Questions relating to Newton's Laws of Motion

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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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particle hitting particles attached with springs [on hold]

In classical mechanics if you have a particle moving and it hits a particle at rest although that particle is attached to a spring that is in turn attached to a third particle. Assume all particles ...
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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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1answer
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mechanics, kinematics motion under gravity [closed]

A particle is thrown vertically downwards from a point h metres above the ground with speed V m/s. A second particle is thrown at the same instant from the ground with speed 5V m/s in the same ...
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23 views

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
14 views

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
29 views

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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22 views

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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1answer
28 views

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
44 views

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
37 views

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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38 views

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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2answers
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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
36 views

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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40 views

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
24 views

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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39 views

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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1answer
33 views

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
19 views

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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2answers
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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 ...
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1answer
43 views

Distance travelled on a curvelinear path and the coordinate of points

A race car travels in a curvilinear path at points A, B, and C. The following data is given: At point A time 0 seconds Speed is 195.1696800 $\large{\frac{m}{s}}$ tangential acceleration is 0.22 ...
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1answer
42 views

Center of mass calculation

Calculate the center of mass for : The area bounded by parabola $y = x^2/b$ and the line $y = b$. I got the following integral I just need verification that my work is correct. First I got ...
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1answer
91 views

Proof of Hamilton's equation from integral invariant

This is from pages 273 - 274 0f Whittaker's book of analytical dynamics. Its in the public domain. Let $q_1,q_2,\ldots,q_N$ be functions of time. And let $p_1,p_2,\ldots,p_N$ also be functions of ...
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1answer
71 views

Trouble with a Statement in Arnold's “Mathematical Methods of Classical Mechanics”

On Pg 6 of Arnold's Mathematical Methods of Classical Mechanics (2nd Edition), there is a line which reads One can speak of two events occuring simultaneously in different places, but the ...
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Forms and conservative forces

According to Tongs notes on Classical Mechanics; a force is called conservative when $F=-\nabla V$ And iff $\nabla \times F = 0$. This is in $R^3$. Also the potential $V=\int_{x_o}^{x^1} F(x)$ $dx$ ...
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Books on Classical Mechanics

Can I get a book for a one semester course in these topics? My requisites are I dont want a very detailed book on these topics.It should just deal with the concepts clearly and provide some solved ...
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1answer
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Which ellipses settle to 1-point contacts within a snow-globe circle?

Suppose you have a solid ellipse with axes $a$ and $b$, $(x/a)^2 + (y/b)^2 = 1$, confined inside a unit-radius circle. You shake the circle like a snow globe, and the ellipse settles to the bottom ...
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2answers
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Position vector is given by r(t)=3sin(4t)i+3cos(4t)j+5tk Determine the velocity and acceleration of the particle at any time t>0.

I understand that the velocity and acceleration are found by the first and second derivatives of the position vector respectively. Also that the magnitude of the velocity is speed, given by ||v(t)||. ...
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How to solve an overconstrained system of equations?

What is the easiest codeable way to solve an overconstrained static model? How does Force Effect https://forceeffect.autodesk.com do it? Given a 10m long bar angled as the hypotenuse of a 3, 4, 5 ...
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2answers
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Determine the resultant force of the inclined plane

So I was given this problem for my statics subject. I'm just having problems getting the x and y components of the force $120$ lb. I know that $60$-lb Force $$F_x = (60\mbox{ lb})\cos20^\circ$$ ...
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32 views

How to derive velocity out of acceleration in a circular motion?

A car starts moving in a circle with a radius of 200 meters. It has a constant tangential acceleration of $1{\text{m}\over {\text{sec}}^{2}}$. a. What is the angular acceleration? b. What is the ...
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Axioms of Newtonian Mechanics

Axiomatically speaking, could Newton's laws be derived (as theorems) from the conservation of momentum and energy -- along with a few suitable definitions of things like an inertia frame and force? ...
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Questions about the mechanics of a spinning CD.

A CD is spinning counterclockwise with a radial velocity of $\omega=30\text{rad}/\sec$. The preface of what I did manage to solve and further details: I was asked what is the period time (12 seconds), ...
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2answers
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How to Show Cotangent Bundles Are Not Compact Manifolds?

Hamiltonian mechanics occurs in a sympletic manifold called phase space. Lagrangian mechanics take place in the tangent bundle of the configuration manifold. Using Legendre transform makes possible ...
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1answer
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Finding the range of a cannonball- proof verification.

I asked such a question before but I do learn best by mistakes and corrections.(I didn't fully understand it yet.) I could really use your verification: A cannonball is being fired with a velocity of ...
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1answer
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How to get a range of a body being shot\thrown? Theoretical question.

I already asked a similar question here, but the answers were technical, leading me to no genuine comprehension of what I am doing.(I am not complaining or anything, I really thanked you then.) ...