Questions relating to Newton's Laws of Motion

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0answers
25 views

Tensorial product, a simple question

I need to find the components of $D$: $$D=a\otimes a$$ where $a$ is a tensor of order 2. Thanks!
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0answers
19 views

Forced harmonic oscillator [closed]

I am completely stuck, any help or advice would be appreciated, big thank you. The motion of a forced harmonic oscillator can be described by the differential equation $$y''(t) + γ\,y'(t) + y = ...
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1answer
59 views

Need some help with an easy mechanics question.

The distance between a boy and the shelf is R. He wants to throw a ball of mass m with an initial speed v such that it hits the top of the shelf of a height h. 1) Show that a ball thrown at the ...
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3answers
48 views

Mechanics Question

A particle $P$ of mass $0.6\text{ kg}$ moves upwards along a line of greatest slope of a plane inclined at $18°$ to the horizontal. The deceleration of $p$ is 4 ms−2 The first part asks us to find ...
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0answers
51 views

fluid dynamics: sketching streamlines of velocity field when there is only one non-zero velocity component

I have been asked to sketch the streamlines in the $x_2$$x_2$-plane for the two-dimensional field: $$v=(x_1x_2,0,0)$$ All the examples I have seen of this kind of question use the $v_1$ and $v_2$ ...
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2answers
89 views

A problem about symplectic manifolds in Arnold's book

There is a problem in Arnold's Mathematical Methods of Classical Mechanics which says that: Show that the map $A: \mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$ sending $(p, q) \rightarrow (P(p,q), ...
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0answers
33 views

Finding the centre of mass and moment of inertia about a point of a laminate

A triangular laminate OAB of uniform area density,$\rho$, has vertices at (0,0), (0,2) and (1,0). The moment of inertia about an axis through O,perpendicular to the plane of the laminate is denoted by ...
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1answer
27 views

Resultant force on a pulley doesn't seem logical to me

I have a simple mechanics problem (to the left) that can be solved using newton's second law. Strings are all the same and are light and inextensible and vertical. When we try finding the tension ...
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1answer
14 views

Vector differential equation problem

I was trying to do this question from a past paper, but I'm not sure how to proceed. The question is: A particle of mass $m$ moves subject to a force $\mathbf F = A(y\mathbf i + x\mathbf j)$ where ...
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1answer
20 views

How do you get a measure space out of a dynamical system?

I'm reading a book on ergodic theory (by Cesar Silva), and also have read Stein and Shakarchi's third book on undergraduate analysis, where there is a section devoted to some ergodic theory. Both ...
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1answer
12 views

When finding the frequencies of normal modes, can you have a negative frequency?

Do you simply just consider the positive solutions? I tried a google search but didn't find anything quickly. The work I am studying is Lagrangian systems.
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0answers
36 views

Linearization of Implicit ODE (Equations of Motion)

let's say we have a system with vector $q_{(t)}$ representing the degrees of freedom (DoF), and state vector $ x_{(t)} = \left \{ \begin{array}{c c} q_{(t)} \\ \dot{q_{(t)}} \end{array} \right \}$ ...
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2answers
61 views

My equations are inconsistent. Could someone help me see the error of my ways?

I'm given the following problem: Now, the following is my attempt at a solution: I have two problems: (1) With my equation for $v$, I end up having to take the log of $0$, which is obviously ...
0
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1answer
34 views

fluid dynamics- is this flow incompressible?

I have been given a flow with Lagrange path trajectories: $$x(\alpha,t)=(\alpha_1\cos(t)+\alpha_2\sin(t),\alpha_2\cos(t)-\alpha_1\sin(t),\alpha_3)$$ and I have to determine whether it is an ...
2
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1answer
30 views

Simple vector cross product question.

How do you compute the cross product of two vectors in the following form: $r_{u} = \cos(\theta)\textbf{x} + \sin(\theta)\textbf{y}$ $r_{v} = -\cos(\theta)\textbf{x} - \sin(\theta)\textbf{y}$ I ...
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0answers
38 views

Finding angular momentum about the center of mass?

If we have a couple of particles of an equal, unknown mass: $r_{+} = (c + e^{-Bt} \cos({\theta}))\textbf{x} + (d + e^{-Bt} \sin({\theta}))\textbf{y}$ $r_{-} = (c - e^{-Bt} \cos({\theta}))\textbf{x} ...
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1answer
35 views

Using conservation of energy on inclined plane [closed]

Hello, I'm trying to solve this using conservation of energy. So for (a) KE+PE at the top = KE+PE at the bottom and then relate to energy alone plane. However, I'm unsure how to state the KE and PE ...
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1answer
24 views

Conservation of Energy

Hello, I am trying to solve the following question using conservation of energy. Part (a) is fine, I have compared it with suvat questions and I have got: $z_{max} = v_0^2sin^2a/2g$ I am ...
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0answers
21 views

Lagrangian of a pendulum inside af disc

I'm currently struggling with a mechanical problem, where I need to find a relationship between the two angles in a mechanical system. The two equations of motions were derived using the Lagrange ...
1
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1answer
39 views

How to calculate this particle's position, velocity and acceleration, each as functions of time.

I'm given a particle of mass $m$, at position $x$, moving through 1-space dimenion with velocity $v=\gamma(d-x)$ for constant $\gamma, d.$ I'm also given that the particle starts from $x=0$ at $t=0$. ...
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1answer
74 views

What are the properties of this Poisson algebra?

I have the following (real) quantities (which are from a Classical Mechanics problem): $$A_1=\frac 1 4(x^2 +p_x^2-y^2-p_y^2 ) \quad A_2=\frac 1 2(x y +p_x p_y)$$ $$A_3=\frac 1 2(x p_y - y p_x )$$ ...
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1answer
59 views

Mechanics - A particle moving in an ellipse

A and B were fine, for C I took componentwise vectors x(t) and y(t), differentiated twice and applied $1/2mv^2$. For (d) I have no idea, and for (e) I applied: $1/2m(v_2^2-v_1^2)$; From then on, ...
0
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0answers
29 views

continuum mechanics: deriving eulerian 'conservation of mass' and 'balance of linear momentum' equations for cylinder with variable cross-section

I know the equations for conservation of mass and balance of linear momentum for a 1 dimensional cylinder with fixed cross-sectional area, A. Is there a way of deriving these equations in fluid ...
1
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1answer
27 views

Euler-Lagrange Eqn to find eqn of a straight line

I'm trying to see how we use the E-L equation \begin{equation} L_x(t,q(t),q'(t))-\dfrac{d}{dt}L_v(t,q(t),q'(t))=0 \end{equation} to find the shortest distance between two points in the Euclidean ...
0
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1answer
17 views

Time taken to reach height below initial point

I am trying to solve the following. It is in my notes, I believe I need to solve: $-H = v_0cos\alpha*t\bf{j}$$+(v_0sin\alpha*t-gt^2/2)\bf{k}$ as a quadratic equation? If so, how do I solve a ...
0
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4answers
106 views

Newtonian Mechanics - Differential equation

If we combine Newton's second law of motion i.e. $F=m\ddot{x}$ and Newton's law of gravity i.e., $$ F=G\frac{mM}{x^2}, $$ where $x$ is distance, we obtain the following equation: ...
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1answer
63 views

Mechanics problem. I solve it with 2 methods and get 2 different answers.

The problem: (I'm having trouble with (ii) but I listed (i) because one of the answers depend on it) Drawing it out would help. The point O is 20m above horizontal ground. A particle is projected ...
0
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1answer
56 views

Cartesian vector field to vector field

Ok so I have a given vector field in Cartesian coordinates, say \begin{align*} \textbf{v}(x,y)=\frac{dx}{dt}\hat{\textbf{e}}_{1}+\frac{dy}{dt}\hat{\textbf{e}}_{2} \end{align*} Where $dx/dt$ and ...
0
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2answers
50 views

Newtons Laws and integration

A particle of mass m moves along a straight line (which, without loss of generality we may consider to be the x-axis) under the infuence of a constant force F. Suppose that the particle starts at x = ...
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1answer
25 views

Continuum mechanics: Find the material (Lagrange) particle trajectories using the (known) Eulerian density and velocity functions

I have been given the Eulerian density function for a one-dimensional flow in the region $x\ge0$,$t\ge0$ to be: $$\rho(x,t)=(t+1)e^{-(t+1)x}$$ and have used the given fact that $v(0,t)=0$ and the ...
0
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2answers
51 views

Inverse of $r sin(\omega t) + v t$?

I am wondering if there is an inverse for this function, $x(t)=r sin(\omega t) + v t$. The inverse function theorem suggests that an inverse for this function does exist, although it may have to be ...
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0answers
26 views

Simple mechanics question - trapeze

A woman of mass 50kg swings on a light trapeze, i.e. a light seat suspended from a fixed point by a light rope. Her centre of mass, G, moves on the arc of a circle of radius 9m and centre ...
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1answer
12 views

Evaluating vectors along paths

If A = (3x$^2$-6yz)$\mathbf{i}$ + 2(y+3xz)$\mathbf{j}$ + (1-4xyz$^2$)$\mathbf{k}$ how do I evaluate $\int$ A $\bullet$ d$\mathbf{r}$ from (0,0,0) to (1,1,1) along the path x=t, y=t$^2$, z=t$^3$?
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3answers
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Moment of inertia of a disc

In my mechanics textbook there is a derivation of the moment of inertia of a disc of mass $m$ and radius $r$ about an axis through its centre and perpendicular to its plane surface, which goes ...
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2answers
33 views

Linear Oscillator without Friction

I already figured out the majority of the solution to this problem but I just need help on the last part. The question is: Consider the linear oscillator without friction: $$m\frac{d^2x}{dt^2}=-kx$$ ...
0
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1answer
54 views

Evaluating space curves

What does it mean to evaluate a function on a space curve? Eg for the following question Consider the space curve defined by the following position vector: $$r(t) = \cos t \ i + \sin t \ j + t \ k$$ ...
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0answers
39 views

Line Integral Problem best or easier solved using geometry?

Does anyone have any recommendation on a line integral problem involving vector fields (aka work) such that evaluating the resulting line integral using parameterization would be significantly ...
0
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1answer
83 views

Differential Equation - Falling Projectile - Help getting started?

Here is the question I'm dealing with: A ball with mass 0.15 kg is thrown upward with initial velocity 20 m/s from the roof of a building 30 m high. There is a force due to air resistance of ...
0
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1answer
38 views

Determine the motion for all time

In the frame $F=[0,\hat{k}]$, a particle of mass $m$, whose trajectory $[0,\infty)\xrightarrow{\rm r}\mathbb{R}$ is $r=z\hat{k}$ moves in response to a force ...
0
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2answers
32 views

Help with basic trigonometric (Physics) problem

I am re-learning basic Physics and I would like to know if I followed the correct steps, so I can continue doing more exercises. The problem says: "A person kicks a ball from the surface of a playing ...
4
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1answer
77 views

How must I understand concepts equations of physics?

I teach myself mathematics, but those days I wanted to learn about General relativity (not to pursue in it but only to have some background), perhaps because I am very curious to learn why exactly We ...
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0answers
40 views

Derivation of Euler lagrange for Yang Mills

I need someone to sketch the conventional steps(from variation to vanishing of arbitrary function chosen , etc, etc) of Classical Yang-Mills. If using exterior product product could you emphasize any ...
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3answers
81 views

Analytical mechanics book

On my PHD I have now one subject: Analysis on manifolds and the fourth chapter of a book I need to learn is Analytical mechanics. But the book is really not good to read, it is too hard. So I need ...
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0answers
162 views

Short proof of Jacobi identity for the Poisson bracket — is this valid?

I've been trying to make a short proof for the Jacobi identity for the Poisson bracket on phase space. My idea goes like this, we know the following: $$ \frac{\mathrm{d}}{\mathrm{dt}} \{f,g\} = ...
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0answers
19 views

Small oscillations, Lagrangian about equilibrium

I have a Lagrangian $$ L = \underbrace{\dot{x}^2 + \frac{1}{2}\dot{\theta}^2 + \dot{x}\dot{\theta}\cos\theta}_{\text{kinetic energy}} \underbrace{ - \frac{3}{4}x^2 + \cos\theta}_{\text{-ve potential ...
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1answer
39 views

Elements in stiffness matrix

I do not know if I am at the right address here, but I'll just ask. Is the following correct? Every element in the stiffness matrix represents the displacement of every element, when exerting an ...
1
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0answers
28 views

Horizontal drift of snowflake

I wonder if the random-walk dynamics of falling snowflakes is understood well enough to estimate the likely sideways drift of a single snowflake falling in a windless environment, from its cloud ...
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0answers
25 views

Spinors and rotations in 3D

I have a cylinder with the axis coincident with the $x$ axis of a $3D$ euclidean reference frame. I have three possible rotations: 1) Roll: rotation of an angle $\Psi$ around the $x$ axis. 2) ...
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1answer
99 views

Am I correct? Distance when velocity depends on position

I hope I'm in the right place to be asking this: I'm looking for somebody that knows better than I who can verify whether or not I've done things correctly. In trying to implement a feature for a game ...
8
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2answers
60 views

Proof of a theorem about oscillation [duplicate]

There is a theorem in page 100 of Arnold's Mathematical Methods of Classical Mechanics, which says that: If $\cfrac{dx}{dt} = f(x) = Ax + R_2(x)$, where $A = \cfrac{\partial f}{\partial x}|_{x = ...