Questions relating to Newton's Laws of Motion

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Motion of particle - Mechanics problem

The problem is: At time $t=0s$ a particle is moving in a straight line and accelerating uniformly at $2 ms^{-2}$. $5s$ later it stops accelerating, but continues to move at a constant velocity ...
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How does rotational energy transfer to linear energy? [migrated]

So I have recently started looking into moments of inertia, and all that stuff. I have come to a question which has a plane inclined at some angle theta and a sphere at the peak. The G.P.E at the top ...
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+50

Showing that an oscillator has its amplitude reduced after completing half-cycle

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{cases} ...
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Moser's Twist Theorem for maps with reflection

Suppose I have a simple two dimensional integrable twist map, such as $x_{1}=x_{0}+y_{0}, \quad y_{1}=y_{0}$. Suppose that I perturb it in such a way that Moser's Twist Theorem is satisfied. What ...
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Equation for position, moving with a value J of the third derivative of position.

Q. An object moves in one dimension (described by an x-value) with a constant value J of the third derivative of position with respect to time. Write an equation for the position $x_0$ and an initial ...
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How do I find the angular momentum and the energy of a central force?

I've been studying classical mechanics with Symon's book and I'm having trouble when I have to find the energy and angular momentum for a given potencial if the particle moving in a circular orbit, ...
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1answer
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Impact of two bodies problem

A body of mass $M$ moving with a velocity $u$ collides with another of mass $m$ which rests on a table. Both the balls are perfectly elastic and smooth and the the body of $m$ is driven in a ...
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1answer
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Derivation of Euler Equation

In the following notes here I don't understand the very last line of proof of theorem 6.1 . We now use the fact that $\frac{\partial}{\partial a}S[x_a(t)]$ must be zero for any function ...
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Does the Euler-Lagrange equation have a series solution?

In classical mechanics the Euler-Lagrange equation of motion is a linear homogeneous ODE of second order, how come we do not have a series solution like other famous differential equations (Legendre, ...
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How much water could be stored?

We have two water storage tanks--- Tank A and Tank B --- on the roof of the upper storey of our two-storey house. The tanks are cylindrical in shape. Each of the two tanks have a circular opening ...
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1answer
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Does the Cauchy-Lipschitz theorem extend to higher order DEs?

The Cauchy-Lipschitz theorem says that the particular solution to a linear first order is unique provided $(x,y)$ are continuous over the domain. Does this extend to higher order DEs? Also does this ...
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1answer
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A Hamiltonian vector field on $\mathbb{R}^{4}$ which has closed orbit but does not have critical point

Is there a polynomial function $H:\mathbb{R}^{4} \to \mathbb{R}$ without critical points but the corresponding hamiltonian vector field possess at least one closed orbit?
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A problem on collision of two elastic spheres

Two elastic spheres, each of mass $m$ collide directly. Show that the energy lost during the impact is $m(u^2-v^2)/4$, where $u$ and $v$ are their relative velocities before and after impact. ...
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1answer
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Mechanics: Projectiles involving a ball shot out of a cannon, moving in the opposite direction of the shot

A child is playing with a toy cannon on the floor of a long railway carriage. The carriage is moving horizontally in a northerly direction with acceleration $a$. The child points the cannon ...
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Periodic phase curves

I'm currently reading Arnolds "Mathematical Methods of Classical Mechanics" and I'm having a hard time solving some of the problems in Chapter 2. I think that the following problem is fairly simple ...
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Significance of 'faces' in Stress tensor components?

I am trying to understand what the significance is of the face for which a force is acting on when talking about a stress tensor. Say we consider the components $T_{xx}$ and $T_{zx}$ of the stress ...
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A problem on Constrained Motion

Q. A particle is moving in a smooth curve under gravity and its velocity varies as the actual distance from the highest point. Prove that the curve is a cycloid. Attempt: The eq. of motion is ...
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what would a planetary orbit look like if gravity had constant magnitude?

Consider a unit-mass particle that is always experiencing a single unit-magnitude force towards the origin. This is a central force, but it is not one of the familiar ones, e.g. gravity whose ...
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Find the other forces on the plate that are equivalent to $210k$ $kN$

Determine the three forces acting on the plate that are equivalent to the force $R=210k$ $kN$ First I tried to find the components of $T1$$T2$ $T3$ So for $T1$ $\frac{-1i+2j+6k}{6.40}$ $T2$ ...
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Couple stress vector confusion

I am trying to sort out the difference between the stress tensor and the couple stress tensor. My previous understanding of the continuum mechanics model only involved the stress tensor. This ...
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1answer
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Can't we really add together two points on a manifold?

Let us consider a classical mechanical system with observables being smooth functions $C^\infty(X)$ on a Poisson manifold $X$. The algebra of observables will be denoted as $A$ Next we can define ...
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Expected number of rolls to roll every number [duplicate]

If I am rolling I die until I roll every number at least once, what is the expected value of times that I will need to roll the die? After a brief computer simulation, I got 15. But why is this the ...
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fluid mechanics help

for part a) i get $$ u = \partial_y \psi, \quad v = - \partial_x \psi$$ I need help with part d, if anyone can show me how to? thanks
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logic verification angular momentum

So I have the following question: Your given a uniform right circular cone with a half angle at the apex of $\alpha$, a height of b and radius of $p_0$. Choose a coordinate system $O_{xyz}$ such that ...
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Why we have to take the vector $\overrightarrow{e}$ ?

The differential equation of the balance of the momentum is $$\rho \frac{\partial{\overrightarrow{u}}}{\partial{t}}=-\rho (\overrightarrow{u} \cdot \nabla )\overrightarrow{u}-\nabla p+\rho ...
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Law of hydrostatic pressure

For a calm fluid of uniform density $\rho_0$, that occupies the space $W \subset \mathbb{R}^3$, and is subject to massive forces (per unit of mass) $\overrightarrow{b}(\overrightarrow{x})$, write the ...
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1answer
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Stress vector - Stress tensor

Is the definition of the stress vector the following? The stress vector is the force per unit surface. The stress tensor is the matrix $\{\sigma_{ij}(x,t)\}$ and its $(i,j)$-component is the ...
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1answer
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Moment of inertia tensor bounds

Can someone explain to me how did they determine the bounds in the following problem I did this in cylindrical coordinates but I would like to also understand this in order to fall back on cartesian ...
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Find the streamlines of the velocity field

I have to find the streamlines of the following velocity fields: $$u=x(1+2t), v=y$$ $$u=xy, v=0$$ I have done the following: $$\frac{dx}{u}=\frac{dy}{v} \Rightarrow ...
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1answer
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Question about moment of inertia calculation and logic

Question: Determine the moment of inertia for a quadrant of a uniform circular lamina of radius b. Here I saw the answer that,however I don't understand it first of all here is the answer and I ...
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1answer
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moment of inertia is same for any axis

A rigid body consists of three thin uniform rods, each of mass $m$ and length $2a$, held mutually perpendicular at their midpoints. Choose a coordinate system with axes along the rods. show that the ...
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Streamlines - Pathlines

Construct and draw the streamlines of the velocity field $u=az-bt, v=\frac{b}{4}z-cy, w=2(a-1)$. Calculate $c$ (as a function of the constants $a$, $b$) such that the flow field ...
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Show the relation $W$ is constant

If the space $W$ is constant (doesn't move with the flow), show that $$\frac{d}{dt}\int_{W}\left (\frac{1}{2}\rho |\overrightarrow{u}|^2+\rho \epsilon\right )dV=-\int_{\partial{W}}\rho \left ...
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The Arbitrary Lagrangian Eulerian (ALE) description

Considering that in an ALE framework, the partial derivative relation $\frac{\partial}{\partial t}=\left.\frac{\partial}{\partial t}\right|_{x}+\underline w.\nabla$ where $w$ is the mesh velocity, ...
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material derivative of displacement

I am slightly confused about what the material derivative of displacement is. $$\frac{D}{Dt}=\frac{\partial}{\partial t}+ v\frac{ \partial}{\partial x}$$ which means that for the displacement we ...
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Visco-elastic fluid reference

What is a good book on visco-elastic fluids for self-teaching after one has studied Gurtin's Intro to Continuum mechanics? Thanks!
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1answer
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How do we get the last relation?

I am looking at the conservation of momentum. The force at $W$ from the tensions at the boundary $\partial{W}$ is $$\overrightarrow{S}_{\partial{W}}=-\int_{\partial{W}}p \cdot ...
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3answers
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Calculate the position of rocket acted upon simultaneously by multiple thrusters?

I'm looking for an equation that will let me predict the position of a rocket after a period of time given that it is acted upon by multiple forces. By multiple forces I mean the main thruster force ...
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Concept of a continuum

I have to explain the concept of a continuum that is used for the description of the dynamic behaviour of the fluids, and to explain how this concept is related on the one side with the laboratory ...
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Deriving energy equation (Kinetic)

A particle of mass $m$ moves on the $x$-axis under a force $$F(x)=-2x+2\epsilon x^2$$ Use newton's second law, $F=m\ddot x$ to derive the energy equation $$\frac{1}{2}m\dot x^2+V(x)=E_0$$ where ...
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1answer
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A generalization of Poincare-Birkhoff theorem

What could be the statment of a possible generalization of Poincare Birkhoff theorem for $M\times [0,\; 1]$ where $M$ is a compact orientable manifold?
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Some derivation in mechanics

I have the following derivation in my physics book I don't know how did they derive them $\frac{d}{dt} \Sigma{_i}[(\vec{r}_{cm} + \vec{r_i})\times m_i(\vec{v}_{cm} + \vec{v_i})]$ = ...
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1answer
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Necc. and suff. conditions for a canonical transformation.

Let $\mathbf{P} = C^{−1}\mathbf{p} + B\mathbf{q}, \mathbf{Q} = C\mathbf{q}$, where $C$ is a symmetric nonsingular matrix. Determine necessary and sufficient conditions on $C$ for the transformation ...
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How would the following graphs differ in shape?

This is a mechanics question but is pretty much mathematical so I figured I should post it here. If I had a particle dropped from rest and it had resistance $mkv$ where mass is $m$, $v$ is ...
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1answer
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Simple Harmonic Motion - Particle Projection

Given $x=A\cos(\omega t) + B\sin(\omega t)$, how do you find the values of constants $A$ and $B$? I am aware that that depends on initial conditions, but I am unsure of the how. The initial conditions ...
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1answer
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Confused on the argument of this function?

So say I wish to go from $$12\sin (t)+4\cos(t)$$ to the form $$A\cos (t+k)$$ by using the double angle formula I can get that $$\cos(k)=4$$ and $$\sin(k)=-12$$ and so we can find ...
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How would you answer this mechanics question?

This is not a homework question, it is from a past paper which I am using to practice. The question is shown in the image below: I really don't know much about mechanics, so I don't even know where ...
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1answer
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Determine the Stress Vector on a plane given the normal?

Let $$\sigma=\left(\begin{array}{lcr} 1 & 1 & 0\\ 1&1&1\\0&1&1 \end{array}\right)$$ be the stress tensor. Find the stress vector acting on a plane through the point whose ...
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Proving that a certain function is an integral of motion for a Hamiltonian

Let $H=q_1p_1-q_2p_2-aq_1^2+bq_2^2$ (with $a,b$ constant) be a Hamiltionian. Show that $G=\dfrac{p_1-aq_1}{q_2}$ is a first integral (integral of motion) of this system. According to the ...
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Deriving an equation for acceleration in circular motion

I have a question: A particle starts to move from rest in a circle of radius 3m, so after $t$ seconds its speed is $5t+1$m/s. Find its acceleration after 1 second. I have tried differentiating ...