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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Amplitude of waves in water after a disturbance [closed]

Suppose I have a perfectly circular pool which is five meters in radius, three meters in depth, and filled with water. Say I drop a steel ball with a radius of five centimeters into the middle of the ...
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How many ways can be used to find the contact force from a sphere which is over another?

The problem is as follows: Two homogeneous spheres are at equilibrium by the action of an horizontal force of minimal magnitude $F= 25\,N$, as shown in the picture from below. The sphere which is in ...
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How to find the elongation of a spring when it is tied to a brass bar over a tilted wall?

The problem is as follows: An homogeneous brass bar which goes from point $A$ to point $B$ has a weight of $300\,N$. The bar is suported over two frictionless surfaces as it is shown in the picture ...
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How to find the speed of a block after the collision with a swinging sphere?

The problem is as follows: A steel sphere has a weight of $40\,N$ and is anchored to a pivot in the ceiling as shown in the picture from below. The wire which connects the sphere to the pivot has a ...
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1answer
16 views

System of equations of non-relativistic scattering in the laboratory system

Considering the system of equations of non-relativistic scattering in the laboratory system: $$\begin{cases} \dfrac{1}{2} m_{1} v_{1}^{2} &=\dfrac{1}{2} m_{1} v_{2}^{2}+T_2 \,,\\ m_{1} v_{1} &=...
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Legendre transforming a differential equation

In classical mechanics, given a Lagrangain $L(x , \dot{x})$, the equations of motion are given by $$ \frac{d}{dt} \frac{\partial L}{\partial \dot{x}}=\frac{\partial L}{\partial x} $$ If we define the ...
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Prove that $\vec a_i^T T \vec a_j=\delta_{ij}$

Given a solution of the small oscillation equations (called normal mode) $$\vec q = \sum_i c_i \vec a_i \cos (\omega_i t - \gamma_i), \ \ \ 1 \leq i \leq n \tag1$$ Where $$\vec q= \begin{pmatrix} ...
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Transformation of spatial coordinates

Please note that the transformed quantities will be indicated by $'$. Let me give some context first. The general approximate form of the potential energy $V$ is given to be $$V^{app} = q^T V q \tag 1$...
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62 views

Is it possible to make velocity time graph periodic?

sorry i know this question is from physics but i believe it uses more of maths. i am a highschool student but here we are not taught Fourier analysis so we can't learn those beautiful curves and ...
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relationship between two system of coordinates

I Have a problem: There are Greenwich coordinates (ECEF),$$\eta_1=2850144.282468, \eta_2=2197962.005840, \eta_3=5248530.328116.$$ Calculate the values of geographical coordinates (Geodetic), for these ...
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Which shape is ideal for an arch?

One of my friends works on a hobby gardening project and she wants to use plastic tubes to have a roof over the plants like in this video: https://www.youtube.com/watch?v=rRpC-zjFUNs Except that ...
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1answer
39 views

Finding a general equation for a particle's motion along a curve in 2D space under the force of gravity

I was trying to find a general equation for a particle's motion along a curve in 2D space. But I'm not satisfied with the final conclusion I get through the derivation below which I believe doesn't ...
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1answer
38 views

Centroid of mass

Is there any good resource for understanding the theory behind the centroid of a mass, $(\bar{x}, \bar{y})=(\cfrac1A\int_{a}^{b}xf(x)dx, \cfrac1A\int_{a}^{b}\cfrac12f(x)^2)$, particularly for someone ...
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spherical coordinates cross prodct

On the one hand, |$\hat{z} \times \vec{r}| = |r(\hat{z} \times \hat{r})|=rsin\theta $. direction ??? On the other hand, $\hat{z} \times \vec{r}=r\hat{z}\times({\hat{\rho}+\hat{z})} = r\hat{\varphi}$. ...
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Stability of non-linear pendulum-like equation

I am working with a differential equation of the from $$ \ddot{x} + a \dot{x} + b\sin{x} - c = 0, $$ where the initial condition is an equilibrium point: $\dot{x}(0) = 0, x(0) = \arcsin{\frac{c}{b}}$,...
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Finite element boundary condition in a weightless vacuum

Without any boundary condition the stiffness matrix $\mathbf{K}$ in a FEM solution for elastic solids $\mathbf{K}\mathbf{u} = \mathbf{T}$ (where $\mathbf{u}$ is vectorised nodal displacements and $\...
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The moment of inertia of a rigid hemisphere about diameter of a base is (a) $Ma^2/5$ (b) $Ma^2/2$ (c) $2Ma^2/5$ (d) More information needed

While solving question, The moment of inertia of a rigid hemisphere of mass $M$ and radius $a$ about a diameter of a base is (a) $Ma^2/5$ (b) $Ma^2/2$ (c) $2Ma^2/5$ (d) More information needed ? I ...
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using Lagrange equations in Kepler problem

What is the use of Lagrange equations in Kepler problem? I read that Kepler problem is an application for Lagrange equations, can someone tell me how it works?
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On method of averaging of ODE's and SDE's

Consider the following ODE for the pair of scalars $(x(t),y(t))$ \begin{align} \dot{x} &= \epsilon f(x,y) \tag 1 \\ \dot{y} &= g(x,y) \tag 2 \end{align} where $\epsilon > 0$ is a small ...
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1answer
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High School Simple Harmonic Motion Question

Could someone please guide me through the following question. I honestly have no clue where to start. Thank you so much! A particle moving in Simple Harmonic Motion starts from rest at a distance 10 ...
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AS Level Mechanics: Pulleys and Acceleration

The diagram shows a tape passing over a fixed smooth pulley. one end of the tape is fixed to the ceiling and the other is attached to a 5kg block. A cylinder of mass 12kg is placed in a loop formed by ...
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1answer
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Understanding the Poisson bracket for this system in $so(4)$

I have the following system of ODEs: \begin{align*} \dot{x}_1 &= x_2x_3A_{32} + x_5 x_6 A_{65}\\ \dot{x}_2 &= x_1 x_3 A_{13} + x_4 x_6 A_{46}\\ \dot{x}_3 &= x_1 x_2 A_{21} + ...
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Definition of Poisson Bracket

Context: Let $f,g :T^*M\rightarrow \mathbb{R}$, the Poisson Bracket was defined classically as $$\{f,g\}=\sum\limits_{i=1}^n\frac{\partial f}{\partial q^i}\frac{\partial g}{\partial p_i}-\frac{\...
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Two OCR A Level questions on $F=ma$

I just asked this on Physics.se, but the question was closed on there for "Homework-like questions and check-my-work questions are considered off-topic here, particularly when asking about specific ...
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How to solve simple pendulum equation and this equation ( $mv' + bv = mg$ )

I would like to know the steps to solve the simple pendulum (with small angle so it will be linear) and the equation of free falling mass with air resistance which is ( $mv' + bv = mg$ ). I just ...
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Problem with Generalized Momentum

I have a little problem: as we all know in Lagrangian mechanics generalized momentum is defined as the derivative in respect to $q$ dot of the Lagrangian. My question is: how can I show that this ...
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Arccosine principal values for a Legendre transformation

Consider the function $f(x) = sin(x)$ . While the function is defined for all $ x \in R $, in the context of a Legendre Transformation, since the function has to be convex, consider $ x \in [\pi,2\pi] ...
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Prove that deviations from minimum are second-order

In this page, Feynman says: https://www.feynmanlectures.caltech.edu/II_19.html “That’s a possible way. But we can do it better than that. When we have a quantity which has a minimum—for instance, in ...
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Differential equation of the central orbit

Why is $$h = v sin(\alpha) b$$, I know that h is a constant $$ = \frac{d\theta}{dt} r^2$$ but in every problem h is taken as the product of the starting velocity and the position and the sine of the ...
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35 views

Time averages for a 2-dimensional harmonic oscillator

I'm studying Ergodic Theory and I think I "got" the concept, but I need an example to verify it. Let's take the simplest possible 2D classical harmonic oscillator whose kinetic energy is $$T=\frac{\...
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Central orbit differential equation problems

I have a problem with the highlighted step in the example, i couldn't quite understand how did he integrate that ?
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How many “linkage shapes” in the plane have n “joints”?

A linkage shape in the plane with n joints means a choice of n points (a.k.a., vertices) in the plane joined by struts (a.k.a., edges). The joints are flexible pivots for the struts connecting them. ...
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Angles of a suspended lamina using centre of mass

A uniform rectangular lamina $ABCD$, where $AB$ is of length $a$ and $BC$ of length $2a$, has a mass $10m$. Further point masses $m$, $2m$, $3m$ and $4m$ are fixed to the points $A$, $B$, $C$ and $D$ ...
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2answers
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Poisson bracket of function of coordinates in terms of canonical brackets

Suppose that we work with a general Poisson structure where we are given the Poisson brackets of the individual coordinates. In practise, how would we be able to use these to determine the Poisson ...
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1answer
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Magnitude of Resistance

question and diagram I have this question (attached image) and I can't seem to work out how to go about answering it, any help would be greatly appreciated!
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Closed bounded orbits in central fields - Arnold

I am reading Arnold's book on classical mechanics and I didn't fully understand his proof of Bertrand's theorem on the central potentials for which bounded orbits of a point mass are also closed. His ...
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How to plot phase trajectories of a spherically symmetric fluid flow system?

I am trying to plot the phase trajectories of a spherically symmetric fluid flow having the following equation of motion: $$\frac{dv}{dr}=\frac{2v}{r^2}\frac{r-1}{v^2-1}$$ Integrating the above ...
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1answer
37 views

When is a multivariable function independent of one of it's input variables?

This question is in the context of variational calculus and Lagrangian mechanics. Consider a function $f(y,y^.,x)$. Here, by $y^. = \frac{dy}{dx}$ If $f$ is independent of $x$, shouldn't this mean ...
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1answer
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Using the Principle of Least Action to find the Constant/Equation of Motion

Suppose that we have a particle with mass $m$ which moves in its plane with its position at time $t$ defined by the planar polar co-ordinated $r, \theta$ (with the notation $r=r(t)$ and $\theta = \...
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Energy conservation and solving for unknowns with a one dimensional string

I am trying to solve for unknowns $R,T$ when I have the following wave ansatz: $u(x,t)=\Re((e^{iwx}+Re^{-iwx})e^{-iwt}$, for $x<0$, and $u(x,t)=\Re(Te^{iwx}e^{-iwt})$, for $x>0$. If I impose ...
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1answer
35 views

Why homogeneous? [closed]

Why in the conservation laws ( I mean energy, linear momentum and angular momentum conservations) we consider time or space to be homogeneous _as the characteristic of inertial frame? Can anyone ...
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4th-order Runge-Kutta method for double pendulum numerical solution

I am trying to numerically solve the equations of motion of the double pendulum system using the 4th order Runge-Kutta method by a C++ code. system diagram Energy equations required equations for the ...
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1answer
23 views

Express the total force as a differential equation [closed]

A particle of mass $m$ moves along a straight line (the $x$-axis) while subject to a force proportional to its displacement $x$ from a fixed point $O$ in its path and directed toward $O$ and resting ...
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Year 12 Maths - Mechanics - Resisted Motion - Getting stuck on the last part

I am getting stuck on part iv) of the question. Below are the question, photos to the solution and my working out respectively. Could someone please help me with this? Thank you so much! A particle ...
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1answer
54 views

Conversion of a vector field from Cartesian to Polar Co-Ordinate Systems

I understand that textbook questions are to be avoided here but I would like to request help from this forum. This a question in mechanics which involves what should be a fairly straightforward ...
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1answer
22 views

How do you find the centre of gravity using moments?

I'm just working through this example (Example 2), using moments and resolving forces to find the centre of gravity of three particles. I understand resolving the forces and finding $\bar{x}$, but I ...
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2answers
16 views

Impulse and change of direction

A particle $P$, of mass $0.5$ kg, is moving with velocity $(4i+4j)$ m/s when it receives an impulse $I$ of magnitude $2.5$ Ns. As a result of the impulse, the direction of motion of $P$ is ...
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1answer
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VERY HARD Collision Questions, (Further Maths A Level)

Ques I've attached the question using the link since it would be too hard to format myself. Essentially, What i've done so far is said; From start to wall 1, the time is $\frac{d}{u}$ From Wall 1 ...
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Mechanics Impulse Deflection Question

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Energy in Lagrangian problems? [closed]

When asked to find energy, when do you use $E=\sum_i\dot q_i p_i−L$ as opposed to $E = T+V$, where $T$ is the kinetic energy and $V$ is the potential energy? This is for Lagrangian problems.

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