Questions relating to Newton's Laws of Motion

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Show that boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$

I was wondering if somebody would be able to help me with this problem. I know how to solve it using dimension arguments but I'm unsure what is meant by 'transform techniques'. Any help would be ...
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Scaling Two Equations

I recently got set this problem and am having trouble scaling the resulting equations. Any help would be appreciated. An incompressible thermal conducting fluid is contained between two infinite ...
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Numerically iterating the dynamics of a constrained Newtonian system

This question is about the dynamics (in classical mechanics) of a rigidly linked chain of $N$ point masses, see figure. Let us say that the masses $m_1,\ldots,m_N$ have initial positions ...
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Flow between two infinite horizontal plates

I recently got set this problem and I was wondering if anyone would be able to give me some hints/intuition on how to solve it. Thanks. An incompressible thermal conducting fluid is contained between ...
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Trouble understanding Poisson Brackets

I'm looking at page 94 here - I understand the definition of Poisson brackets at the top of the page (which uses summation convention) but I don't get why the calculations in (4.61) are true. I'm ...
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2answers
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A question on the conservation of energy between two objects.

A wedge of mass $M$ rests on a smooth horizontal surface. The face of the wedge is a smooth plane inclined at an angle α to the horizontal. A particle of mass $kM$ slides down the face of the wedge, ...
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How do you go about finding the normal acceleration of a plane associated with polar co ordinates?

Question A plane, having just taken off, has a constant speed of v=94.3m/s. When $\theta=20^{o}$, the plane is climbing at an ever steepening rate of 0.17 rad/s. What is the normal acceleration of ...
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Mechanics question about energy

A body of 10 kg weight drops from a height of 60 cm upon a spring whose modulus is 60 kg/cm . What will be the speed of the block when the spring is deformed 2 cm?
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Mechanics - Conservation of Energy & Forces

The Question: Steel balls of mass=0.10kg are dropped from a height of ℎ=5.0m onto a smooth steel plate inclined at 45∘ to the vertical. If the balls are dropped one at a time at a rate of n=100 ...
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Derivation in eulerian angles

In the following link page 393 of the book http://uqu.edu.sa/files2/tiny_mce/plugins/filemanager/files/4320535/Analytical_Mechanics_Fowles_7ed__English.pdf I don't understand at all how did they ...
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Winch bridle force calculation for performer flying [migrated]

Hi this is my first post o this site so please forgive me if I have got anything wrong in the process. I am trying to do some calculations on the forces some performer flying winches are seeing. ...
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1answer
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About Noether's theorem - question about how can I do such that I can study this theorem

My teacher asked me to try to write the main important steps which I have to learn such that I can understand the Noether's theorem. Some few steps are: Lagrangian Formalism Hamiltonian Formalism ...
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Fraction of original velocity in series of elastic collisions

I need to find the answer to this question: Three particles A, B, and C, with masses $m$, $2m$ and $3m \, \mathrm k \mathrm g$ respectively, lie at rest in that order in a straight line on a ...
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1answer
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Kinetic Energy Alternative Expression

Attempt: I've managed to show everything except for the last part. I need to show an alternative expression for $T$ as shown below. I've tried showing the relevant scalar product is the same as ...
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1answer
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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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1answer
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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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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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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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1answer
45 views

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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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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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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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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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!