeps its beam of light around in a circle at a constant rate of 1 ...
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0answers
35 views

About periodic trajectories of a Hamiltonian system

Consider a Hamiltonian system with Hamiltoniana $H (\mathbf{q}, \mathbf{p})$, where $H$ doesn't depend on time $t$. It is known that in some domain of phase space the trajectory of system are peiodic. ...
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2answers
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Compute the following line integral along a path of your choice (Finding potential)

Consider the following vector field: $$\vec A(x,y,z)=(yz)\hat i+(xz)\hat j+(xy)\hat k$$ Compute the line integral of $A$ along a path of your choice connecting $(0,0,0)$ to $(1,1,1).$ I recognise ...
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2answers
43 views

Second order non linear differential equation: Central force question

The problem is as below: I have derived that the particle satisfies the motion equation $$ \frac{d^2u}{d \theta ^2 } + u = \frac{F(1/u)}{mh^2u^2} $$ by Newton's Law, $u= 1/r$ and $h = r^2 \frac{d \...
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1answer
42 views

Stream function and Vorticity relationship with Streamlines

For a two-dimensional flow, the velocity is given by $\textbf u= (u(x,y,t),v(x,y,t),0)$. Define the stream function $ψ (x,y,t)$. Evaluate $(\textbf u·∇) ψ$ and deduce a relationship between the stream ...
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Time-$t$ map of a Hamiltonian flow: how to check twist property?

I would like to obtain a general formula to verify if a certain time-$t$ map of a Hamiltonian flow is twist. I have a Hamiltonian $1$ degree of freedom system $H=H(q(t),p(t))$, such that all orbits ...
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0answers
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How to model a point force with uncertain concentration point?

I consider a beam which is bent under influence of a point force concentrated at some point $\xi$ of the beam. The exact co-ordinate of $\xi$ is not known, but it is known a neighbourhood $(l_0,l_1)\...
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1answer
23 views

Conservation of Net Mechanical Energy in SHM

I have a question I wish to answer: Show that the simple harmonic motion solution of the simple pendulum in the form $$\theta (t) = A\cos ({\omega _0}t)$$ (constant A) conserves net mechanical energy ...
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0answers
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A soft question on Gauge Equivalence in Integrable Systems

I have a question about two well-known spectral problems in Integrable Systems. These are the Dirac and the ZS-AKNS spectral problems. They are are known to be gauge equivalent (please see equations (...
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0answers
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Calculate Universal Time for when an object in orbit reaches a given radius / altitude?

Assuming that an object in orbit WILL reach a given radius / altitude at some point in the future, how can I work out the exact time it will reach that point? Assume that the object is a Satellite in ...
2
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1answer
26 views

Gravitational Potential Energy Near the Surface of the Earth…

Okay so for an object of mass m at a distance r from the Earth's centre, the GPE is $$U(r) = {{ - GMm} \over r}$$ For an object at height z above the surface (which obviously means at radius $r = {r_{...
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1answer
47 views

Finding pressure using Bernoulli's Theorem

The inviscid irrotational flow around a circular cylinder of radius $a$ is described by the complex potential $$w =Uz+ \frac{Ua^2}{z}$$ where $U$ is a positive constant. I found $\psi=U \cos \theta (...
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1answer
22 views

Why is $\omega $ the natural/angular frequency?

Pardon me cause I'm a little confused. If we have something like: $y=A\sin \left( \omega t-\delta \right)$ why would $\omega$ be considered the natural frequency? I always thought the frequency of a ...
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1answer
66 views

Finding stagnation points and stream function

Sorry for the lack of latex. The question I want to ask would need all this info and it would take very long to write it. (a) Irrotational flow means $\nabla \times \textbf u =0$ so we can define ...
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1answer
22 views

Viscous fluid boundary condition

Consider an incompressible viscous fluid of kinematic viscosity $ν$ , dynamic viscosity $µ$ and density $ρ$ . A viscous boundary layer is located over a solid surface at $y = 0$ and $x > 0$. The flow ...
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1answer
84 views

Employing Newton's Laws with differential equations [closed]

Going through some problem sheets from previous semesters and can't find a full solution for this question so was wondering what the answers might be. A particle of mass $m$ moves on the $x$ axis ...
1
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1answer
25 views

Manipulating vorticity equation

We have $\omega = (0,0, \xi(x,y,t))$ and $\textbf u =(u(x,y,t),v(x,y,t),0)$ and that $$\frac{\partial \xi}{\partial t} +u \frac{\partial \xi}{\partial x} +v\frac{\partial \xi}{\partial y}=0$$ is a ...
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1answer
48 views

Finding pdes of velocity component and pressure

An incompressible viscous fluid of constant density $ρ$ and kinematic viscosity $ν$ occupies the space above a solid boundary at $y = 0$ in two-dimensional Cartesian coordinates $(x,y)$. For time $t &...
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1answer
25 views

Finding an expression for velocity [closed]

Consider an annulus formed by two circular cylinders, with one cylinder inside the other. The inner cylinder has radius $a$ and the outer cylinder has radius $b$. The cylinders have a common axis, and ...