Questions relating to Newton's Laws of Motion

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0answers
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Horicontal tension in catenary [on hold]

How is the tension at vertex equal to $T^o = wc$. Where $c$ is parameter and $w$ is weight per unit length.
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1answer
22 views

Does the momentum of a particle depend on its position [on hold]

By definition: $\displaystyle \dot{x}=\frac{p}{m}$, where $p$ is the momentum of the particle $m$ is the mass, and $\dot{x}$ the velocity. As the velocity depends on the position of the particle(?) ...
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2answers
39 views

Is velocity a function of displacemnt?

The velocity $\displaystyle\vec{v}$ of a particle $=\frac{d\vec{x}}{dt}$. So surely this means that $\vec{v}$ is dependent on the position of the particle?
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0answers
16 views

Trajectory of an object under gravity

Is there an equation (cartesian/polar)depicting the trajectory of the motion of an object relative to another (in a two body system) under gravity?
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0answers
17 views

Simple Harmonic Motion; Tension in Elastic rope

I'm struggling to model this question out correctly. A glider and its pilot have total mass $230$ kg. The glider lands on a horizontal airstrip and when its speed is $16$ m/s it hooks on to the ...
1
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1answer
55 views

How does angular acceleration change with revolutions?

So, for a section of my EPQ (A-Level, Extended Project Qualification), I am trying to analyse a hypothetical circular accelerator using the angular motion equations for constant angular acceleration. ...
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3answers
37 views

Question on finding moment of inertia

The question is : "the flat surface of a hemisphere of radius 'r' is cemented to one flat surface of a cylinder of the same radius and of the same material. If the length of the cylinder be 'l' and ...
0
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0answers
22 views

Non-integrability and splitting of separatrices

It is well-known that the (first-order) Melnikov method is the standard technique to detect non-integrability of a perturbed system of ordinary differential equations or maps. Namely, the unperturbed ...
1
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1answer
28 views

Integrating the separable, first-order ordinary differential equation $m \frac{dv}{dt} = mg - av$

I can't solve the very first problem from Slater & Frank's book, and have no one to help me (I'm self-studying it in these vacations): A particle moves in a vertical line under the action ...
1
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1answer
50 views

Boundary conditions for vibrating beam

I'm solving the equation for the transverse vibrations of a Euler-Bernoulli beam fixed at both ends and subject to axial loading (as per this diagram). It's a similar problem to that described by Rao ...
-1
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0answers
21 views

How to estimate a mass-spring-damper parameters in MATLAB using RLS and OLS?

Assume that we have the differential equation of a mass-spring-damper model as follows: $$m\dfrac{d^2 y}{dt^2} + c \dfrac{dy}{dt} + ky(t) = F(t)$$ where m is the mass, k the spring's stiffness ...
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1answer
18 views

How to derive the equilibrium length of a spring?

A straight wire rotates with constant angular speed $\omega$ about one of its end points (the origin $O$) in a horizontal plane containing $e_1$ and $e_2$. A bead of mass $m$ is free to slide along ...
0
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2answers
36 views

Position vector of a particle moving with constant speed on a straight line

Suppose we have a particle which starts from a point $A$ and moves with constant speed $u$ along the line $AB$. One wants to show that the position vector $\mathbf{x}$ of the particle at time $t$ is ...
1
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1answer
18 views

analytical hard sphere collision condition with periodic boundary conditions

Hello Stack Exchange Mathematics, I am curious if there is an analytical or efficient numerical solution for the collision of hard spheres in a rectangular unit cell with periodic boundary ...
0
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2answers
38 views

Applied Mathematics: Two particles of mass $m$ moving smoothly along the $x$-axis connected by a spring?

Two particles of mass $m$ can move smoothly along the $x$-axis and are connected by a spring of natural length $l$ and spring constant $k$. Here are the questions and my proposed answers. I'm stuck ...
1
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1answer
31 views

What is a Hamiltonian in a Poisson algebra?

Classical physics on the phase space $T^* M$ (with $M$ a smooth manifold) is done mostly in the following way: one endows $T^*M$ with a Riemannian structure $g^*$ (that will give the kinetic term) and ...
0
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2answers
44 views

Is this the pullback of a Hamiltonian flow?

In this reference just in the beginning the author gives the theorem (Theorem 1) of the conservation of a Hamiltonian flow $\phi_t$. According to it this means that $$ \frac{d}{dt}\phi_t^* H = 0$$ I ...
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2answers
71 views

What is “Phase Space” in differential equations and classical mechanics?

I started reading a book on ordinary differential equations by Vladimir Arnold. He started his book of with the idea of phase space and phase points. I seem to be confused what the general idea of ...
5
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1answer
63 views

Periodic orbits on centre manifold

I am interested in periodic orbits of mechanical systems of second-order dynamics with no damping, i.e. governed by an equation of the type \begin{equation}(1)\quad \ddot x + f(x)=0 \end{equation} ...
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0answers
30 views

Question on inclined plane with velocity of projection $u$ both up and down plane.

Show that for a given velocity of projection the maximum range down an inclined plane of inclination $\alpha$ is greater than up the plane in the ratio $$\frac{1+\sin(\alpha)}{1-\sin(\alpha)}$$ Let ...
2
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2answers
59 views

How to calculate the velocity for this [closed]

I ve tried to solve this problem in so many ways but still didn't manage to do it... What would be the correct way to solve it please? This arm of this mechanism has a length of 0,2m. The piston ...
2
votes
2answers
43 views

Calculate velocity of a mechanism

I have tried many times to solve this problem in different ways but with nos success: The angular velocity of the shaft AB is 3rad/s counterclockwise. Calculate the velocity of the shafts BD and DE. ...
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1answer
79 views

How to calculate the velocity for such a situation [duplicate]

I am not sure whether this is the correct sub stackexchange to ask my question but I ll have a try. I ve tried to solve this problem in so many ways but still didn't manage to do it... What would ...
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0answers
19 views

Pullback of a Hamiltonian

I understand that a Hamiltonian vector field $H$ creates a Hamiltonian flow $\phi_t$. Now, in order to prove that the Hamiltonian is conserved one uses the following \begin{eqnarray*} ...
1
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0answers
29 views

Everything about Legendre transform

The Legendre transform, or transformation, seems to have many properties which are useful in different fields. For example: It switches between Lagrangian and Hamiltonian formalism in mechanics / ...
2
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2answers
53 views

Elastic Strings and Simple Harmonic Motion

The Ceiling of a hall is 15m above the floor. A vertical elastic string of natural length 5m and modulus of elasticity 6N has one end attached to the ceiling and the other end attached to the ...
3
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3answers
362 views

How to solve harmonic oscillator differential equation: $\dfrac{d^2x}{dt^2} + \dfrac{kx}{m} = 0$

I am able to solve simple differential equations like : $$\dfrac{dy}{dx} = 3x^2 + 2x$$ We simply bring $dx$ to other site and integrate. But how do we find solutions of differential equations like ...
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0answers
26 views

Is time-1 map of a Hamiltonian vector field defined on a cylinder always twist?

Suppose I have a one degree of freedom analytic Hamiltonian $H(q,p)$ defined on a semi-infinite cylinder, i.e. $(q,p) \in \mathbb{T} \times \mathbb{R}^{+}$, such that all level sets $H(q,p)=c$ are ...
1
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1answer
23 views

Find the starting height from which a particle projected horizontally at $40$m/s travels $100$m [closed]

This question is in my homework: A particle is projected horizontally with speed 40m/s from a point A. It hits the ground 100m horizontally from A. Find the height of A. Would it be possible for ...
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votes
1answer
11 views

Find the time it takes for a particle to be traveling perpendicular to original projection

I received this question in my recent homework and don't know how to approach it: "A particle is projected from a height of 30m above the ground, with initial velocity 3i+4j. Find the time it takes ...
0
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1answer
30 views

Classical mechanics principle of least action

I don't understand here what does the book mean by expanding in terms of $\delta{q}$ and $\delta{\dot{q}}$ can someone explain that part. I don't understand how did he get that final step? The ...
2
votes
1answer
41 views

Applied Mathematics: Spherical Polar Coordinates and Newton's Second Law

I've been attempting this question but can't seem to find a solution. Question: A particle of mass $m$ moves under the influence of a force which, in spherical polar coordinates, only acts in the ...
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1answer
50 views

Mechanics derivation that I don't understand

I am reading the section of method of calculus of variations from Goldstein, where he tries to find a curve for which given line integral has a stationary value. After some steps into the derivation ...
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2answers
19 views

Finding a constant in a particle motion problem using the energy equation

I have found V in terms of x, and then I have found the energy equation for x=1 and x=2. I've then set them equal to one another and solved, finding lambda = 20. I didn't use the values v=4 and v=2; ...
0
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0answers
35 views

Rope wrapped around pole Friction

A rope is wrapped $M$ whole turns round a cylindrical post, the two ends of the rope going in opposite directions. The coefficient of friction between rope and post is $0.25$. It is desired that by ...
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1answer
18 views

Resisted motion involving densities? [closed]

A space craft in the shape of a cylinder has mass $N$ and the area of its cross section is $B$. It is moving at constant velocity but meets a dust cloud, with the dust sticking to the spaceship. If ...
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0answers
26 views

Definition of Work Done

I am trying to make sense of the meaning of the definition of work. The original definition of work was also known as "the weight lifted through a height." I was hoping that our mathematical ...
1
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1answer
49 views

Simple Harmonic Motion (SHM) With Dry Friction

Consider a mass $m$ at position $x(t)$ on a rough horizontal table attached to the origin by a spring with constant $k$ (restoring force $−kx$) and with a dry friction force $f$ \begin{equation} f= ...
2
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0answers
32 views

Is the Hamiltonian conserved or not?

The question is the very last sentence at the end of this post. In this post, I'll first show that the Hamiltonian is conserved since it does not have explicit dependence on time and then show that ...
0
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0answers
26 views

What is this dynamics question asking me to do?

I've made it to the the final part of this question and got the expression with arctan in it: However, I'm not sure what the final part is asking me to do. I know I've got to plug the new values ...
0
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0answers
29 views

Kepler orbital elements from state vectors

Well given the 6 common Kepler orbital elements: Eccentricity $e$ Semimajor axis $a$ inclination $i$ Longitude of ascending node $\Omega$ Argument of periapsis $\omega$ True anomaly $\nu$ As can ...
0
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0answers
15 views

Transforming an infinitesimal line element, dx, to 1/2(curl(u)/\dx)? What does this mean physically?

Consider transforming an infinitesimal line element,say dx, to 1/2(curl(u)/\dx)? Where curl denoted /\ here, and dx is an infinitesimal 3d vector, and u is the displacement vector --What does this ...
0
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0answers
36 views

Tension on the lower rod of a governor for an engine.

A governor is used for controlling the speed of an engine. This consists of a framework set in a vertical plane which rotates with the vertical shaft driven by the engine. $A$ is a fixed point on ...
0
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1answer
18 views

A bead is threaded on a friction-less vertical wire loop of radius $R$.

The question is the very last sentence at the end of this post. In this post, I'll demonstrate how I reach to a contradiction(the conditions mentioned in conjecture 1 should be satisfied by all ...
1
vote
1answer
28 views

How to find the instantaneous angular velocity vector?

Let $(R_0): O_0 \vec x_0 \vec y_0 \vec z_0$ and $(R): O\vec x \vec y \vec z$ be two given orthonormal frames. The unique vector $\Omega_{0,1} = \Omega_{0,1}(t)$, given by the following three ...
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1answer
66 views

Differential equations and classic differential geometry are mostly impossible to understand [closed]

I need advice on my studies of mathematics... I'm really depressed because it's impossible for me to understand many important parts of books such as Tenenbaum & Pollard "Ordinary Differential ...
1
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1answer
49 views

Why is horizontal tension constant in a hanging chain (catenary)? [closed]

Why is the horizontal tension always constant for all points in a hanging chain, which forms a catenary? Also, shouldn't the lowest point of the catenary be free hanging and not have any tension??
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0answers
5 views

equations seperating and quadratures?

I am solving the following problem. It it sometimes occurs that the generalized coordinates appear separately in the kinetic energy and potential energy in such a manner that T and V maybe written in ...
0
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0answers
49 views

Lagrangian in pseudo-Riemannian manifold and geodesics

I'm trying to solve the following problem without success. Let $V$ be a smooth function on a pseudo-Riemannian manifold $(M, g)$, which is either bounded from above or from below. Show that there ...
0
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1answer
20 views

Question about a g-force simulator in question 6, part b, regarding the moments in a balanced system at rest.

The following is regarding question $6$, part $b$, in the following link: https://thol.sunway.edu.my/examdbase/alv/math/p3/math_p3_j96.pdf Using the principle of moments and considering the case ...