Questions relating to Newton's Laws of Motion

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2answers
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Differential Geometry and classical mechanics basics.

This is so far my understanding of classical mechanics, Interspaced are a few questions where I am still not entirely sure what is going on. Thank you for your help !! A tangent space can be thought ...
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1answer
27 views

Mechanics question involving integration.

a particle P moves so that its position vector r satisfies the differential equation $$\frac{dr}{dt}= c \times r,$$ where c is a constant vector. Show that P moves with constant speed on a circular ...
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0answers
24 views

Lax Pairs and constant eigenvalues

Can someone tell me whether the following is true, and if so a hint the proof? If we have a Lax Pair $\dot{L} = [A,L]$ then the eigenvalues of $L$ are constants of the motion. (The opposite ...
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1answer
33 views

Vector spaces and bundles in classical mechanics

My understanding so far is that if we have a manifold of coordinates, the velocity field is more generally called a tangent space $T_xQ$, further the space described by position and momentum is a ...
1
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1answer
28 views

Vector space and form?

What is the significance of equipping the vector space with a form? For instance a symplectic space has a symlectic two form? Why does it need it + what does having it allow/benifit us?
3
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1answer
20 views

mechanics piston problem involving rotational motion.

The above figure shows a piston driving a crank OP pivoted at the end $O$. The piston slides in a straight cylinder and the crank is made to rotate with constant angular velocity $ \omega $. Find ...
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0answers
17 views

representation of Eulers's equation in biharmonic form

As we know the Euler's equation $${\rm div}{\rm div}(\frac{\nabla^2F}{\|\nabla^2F\|})=0$$ Can be written in biharmonic equation form $$\Delta^2F+ (something)=0$$ I want to know in the context of solid ...
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1answer
31 views

Simple Forces - Finding Force and Tension for self learner

Problem A particle of mass 4kg is suspended from a point A on a vertical wall by means of a light inextensible string of length 130cm. a) A horizontal force, P is applied to the particle so ...
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0answers
42 views

Balance of forces in a mechanics problem

I tried to solve a particular problem of mechanics and found some difficulties in the vector analysis part that I can't get rid of. It's probably some stupid mistake I made, but I can't see it now, ...
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3answers
30 views

Basic problem in Mechanics

A body starts from rest and builds up to a velocity of 7.2km/hr in half a minute. A) What is the acceleration? B) What distance is travelled in the half minute? Apparently the answer is ...
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0answers
20 views

At how much horizontal distance the body will strike the ground?

A very smooth plane of length 40m is inclined to horizon at an angle 30 degree . From the foot of this plain , a body starts from rest and moves with an acceleration $10m/s^2$ . After moving a ...
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1answer
24 views

Determine coefficients so that $f(x,y)$ is a conserved quantity

Given the function $f(x,y) = ax^2y + by + ct^2 $ with $x(t)=2t$ and $y(t)=3t^2$ I shall determine the coefficients a,b,c so that $f(x,y)$ is a conserved quantity. My approach is $$\frac{df}{dt} = ...
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0answers
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Why does this graph produce a straight line? [duplicate]

When we graph the sin and cos of theta against the range of a projectile, we get a straight line. When we graph range against angle, we get a hyperbola. Why does the sin and cos of theta against ...
0
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1answer
32 views

Conservation of momentum 1-D problem: track is not level

This problem is a standard 1-D problem. You have 2 equal carts approaching each other and collide on linear air track. BUT the person setting up the track did NOT do a good job at making the track ...
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0answers
16 views

Showing that the vivial theorem holds for the case of a particle in a closed orbit with (f = -kr) in terms of E

I'm trying to show that the vivial theorem holds by explicitly calculating the time average of potential in terms of the total energy E and showing they're the same computed both ways.This is for a ...
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0answers
13 views

Show that after the deformation the line element is given by the following equation

Assume throughout that $X_i,x_i$ (i=1,2,3) are respectively the material and spatial coordinates of a point referred to a common rectangular Cartesian coordinate system with origin $0$, and the the ...
0
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1answer
25 views

How Lagrange equations imply Newton equation

Excuse my english (it's not my first language) I'm self-studying Mechanics and i have a little problem : We can see that in Landau's book or in wikipedia that when we inject the lagrangian in Euler ...
2
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2answers
72 views

Rigid Body Motion with Fixed Point

Suppose that $ T : \mathbb{R} \longmapsto \mathbb SO_{3} $ is a differentiable function. This means that $ T(t) $ is differentiable component wise. Does there exist a function $ \Omega : ...
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0answers
30 views

Using Feynman's Subscript Notation

I have a homework problem that wants me to calculate the force $\vec{F} = \vec{\nabla}_{\vec{X}}U + \frac{\mathrm{d}}{\mathrm{d} t} \left(\vec{\nabla}_{\dot{X}} U\right)$ where $U(\vec{X}, \dot{X}, ...
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1answer
18 views

Circular Motion and Conservation of Energy Question

A particle $P$ of mass m is connected to a fixed point $O$ by a light inextensible string $OP$ of length $r$ and is moving in a vertical circle, centre $O$. At its lowest point, $P$ has speed $U$. ...
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0answers
15 views

Bullet hit, calculate pressure

I wanna calculate the stress that is there when a bullet hits a metall plate. i know the formula pressure = force / area But how can I calculate the force?
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0answers
27 views

Lagrangian of particle on a horizontal, square rotating hoop.

The problem of interest is worded as follows: A horizontal square wire frame with vertices $ABCD$ and side length $2a$ rotates with constant angular frequency $\omega$ about a vertical axis through ...
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2answers
47 views

Classical Mechanics Question

A toboggan travels along the path ABC shown in the diagram. The path lies in a vertical plane, and consists of two circular arcs $AB$ and $BC$. The line ABC is horizontal and there is no friction ...
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2answers
86 views

Period of a mechanical system

Im trying to solve the following problem. Consider $\mathbb{R}^{2}$ with coordinates $(x,y)$. Let $H$ be a smooth function on $\mathbb{R}^{2}$. Also, consider the Hamilton equations: ...
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1answer
77 views

Cauchy Momentum Equation - Stress Tensor

I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation ...
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0answers
9 views

Estimate the period of motion of a particle

a) Evaluate the period of motion of a particle in the potential in the field $U(x)$ if it's energy value lies in the vicinity of $U_m$ (see the picture below). b) Estimate at what part of of period ...
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1answer
21 views

projectile motion (dynamic) stone

2) A stone is projected downwards with a velocity of 20m/s at an angle of 30degrees below a horizontal line through the point of projection. Find the velocity of the stone after 2 sec. t=0 => ...
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1answer
38 views

complicated projectile motion about throwing thing [closed]

A stone is projected with a velocity of $20$ m/s at an elevation of $\theta$, given $\tan(\theta)=3/4$, from the top of a wall $9$m high. Find the range of the horizontal plane through the bottom ...
5
votes
1answer
75 views

projectile motion (with height) complicated

When a child standing on a horizontal path throws a ball, it leaves her hand from a point that is 90cm vertically above the path. The ball clears a 4.5 m high wall that is 10.5 m away from ...
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1answer
32 views

Show that the action variable is $J = \sqrt{A^2 + 2E} - A$

I have the hamiltonian: $$H = \frac{1}{2}p^2 + \frac{1}{2}A^2 \tan^2(q)$$ And I would like to show that the action variable is $J = \sqrt{A^2 + 2E} - A$, where $E$ is the energy. I'm having a ...
0
votes
1answer
13 views

Vector finding out the value of P?

Given two force 2P and P , when the first one is doubled , i.e 4P and the 8 is added to the second one i.e P+8 , the direction of the resultant remains unaltered . What is the value of P ? N.B : I ...
4
votes
1answer
62 views

How to derive the hamiltonian from a non-classical lagrangian

For the non-classical lagrangian of a hydrogen atom: $$L = -mc^2 \sqrt{1-\frac{v^2}{c^2}} + \frac{e^2}{4 \pi \epsilon r}$$ We get that two conserved quantities are: $J = \gamma mr^2 \dot{\phi}$ and ...
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1answer
25 views

Find the smallest $t$ such that the object reaches the height of $96\text{ feet}$ at time $t$.

An object is tossing upwards with an initial speed of $64 \text{ feet/sec}$.Suppose the gravitational acceleration is $32\text{ feet/sec}^2$. Find the smallest $t$ such that the object reaches the ...
0
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1answer
23 views

Workout initial velocity of projectile without knowing launch angle

I am trying to work out the speed of a tennis serve. I know the following: X Distance: 18m, Time: 0.5s, g: 9.8, Launch height: 3m. Is it possible to work out the initial velocity of the serve ...
2
votes
1answer
14 views

Problem on Finding the speed using Intertia

I did the first part (using parallel axis theorem) and showed that intertia. The problem is in the second part, I know that $C=I\frac{d^2\theta}{dt^2}$ , where C is the moment. So in this case it ...
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0answers
81 views

What are sliding vectors mathematically?

What is the mathematical definition of sliding vectors and their operations, as used in mechanics? What kind of mathematical structure do they form? Does the operation of constructing the "space" of ...
3
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2answers
97 views

Find the time interval between oscillations of SHM.

Parts i) and ii) I can solve. But for part iii) I can't do, as I don't know which equation describes the SHM motion? Is it $y=0.5sin(1.2t)$ or $y=0.5cos(1.2t)$ or $x=0.5sin(1.2t)+2.5$? I thought ...
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0answers
19 views

Deriving the $F_3$ type generating function in Hamiltonian formulation

I'm working on some practice questions and I am a bit confused with this one: Generating functions of the type $F_1(q,Q)$ satisfy the condition: $$pdq-PdQ = dF_1$$ Starting from this condition ...
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1answer
33 views

Clarifying understanding of Poisson Brackets in Hamiltonian Dynamics

I'm just reading through my textbook and would like to clarify my understanding of 'Canonically related variables'. In my textbook, it says that if $Q_i$, $P_i$ are related to $q_i$, $p_i$ by a ...
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1answer
29 views

Simultaneous Suvat help [closed]

I learnt this in college but can't for the life of me remember how to do it. I've searched stack exchange and the internet for answers but it isn't clicking. It doesn't help my teacher has decided to ...
0
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2answers
64 views

Electromagnetic Theory Question involving Coulomb's Law [closed]

Question: Two light pith balls, each of mass m are suspended from a point by threads of length $l$. Each ball carries a charge $Q$ and the resultant repulsion forces them a distance $r$ apart. By ...
0
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1answer
19 views

Intersection of arbitrary oscillation

I am looking for a solution to calculate the intersections of arbitrary (harmonic) oscillations. For example Biorythm with constant amplitude and different periods. Especially my problem with ...
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3answers
33 views

Asymptotic behavior of $-gTt-gT^2e^{\frac{-t}{T}}$ for small $t$

I want to solve this using Taylor series expansion of $e^{f(x)}$ $$\begin{align}x=-gTt-gT^2e^{\frac{-t}{T}}+gT^2+x_0\end{align}$$ Show that for small values of t $(t\ll T)$, the equation for ...
0
votes
1answer
34 views

Find the magnitude of the total force exerted on the hinge by the wall?

A regular platform $ABCD$ of weight $200N$ is smoothly hinged, along its edge $AB$, to a verical wall, The platform is kept horizontal by two parallel chains inclined at $45^o$ to the horizontal ...
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1answer
78 views

Lagrangian equivalence up to total time derivative: dependence on higher derivatives

I recently encountered the problem Show that the Euler-Lagrange equations of motion for $L_1$ and $L_2$ are the same when $$L_2(\ddot{q},\dot{q},q,t) = L_1(\dot{q},q,t) + \frac{d}{dt} ...
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1answer
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Vectors Statics and Forces

The following forces are applied to a wall bracket: $F_1 = 100N$ at $30$ degrees above the $x$-axis, $F_2= 80N$ at $20$ degrees below the $x$-axis. Find the resultant force and its direction. I have ...
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0answers
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Matrix linearization of the Lagrangian points.

I have to solve a long problem, and I´m in trouble in a step. The step is to linearize the next differential equation, by writtin its correspondient Jacobian, and then, calculate the eigenvalues of ...
0
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1answer
35 views

Prove scalar product is distributive

The scalar product is defined as r*s = the sum of all r*s. Using this definition, prove that r*(u+v) = r*u + r*v. Also, if r and s are vectors that depend on time, prove that the product rule for ...
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1answer
41 views

How to find time it takes for an object to slide on an incline ramp [closed]

So I hope I am asking this question in the correct spot. Here is my question: if there is an incline at $70$ degrees, the object's friction is $\mu = 0.1$, and the incline is $1$ meter long, how long ...
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votes
5answers
65 views

If angular velocity $\omega=\sqrt{\frac{3g\sin\theta}{2a}}$ can I find angular acceleration $\alpha$ by differentiating $\omega$?

It was my understanding that angular acceleration is the derivative of angular velocity. The reason I ask is Thanks.