Questions tagged [classical-mechanics]

For questions on classical mechanics from a mathematical standpoint. This tag should not be the sole tag on a question.

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Stieltjes integral and Riesz representation theorem

Would you please explain how I can derive eq(1.2) from eq (1.1) by using Stieltjes integral in the Riesz representation theorem?? And is it necessary to suppose that the time interval begins from -$\...
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Rotations in 3D space: order of concatenated rotations.

I am currently doing an online course in Udemy to clarify some things for a project I'm currently doing on solid rigid motion and simulation in University, although understanding some of the things ...
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Are these equations "properly" defined differential equations?

Are these equations properly defined differential equations? Modifications were made to a deleted question to re-focus it Intro Recently, in this answer I figure out that the following autonomous ...
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How to change coordinates/variables?

I want to model the dynamic behaviour of a point-mass particle moving along a 2D surface given by $h(x_1,x_2)$. I made a question in physics stack exchange, where i found the error in my modelling, ...
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Is there a method to calculate the eigenvalues for this a n x n symmetrical matrix

I'm working with a mechanics problem where I try to find the eigenmodes of the system. The system contains of $n$ masses all connected with springs to one another (same spring constant $k$), the outer ...
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Why is the space of all frames in $\mathbb{R}^3$ the space $\mathbb{R}^3 \times \operatorname{SO}(3)$?

The following regards the textbook "Mathematical Methods of Classical Mechanics" by V.I. Arnold. Definition. The configuration space of a system of $n$ points is the direct product of $n$ ...
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Is there an easier way to derive this equation for this mechanics problem?

We're given a semicircle whose total mass $M$ is evenly distributed along its contour and a little particle of mass $m$ is dropped from its top left corner as in the image below all over an horizontal ...
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Dimensional Analysis - Simple Pendulum

For a simple pendulum consisting of a spherical mass $m$ anchored to a fixed point $O$ via a mass-less rod of length, perform dimensional analysis on the problem to find the period of oscillation τ (i....
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2n independent variables and DOF = n in Hamiltonian Mechanics

I’m studying Lagrangian and Hamiltonian mechanics for the first time. In the Lagrangian approach, that is the one I’ve studied first, a fundamental point is the definition of the degree of freedom of ...
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Find the equation for the angle in which the particle abandons the hemicircle. No friction.

I think I missed something in this mechanics problem. We're given a polish and homogeneous hemicircle which has mass $M$ and a particle of mass $m$ laying on the top of it. There is also no friction ...
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Levi-Civita Connection with Geodesic Spray Containing Flow Lines of Time-Independent Vector Field?

Let $Q^n$ be a closed manifold and $M = TQ$ be its tangent bundle. In [1], it is worked out in Equation 5.31 that there is a kinetic energy Riemannian metric $g$ on $Q$ with Levi-Civita connection $\...
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The material point M (m=1-mass) in the Oxy plane moves on the effect of a force field which components are Fx=y,Fy=x.

Can someone help me with this exercise? The material point M (m=1) in the Oxy plane moves on the effect of a force field which components are $F_{x}=y$,$F_{y}=x$. Is there a power function and if so, ...
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How to write the Lagrangian of a cyclic tri-atomic molecule?

I'm familiar with Lagrangian of a $\bf{linear}$ tri-atomic molecule moving only on 1-dimension.like this The kinetic energy is written as $$ T = \frac{1}{2}(m\dot{x}_{1}^{2} + M\dot{x}_{2}^{2} + m\dot{...
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Why does the solution to pendulum problem with the geometric approach of Jacobi metric does not correspond to the solution with Lagrangian approach?

When we solve the pendulum problem with EL equation, we get to the differential equation $\ddot{q}+\frac{g}{l}\sin q=0$ but when I apply the substitution $t \rightarrow t\sqrt\frac{g}{l}$ and ...
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3 votes
2 answers
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Find the eccentricity of the elliptical orbit when the central force is changed from one focus to another

A body describing an ellipse of eccentricity e under the action of a force directed to focus when at the nearer apse , the centre of force is transferred to the other focus . Prove that eccentricity ...
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Counter-intuitive result for resisted motion

A suppose a particle of mass $m$ is projected horizontally across two surfaces $S_1$ and $S_2$ with resistive forces $mkv$ and $mkv^2$ respectively where $k$ is a constant and $v$ is the velocity of ...
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Atwood system of two pulleys

A light string passes over a light fixed pulley carrying a mass P at one extremity and a light pulley at the other. Another light string passes over this second pulley and has masses Q and R at its ...
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Is there a version of Poincare recurrence theorem on attracting set?

In this paper, on the left-side on page 2, it has a statement of poincare recurrence theorem, which I quoted here: Statement 1: Our algorithm relies on the Poincaré recurrence theorem, which states ...
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Equation of motion with non-linear drag

Consider a particle of mass $m$ and electric charge $q$, initially at rest (zero velocity, $v(0) = 0$), placed in a unidirectional potential $ψ(x) = −E_0qx \hspace{0.05in} \text{sin}(ωt)$. Assume that ...
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At what point of the pole should the rope be tied so that the man can turn the pole upside down easily?

Question: One end of a $20m$ long rope is tied to a telegraph pole standing perpendicularly on the ground and its other end is being pulled by a person with a definite force. At what point of the pole ...
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Loss of kinetic energy in oblique collision of spheres

I am doing this question from the Edexcel M4 syllabus for a student: I correctly obtained the velocities of A and B after the collision respectively as $3{\bf i} + 2{\bf j}$ and $2{\bf i} - {\bf j}$, ...
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clarification for the tangential and normal components of acceleration

please help me understand this bit from the book: "Vector Mechanics for Engineers Statics and Dynamics by Beer, Johnston(12th edition)"(page 692). why or how is ${\it \Delta}e_t = 2\sin(...
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2 votes
2 answers
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maximum range of projectile launched from elevation

I am trying to find the maximum range of projectile from an elevation. I found the answer in this question, but I have two questions: Why does $y$ need to be $0?$ Why do they differentiate with ...
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How did they prove $\frac{1}{2}P=Q=R$?

Question: Three forces of magnitudes $P, Q$, and $R$ act along the three sides of an equilateral $\triangle ABC$. If their resultant passes through the centroid and is parallel to the side $BC$, prove ...
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3 votes
3 answers
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Formal definition of virtual displacement

I'm trying to read a book on classical mechanics, and I'm having a hard time trying to know what exactly is a virtual displacement, sometimes called a variation. In the Lagrangian "formalism"...
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A mechanics question with "constant acceleration"

Main problem is with part (b). I got it right up to the accelerations with -11.6 and 4.4 respectively. Then I dont really see the reason why we have to break it up to two parts as given in the ...
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Typo in question involving velocity and acceleration vectors?

I need to show that $\underset{\sim}{v}=e^{-0.1t}(\underset{\sim}{u}+\underset{\sim}{j})-\underset{\sim}{j}$ satisfies the equation $\underset{\sim}{a}=-0.1\underset{\sim}{v}-10\underset{\sim}{j}$ ...
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Mechanics question regarding max height and limiting motion.

$\renewcommand{\vec}[1]{\underaccent{\tilde}{#1}}$ I'm trying to do a three part mechanics question that goes as follows: i) If $\frac{du}{dt}+uf(t)=g(t)$ and $F'(t)=f(t)$, by considering $\frac{d}{dt}...
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How to prove that $\int_{\theta_0}^{\pi} \sqrt{\frac{1-\cos{\theta}}{\cos{\theta_0}-\cos{\theta}}} \, d\theta=\pi$? [duplicate]

I was solving a physics problem regarding the traveling time on Tautochrone curves, and I arrive at the following integral: $$\int_{\theta_0}^{\pi} \sqrt{\frac{1-\cos{\theta}}{\cos{\theta_0}-\cos{\...
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How to replace these two forces with one force? [duplicate]

[Please also refer to the EDIT appended below.] Here, $P>Q$. $O$ is the center of mass of the rigid and uniform bar/stick. As $P>Q$, the resultant is situated to the right of $\vec{P}$ and is ...
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In the calculation of precession of Mercury, why we can evaluate a definite integral using a contour integral

In the calculation of Mercury using perturbation, we have to evaluate the integral: $$ \delta \phi =-\frac{GMm}{2a^3}\sqrt{\frac{m}{2E}}\frac{\partial}{\partial J}\int^{r_{max}}_{r_{min}}\frac{r^3 dr}{...
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How will the force act outside the stick?

Scenario 1: Here, $P>Q$. $O$ is the center of mass of the rigid and uniform bar/stick. The resultant acts to the right of $\vec{P}$ as $P>Q$. Scenario 2: Here, $P>Q$ also. $O$ is the center ...
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Where will the projectile strike the plane??

A particle is projected with velocity $V $,at an angle of $75^0$ to the horizon,from the foot of a plane whose inclination is $30^0.$I have to find where it will strike the plane. In the given ...
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Implications from the form of Binnet equation

Binnet equation, $$\frac{d^2}{d\theta^2}\left(\frac{1}{r}\right)+\frac 1r=-\frac{m}{L^2}F(r)r^2,$$ where $m$ and $L^2$ are positive constants, can be used to give the orbit $r(\theta)$ of a particle ...
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Why did my book consider the ratio $7:5$ when it also could've considered $5:7$? Was it just a matter of choice?

Question: Four forces $P$, $2P$, $3P$, and $4P$ act respectively along the sides of a square taken in order. Find the magnitude, direction, and line of action of their resultant. My book's attempt: (...
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1 vote
1 answer
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Why keep two pillars when one pillar can do the job?

Question: A uniform plank $AB$ of $4m$ length and $15kg-wt$ weight rests horizontally on two supports/pillars: one at $A$ and the other at a point $0.5m$ from $B$. Find the weight of a boy if he can ...
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How to solve second order ODE from a simplified version of the two body problem?

While thinking about the motion of a particle being attracted to another massive particle and considering that the acceleration of that particle is not constant, I found myself trying to solve this ...
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1 vote
1 answer
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If two unlike forces act on two different points of a rigid body, won't the body just start to rotate?

Question: Two unlike parallel forces $P$ and $Q$ $(P>Q)$ act at $A$ and $B$ respectively. If P and Q are both increased by R, show that the resultant will move through a distance $d=\frac{R}{P-Q}$. ...
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1 answer
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Horiozontal throw dimensions analysis

I have a question about dimensions analysis of a horizontal throw: A ball is thrown from origin where the $x$-axis points in the horizontal direction and $y$-axis vertically upwards. The ball is ...
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4 votes
1 answer
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How can a force vector be shifted laterally when it is a sliding or a bound vector?

Question: One end of a rope of length $l$ is attached to a vertical wall, the other end being attached to a point on the surface of a uniform sphere of radius $a$. If the weight of the sphere is $W$, ...
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1 vote
1 answer
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Didn't my book draw this figure wrongly?

Question: A body of $20\ lb-wt$ is suspended by two strings that are $6ft$ and $8ft$ long, their other ends being tied to the extremities of a rod of length $10ft$. If the rod was so held that the ...
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1 vote
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Integrability of Euler top

I'm taking an integrability course, it's a bit too fresh and my physics a bit rusty. I'm stuggling with an exercise about the Euler top. I'd appreciate a hand with some questions, or a least some tips....
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What does it mean by a force acting horizontally?

Question: A body of $10kg-wt$ is kept in an equilibrium condition by a force of $5kg-wt$ acting horizontally, a force of $F\ kg-wt$ acting at an angle $60^{\circ}$ with the horizon and a force of $R\ ...
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3 votes
2 answers
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Phase portrait of the simple pendulum. How much time does it take to get from one unstable equilibrium to the other?

Consider a simple pendulum. I am trying to understand the phase portrait of its dynamics. I read that the "time" in which the dynamical system goes from the the unstable equilibrium $\theta ...
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2 votes
2 answers
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Mechanics exercise using Newton’s second law.

Could anyone help me with this question?- Suppose a particle of mass m with position $x>0$ moves in 1D space under the influence of the gravitational force of another point particle with mass $M$ ...
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Can the angle between two vectors be negative? [duplicate]

Question: The angle between two vectors $\vec{P}$ and $\vec{Q}$ is $\alpha$. If $\tan\alpha=-\sqrt{3}$, determine $\alpha$. My attempt: $$\tan\alpha=-\sqrt{3}$$ $$\alpha=\tan^{-1}(-\sqrt{3})$$ $$\...
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Why is this angle taken as negative?

Question: The resultant of two concurrent forces, $\vec{P}$ and $\vec{Q}$, $[P>Q]$, trisects the angle between them. Show that the angle between them is $3\cos^{-1}\left(\frac{P}{2Q}\right)$ and ...
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1 vote
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How is my book finding the magnitude of the resultant differently?

Question: Forces of magnitude $Q-R$, $Q$, $Q+R$ act at a point in directions parallel to the sides of an equilateral triangle taken in order. Find the magnitude of their resultant. My attempt: Let $$...
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find the angle of a suspended lamina, check the photo for the full question

Centre of mass related question I don't understand how to tackle this question, I know how to find the centre of mass of the shape but don't know what is the next step.
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1 answer
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$\sqrt{\sin^{2}(\frac{\alpha}{2})}=\sin(\frac{\alpha}{2})$?

Question: Two forces $\vec{P}$ and $\vec{Q}$ acting at a point have a resultant $\vec{R}$ and the resolved part of $\vec{R}$ along $\vec{P}$ is equal to the magnitude of $\vec{Q}$. Prove that the ...
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