Tagged Questions

Questions relating to Newton's Laws of Motion

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0answers
13 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 ...
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1answer
11 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 ...
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1answer
51 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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0answers
22 views

Show the conditions for circular orbit

I am working through this exam practice question, and just need a bit of guidance with the last few parts, would greatly appreciate any help: The relativistic Lagrangian to describe the hydrogen atom ...
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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 ...
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1answer
16 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
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1answer
13 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
19 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
96 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
10 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
25 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
24 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
57 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
17 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 ...
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1answer
21 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 ...
1
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1answer
71 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
16 views

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
11 views

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 ...
2
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0answers
33 views

Liouville's theorem (Hamiltonian) [closed]

can some one give me a link for a rigorous proof for Liouville's theorem (Hamiltonian) thanks
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1answer
32 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
31 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 ...
2
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5answers
58 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.
2
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1answer
35 views

At the instant of release of an object from rest. Is the only force that can act its weight? [closed]

This is the third question from a mechanics exam past paper: I can do parts i) and ii) but for iii) in finding the angular acceleration, i used $C=I\alpha$, where $C$ is the applied couple or ...
0
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2answers
44 views

Why does resolving forces in one direction give a completely different answer to resolving the opposite way?

I can solve parts i), ii) and am able to show that $R=0$ for part iii). In this question $g$ is the acceleration of free fall taken to be $9.8$ Using Newtons 2nd law [$F=ma$] for the last part I ...
1
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1answer
51 views

What is the average velocity of the motorcycle?

The position of a person riding in a motorcycle race is give by $s(t)=4t^2+3t$, where $t$ measures time in seconds since the race began, and position is measured in feet beyond the starting line. ...
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2answers
93 views

Finding the equation for a (inverted) cycloid given two points

If I have two points on a Cartesian plane, and I know that they are connected by a cycloid, then how do I find the equation for that cycloid? For background information, I have been playing around ...
2
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2answers
229 views

Finding the position of a moving point [closed]

A point is moving on a given curve. For example, curve equation is: $$x^2 + y^2 - 10y = 0,$$ which is a circle with $5$ meter radius. If point is on $(0,0)$ at $t = 0$ and is moving on the curve ...
4
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1answer
88 views

Geometric meaning of a matrix decomposed into its symmetric and skew-symmetric parts

What's the geometric meaning of a matrix decomposed into its symmetric and skew-symmetric parts? For example, a skew-symmetric matrix on its own can be interpreted as an infinitesimal rotation. As ...
3
votes
1answer
158 views

Erroneous calculus of variations reference in V. I. Arnold's Mathematical Methods of Classical Mechanics?

The beginning of section 12, Calculus of variations (chapter 3, Variational principles) in V. I. Arnold's Mathematical Methods of Classical Mechanics (2nd edition, p. 55) reads: For what follows, ...
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2answers
40 views

Is it possible to write the curl in terms of the infinitesimal rotation tensor?

Is it possible to write the curl in terms of the infinitesimal rotation tensor? Basically, we can write the curl as a matrix operator $$ curl=\begin{bmatrix} 0 & -\partial z & \partial ...
4
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2answers
98 views

How is partial time derivative $\frac{\partial}{\partial t}$ defined for vector flows?

This question emerged when I was thinking about Liouville's theorem of classical mechanics. As far as I understand, the change of any function along the integral curves of Hamiltonian vector field is ...
3
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4answers
768 views

A 6 meter ladder…

A $6$ meter long ladder leans with a vertical wall and top of the ladder is 3 meters above the ground.If it slips at a rate of $2$ m/s then how fast the level is decreasing from the wall? My ...
3
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1answer
142 views

Littlewood's orbital joke

In A Mathematician's Miscellany, Littlewood offers this item: To determine the orbit of a planet or comet 3 observations, each of two (angular) co-ordinates and the time t, suffice. It is ...
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0answers
43 views

Newton's method for the brachistochrone

Consider the potential $V(x,y)=-y$ and a particle at rest in the beginning of the coordinate system. We are going to examine the brachistochrone - the smooth curve of fastest descent. Assume we are ...
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0answers
42 views

Analyze rotation of satellite orbit due to transverse acceleration.

Consider a small satellite which moves in a 2D elliptical orbit around a much larger body (e.g. the Sun) under the influence of Newtonian gravitational acceleration $$Ar=G.M/d^2$$ QUESTION:- Is ...
1
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2answers
30 views

The volume is preserved by the flow: where is the absolute value?

Consider the following excerpt of the Liouville's theorem proof taken from "Arnold - mathematical methods of classical mechanics": In changing the variables in the integral, I don't understand why ...
1
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1answer
21 views

constant deceleration problem

A train bracking with constant deceleration covers $1km$ in $20s$, and a second kilometre in $30s$ find the deceleration. Here is what I did: Since the deceleration is constant it would reach the ...
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0answers
29 views

Curvilinear Coordinates and basis vectors

In the following notes, http://www.maths.manchester.ac.uk/~wparnell/MT20401/MT20401_lecture3.pdf $\frac{\partial \vec{r}} {\partial q_i}$ forms a basis set for the vector space. How does this happen? ...
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3answers
74 views

Angles and forces

Two forces of 40 pounds and 28 pounds act on an object. The angle between the two forces is 65 degrees. Find the magnitude of the resultant force to the nearest pound. Using this answer, find the ...
1
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1answer
27 views

Simulation of orbiting bodies

I am writing a computer program to simulate orbiting bodies such as planets and stars. I wish to have a starting point in which a number of bodies are randomly scattered around a central heavy body. ...
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0answers
34 views

Is this divergence-free? (Double Pendulum)

Concerning this page http://scienceworld.wolfram.com/physics/DoublePendulum.html for the double pendulum the moving equations are given by $$ ...
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1answer
31 views

Why is are these expressions in Leibniz notation not equivalent?

I have this problem, a projectile is fired into fluid with a rate of deceleration $a=-0.4v^3$ and in initial velocity $v_0=60$. We are to find how fast its going after $t=4$ seconds. If one starts ...
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1answer
47 views

Why is this integral not returning to the original equation when derived?

A projectile is fired into fluid at a rate of $60$ (nevermind the units on this one.) It decelerates such that $a=(-.4)v^3$. This is all fine and dandy. The book provides this solution. ...
2
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1answer
19 views

Question regarding Energy Conservation

A point mass $m$ is projected from the earth surface with speed $v_0$ and at an angle $θ$ above the horizontal. Assume that the gravitational acceleration is constant and has the absolute value ...
0
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1answer
56 views

Which is the equation of the trajectory of a particle?

The solution to this question has left me rather confused... Question: A particle moves with the known constant speed $v = |d\vec{r}/dt|$ on a helix. Its position is given by ...
4
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0answers
65 views

Angular momentum cylindrical coördinates

From "Classical Mechanics" - Taylor, problem 3.30 Consider a rigid body rotating with angular velocity $\omega$ about a fixed axix. (You could think of a door rotating about the axis defined by ...
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2answers
399 views

How does energy conservation follow from Newton's second law?

Question : Show in the one-dimensional case, how for potential forces $F(x) = \dfrac{−dV (x)}{dx}$, energy conservation follows from Newton’s 2nd law From Newton's second law we know ...
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1answer
60 views

Mathematics/Mechanics Problem

I would like to ask you if anybody could help me with this problem. So far i know that the positions where B and A have to meet are at distances L and L+2r
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1answer
71 views

Calculating a double pendulum

consider the following situation of a double pendulum. We found the moving equations as $$ \ddot{\theta_1}=-L_1\sin\theta_1 + \frac{m_2}{m_1}\cos\theta_2\sin(\theta_2-\theta_1),\\ ...