Questions relating to Newton's Laws of Motion

learn more… | top users | synonyms

0
votes
0answers
24 views

How do i prove this formula? [on hold]

How can i prove this formula? $\dot{J}(\vec{X},t)=J(\vec{X},t)\dot{}div(\vec{V},t)$, Where $J(\vec{X},t)=det(\nabla_{\vec{X}}{\lambda_{t_0,t}(\vec{X},t)})$ and $\lambda_{t_0,t}$ is the movement ...
0
votes
0answers
6 views

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 $F[B] = [0,2]$. ...
0
votes
0answers
17 views

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 ...
5
votes
5answers
62 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
votes
1answer
21 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 ...
0
votes
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 ...
0
votes
0answers
14 views

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 ...
1
vote
3answers
52 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 ...
0
votes
0answers
32 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. ...
1
vote
2answers
34 views

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 ...
-1
votes
0answers
32 views

How to prove the orbit of planet is a circle or ellipse?

I think it is enough by $F=ma$ and $F=\frac{GMmr}{|r|^3}$.But I get stuck in a ODE $$ x'(t)=\frac{-GMx(t)}{(x^2(t)+y^2(t))^{3/2}} $$ How to deal it ? Or how to prove the orbit of planet is a circle ...
2
votes
2answers
40 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 ...
1
vote
1answer
38 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 ...
0
votes
0answers
30 views

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 ...
0
votes
0answers
11 views

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 ...
0
votes
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 ...
1
vote
0answers
48 views

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 ...
0
votes
0answers
4 views

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
votes
1answer
25 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 = ...
0
votes
1answer
46 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 ...
0
votes
1answer
19 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 ...
1
vote
1answer
64 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 ...
1
vote
1answer
21 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 ...
-1
votes
1answer
79 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
vote
1answer
24 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 ...
2
votes
1answer
46 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 ...
-1
votes
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 ...
1
vote
1answer
83 views

Figuring out velocity,acceleration, work of a particle given that we know its position vector.

Recently this question came up in a problem class of mine. A particle moves in such a way that its position vector at any time $t$ is $\vec{r}(t)=\pmatrix{A\sin{\omega t}\\A\cos{\omega t}\\Bt^2}$, ...
0
votes
2answers
18 views

Resolving Forces in an inclined plane ( Mechanics )

I have exams after a few days and I'm doing all I can to understand the concept of resolving forces. With hard luck and a few hours of devotion, I acquired basic knowledge on Resolving Forces and was ...
0
votes
1answer
27 views

Understanding shift to polar coordinates in the newtonian central force system of ODE's

This is from Hirsch, Smale and Devaney chapter 13. The larger context is moving towards blowing up the singularity at the origin of the system. The second order ODE is defined, $X:t\rightarrow ...
0
votes
1answer
23 views

What direction would a hinge's reaction force point?

In my homework question there is a ladder with the bottom touching the smooth floor and the other end is attached to a hinge. I need to draw a force diagram and use that to find the normal. There is ...
0
votes
1answer
25 views

Object of unknown mass in simple harmonic motion… [closed]

An object with mass m in simple harmonic motion on a vertical spring is observed for 5 full oscillations. The time is measured to be 13 seconds. What is the angular frequency (3dp)? A previous ...
0
votes
1answer
22 views

A question about a pendulum moving in simple harmonic motion

For a 2.7kg mass oscillating in simple harmonic motion (spring constant: 360N/m) with an amplitude of oscillations measured at 3.4cm. How do I calculate the total mechanical energy, maximum speed and ...
0
votes
1answer
16 views

Integrating equation with square on the bottom.

Say you are working with acceleration as a function of displacement and you are using calculus. $a = \frac{1}{(s - 600)^2}$. If you wanted to obtain velocity you'd use $a = v\frac{dv}{ds}$ so $vdv = ...
0
votes
0answers
6 views

Angular velocity using Euler's dynamical equation

using Euler's dynamical equation for force free motion of a rigid body,symmetrical about Z-principal axis,with angular velocity $W=(w_1,w_2,w_3)$ where $I=1,2,3$ are the components along the three ...
-1
votes
1answer
21 views

Motion of a pendulum equation in the George Simmons book on differential equations [closed]

I just can't understand the transition between this two formulas, why $dt$ becomes $T/4$. Can anybody help me with that?enter image description here
0
votes
0answers
22 views

Sketch the phase portrait

I'm asked to sketch the phase portrait for the potential given below. Also below is my attempt of sketching the phase portrait. I appreciate that my sketch is not fantastic but is it correct? phase ...
3
votes
2answers
31 views

What does it mean to use levi civita symbol with Poisson brackets in this way

I'm doing some studies in mathematical methods for physics and I came across something that I don't really understand. I have only been using the $\epsilon_{ijk}$ when I cross some vectors or ...
0
votes
0answers
28 views

Projectile with Friction

I am trying to find the optimal angle for a projectile to travel a maximum horizontal distance. I've been given the projection has mass $1kg$ and initial velocity $150m/s$. The equation for air ...
1
vote
3answers
32 views

Relative velocity from points' point of view

Edit: The terminology might be imprecise. Please pay attention to the picture I drew to explain my problems. I will appreciate an edit that will ensure the terminology is no longer disputable. This ...
0
votes
1answer
35 views

configuration spaces in mathematics and in physics

On the Wekipedia website Configuration space , there are two configuration spaces defined. One is Configuration spaces in physics, the other is Configuration spaces in mathematics. Question. Do ...
1
vote
1answer
20 views

Find the equilibria

Consider the equation $\ddot s = s-s^3.$ Let $m=1.$ 1) Write this as a first order system. Let $\dot s=v.$ Then we get $\dot v=s-s^3.$ So first order system is $$\begin{pmatrix} \dot{s} \\ \dot{v} ...
0
votes
0answers
20 views

Find the potential energy and sketch it

I'm given the equation $\ddot s=s-s^3.$ I'm asked to compute the potential energy and sketch it. I'm also given $m=1.$ To do this I have done the following: $$F=ma=\ddot ...
0
votes
2answers
27 views

Write the following equation as a first order system, then find the potential energy and sketch it

I'm asked to write the following as a first order system: $$\ddot s=s-s^3$$ In order to do this I have set $v=\dot s.$ This then gives me $$\dot v=s-s^3.$$ Is this correct? Next I'm asked to ...
1
vote
1answer
29 views

Mechanics ODE Problem Particle

I have a particle of mass $M$ which moves with a velocity $v$, such that; $$Mv' = -Mg - kv^2$$ where $g$ and $k$ are positive, and its initial velocity is $U$, i.e $v(0) = U$. I am then told that ...
0
votes
1answer
44 views

How to rewrite/solve this differential equation

\begin{equation} \sin(\theta + d\theta) = \sqrt{1 + \frac{dy}{y}}\cdot{\sin(\theta)} \end{equation} I think this is a non-linear and non homogeneous first order equation. I found this whilst trying to ...
1
vote
1answer
29 views

Finding fixed points of an equation when the derivative is not defined

For a dynamical system governed by the equation $f(x) = \mathrm{d}y/\mathrm{d}x = 2(1-x^2)^{1/2}$. Find stable and unstable fixed points. The fixed points for the above equation are $+1$ and ...
0
votes
1answer
16 views

MOI about a diagonal

If by taking a thin rod, and finding its Moment of Inertia about an axis, say through the mid point of its side, one can observe that stretching the rod uniformly along the axis of rotation will give ...
0
votes
0answers
30 views

The definition of the First Variation - Calculus of Variation

I have the following definition of the functional derivative $ \frac{\delta S}{\delta\gamma}$, where $S$ is my functional and $\gamma$ is a curve: $$\tag{1} \int^B_A \frac{\delta S}{\delta\gamma} ...
1
vote
3answers
39 views

Moment of Inertia (Square Laminas)

If I have a uniform square lamina of side length 2a and intend to find its Moment Of Inertia about a perpendicular axis to its plane, is there a general formula for this? If there isn't, I have tried ...