New answers tagged classical-mechanics
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Constraint equation for an elastic pendulum
If you need to evaluate the reaction at the constraint with a Lagrangian approach, you need to add the constraint in the computation of the stationary point of the action.
Let's call $P$ the mass ...
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Constraint equation for an elastic pendulum
The elastic pendulum have two generalized coordinate
$~x,y~$
from here the constraint equation
$$\tan(\theta)=\frac xy\quad,\omega\,t=\arctan(x/y)$$
and with $~x=r\,\sin(\theta),y=r\,\cos(\theta)$
you ...
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"Natural frequency" seems to be a poorly defined concept
The Wikipedia definition of natural frequency is a poor one, since (as you point out) it implies that a system can only oscillate at its natural frequency if there is no driving force. This is ...
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"Natural frequency" seems to be a poorly defined concept
The problem with an eigen-frequency is that it is exact, and the phase space to hit an exact frequency is 0, but if you do hit it, the response diverges...classically.
In quantum mechanics, the eigen ...
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Solution as the real part of complex exponential for simple harmonic motion
Let us set $ z(t) = \left(C_{1} e^{i \omega t}\right)$. Also, as stated by @Michael Seifert
$$ \left(C_{1} e^{i \omega t}\right)^{*}=C_{1}^{*} e^{-i \omega t}$$
and therefore
$$ x(t) = z(t) + z^{*}(t) ...
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Fundamental "definition" of momentum
When we say that massless particles have "momentum", are we even talking about the same thing as when we say that particles with mass have momentum?
The motion of the electric field in an EM ...
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Finding the tipping force for a cantilever table
We can look at the torque. We have a pivot point r that everything will go around. And we can see three forces $F_1$, $F_2$ and $F_3$. $F_1$ twisting one way (the sitter), $F_2$( the centre of mass of ...
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Help in understanding this derivation of Lagrange Equations in Non-Holonomic case
The quoted text says that the vector ${\bf Q}$ with components $Q_r$ is perpendicular to all vectors ${\bf dq}$ with components $dq_r$ that are themselves perpendicular to all vectors ${\bf A}_k$ ...
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Help in understanding this derivation of Lagrange Equations in Non-Holonomic case
This follows from the fact in linear algebra that if $V$ is an $n$-dimensional vector space, and $$Q, A_1, \ldots, A_m \in V^{\ast}$$ are elements in the dual vector space, then
$$ {\rm ker}(Q) ~\...
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Solving forced harmonic oscillator differential equation using fourier transform
It is often helpful to add a regularization factor $e^{-\epsilon \left| t \right|}$ to the integrands in the Fourier transformation to ensure convergence. You then take the limit $\epsilon \to 0$ ...
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Solving forced harmonic oscillator differential equation using fourier transform
Without any claiming to be a complete answer with all the details you need, you could start using the (unilateral) Laplace transform
\begin{equation}
F(s) := \int_{t=0}^{\infty} f(t) e^{-st} dt \ .
\...
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Why does $\vec{r}\cdot\dot{\vec{r}}=r\dot{r}$?
the position vector $~\vec r~$ is
$$\vec r=r\,\vec e_r\quad, \text{with}\quad \vec e_r\cdot \vec e_r=1 $$
thus
$$\vec v=\dot r\,\vec e_r+r\,\vec{\dot{e}}_r$$
and
$$\vec r\cdot \vec v=r\,\vec e_r\,(\...
- 11.3k
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Independence of generalized coordinates in the derivation of Lagrange equations from d'Alembert's Principle
OP has a point. Already above eq. (1.38) Ref. 1 writes that the generalized coordinates generalized coordinates $q^1,\ldots, q^n,$ are independent variables, and this fact is then used repeatedly e.g. ...
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Lagrange multipliers method for a bead on a parabola-shaped wire
Observe that depending where the bead is on the parabola, the constraint force of the parabola changes its direction and magnitude. That magnitude is related to $\lambda$. So lambda changes with time. ...
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Classical Mechanics, The Theoretical Minimum: angular momentum conservation for the double pendulum without gravitational field
A few pages earlier in the book, we proved that $\sum_{i}p_if_i(q)$ is conserved.
So, for the double pendulum example
The generalized coordinates {$q_i$} are {$\theta$, $\alpha$}
The functions {$f_i$...
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Why does $\vec{r}\cdot\dot{\vec{r}}=r\dot{r}$?
The thing that is missing in the statement is the angle between two vectors. So the complete equation should be:
$\vec{r}\cdot \dot{\vec{r}} =r\dot{r} \cos\alpha$.
$\alpha$ is the angle between the ...
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Kinematics - confusion about signs of angular velocity and acceleration (general rule ?)
In order to approach this kind of problems, the first thing you should do is to choose a reference frame. As an example, you can choose a Cartesian reference frame with the origin in $O$, with an ...
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Why does $\vec{r}\cdot\dot{\vec{r}}=r\dot{r}$?
To sum up the previous answers: it is indeed true that $\vec{r}\cdot\dot{\vec{r}}=r\dot{r}$, in fact:
\begin{equation}
2r\dot{r}=\frac{\mathrm{d}}{\mathrm{d}t}(r^2)=\frac{\mathrm{d}}{\mathrm{d}t}(\vec{...
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Getting different answers by different methods for angle made by a pendulum moving with constant acceleration
By your second way you do not get the equilibrium position, but the maximum angle you get when you accelerate the car. the pendulum goes to the double angle and than drops back, without friction it ...
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Why does $\vec{r}\cdot\dot{\vec{r}}=r\dot{r}$?
The reasoning using spherical coordinates is correct as it applies to all kind of motions, is just a description in polar coordinates but nothing is assumed about the particular kind of motion. In ...
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Why does $\vec{r}\cdot\dot{\vec{r}}=r\dot{r}$?
What's wrong with your argument is that $\dot r \neq v$. In your example $\dot r$ is indeed zero (because velocity is perpendicular to $r$, therefore, the magnitude of $r$ doesn't change) and so is $\...
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Question about velocities in different reference frames
Short answer. The velocity and the acceleration w.r.t. a reference frames are defined here to be the product of the time derivatives of the components and the vectors of the basis. There is no ...
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Question about velocities in different reference frames
There are two things which are changing.
First the position vector as measured in the two frames and secondly the coordinate system in the moving frame relative to the coordinate system in the ...
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Doubt in fictitious forces chapter in Morin
but isn't dA/dt the real thing, the physical thing?
Let's write $\frac{d\mathbf{A}}{dt}=\frac{d}{dt}\mathbf{A}(t)$. $\mathbf{A}(t)$ is just a symbol, we use it to represent the physical vector (which ...
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Lagrangian mechanics and generalized coordinates
As stated in an earlier answer, there is no general algorithm for arriving at the most suitable generalized coordinates (in cases where using some form of generalized coordinates is beneficial). By ...
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Confusion about torque
the torque $~\vec\tau~$ is
$$\vec\tau=\vec{r}_{Ao}\times \vec F=r\,F\,\underbrace{(\hat e_{Ao}\times \hat e_F)}_{\hat e_\tau}$$
where A is arbitrary rigid body point
thus the rotation of the rigid ...
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Confusion about torque
The body will rotate about the centre of mass. The body will also undergo translation. If the force is applied at the centre of mass, the body will undergo pure translation.
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Lagrangian mechanics and generalized coordinates
Is there any systematic/clever/cheating way that could be used to determine these gc's?
Not really. This is kind of an art. On the one hand, it doesn’t matter, any coordinates will do. On the other ...
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How do we know what physics or science textbook said is correct?
THe only way to know is by doing the experiment. There really isn't a substitute. Logical thought, exclusively, leads to multiple theories that typically result in conflicting results; and the only ...
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How do we know what physics or science textbook said is correct?
I think some of the challenge here is that we typically use a scientific-realism voice in textbooks. They state "gravity does this," "heat does that." Which leads to the sort of ...
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How do we know what physics or science textbook said is correct?
"The best teachers are those who show you where to look, but don’t tell you what to see." - (maybe) Alexandra K. Trenfor
The goal of a physics (any science) book is not to claim, but to ...
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Why does a higher frequency mechanical wave have more energy?
But if I just shorten the wavelength (elevating the frequency) why does that one period now contains more energy?
In a mechanical wave, the oscillation corresponds to some periodic bending or stress ...
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How do we know what physics or science textbook said is correct?
In addition to what the other answers have said about trusting the wider scientific community being necessary at some level (since none of us have the time or resources to reproduce all experiments), ...
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Why the dynamic nonlinear differential equations of plate are said to be of eight order?
The author mentions that the order of the system is 8 because it is formed by two fourth order differential equations that are coupled.
If you were considering plates, three simplest theory is of ...
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Obtaining Equations of Motion in a Rotating Coordinate System using Lagrangian
for 3-D rotations, it is often useful to note that $\pmb{\omega}$ is more properly represented as a skew-symmetric 2-tensor $\pmb{\Omega}$ such that $\pmb{\Omega}\cdot \pmb{u}= \pmb{\omega}\times \pmb{...
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Virtual work of constraints in Hamilton‘s principle
For what it's worth, the holonomic constraints (that Ref. 1 is here talking about) are the ones used to define the generalized coordinates $q^1,\ldots, q^n,$ from the positions ${\bf r}_1, \ldots, {\...
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Why does a higher frequency mechanical wave have more energy?
In the case of stationary waves, as the vibrating strings in a guitar, the situation is similar to an oscillator. If one fret is pressed, so that the vibrating string has half of its length, the ...
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Can you prove the possibility to rewrite any lorentz invariant equation as the component of a 4-tensor?
Can this be proven in any way?
Yes, physics is independent of the coordinates in the sense that the mathematical operation of changing your coordinates does not change any of the physical outcomes of ...
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How do I show that there exists variational/action principle for a given classical system?
Suppose you have a system whose kinematics are given by a list of $N$ "configuration" variables $q = \left(q^a: 0 ≤ a < N\right)$, for some number $N = 1, 2, 3, ⋯$ of "degrees of ...
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Why does a higher frequency mechanical wave have more energy?
Your basic error is that you are treating the wave as though it was a system of independent particles and this is not correct.
The particles are communicating with one another in some way, eg via ...
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Confusion of variable vs path in Euler-Lagrange equation, Hamiltonian mechanics, and Lagrangian mechanics
All the problem, in my view, arises form the fact that the EL equations are introduced in a too sloppy way. (The variational approach makes even more obscure an obscure setup.)
Actually,
The ...
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Confusion of variable vs path in Euler-Lagrange equation, Hamiltonian mechanics, and Lagrangian mechanics
On one hand, the Lagrangian $L(q,v,t)$ is a function of the position$^1$ $q$, the velocity $v$ and the time $t$.
On the other hand, the Lagrangian action
$$S[q] ~:=~ \int_{t_i}^{t_f}\mathrm{d}t \ L(q(...
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Accepted
Some books write $V(\vec{r})$ instead of $V(r)$ as a notation for the electric potential, so which one is right?
In general, electric potential is a function of the vector position. Sometimes the vector notation is reduced to a scalar due to spherical symmetry.
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Confusion of variable vs path in Euler-Lagrange equation, Hamiltonian mechanics, and Lagrangian mechanics
The Lagrangian is a function, not a functional. The action is a functional, and is defined as
$$S[q; t_0,t_1] = \int_{t_0}^{t_1} L\big(q(t), \dot q(t), t\big) \mathrm dt$$
The partial derivatives ...
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Does work depend on a point of reference?
I am so fit that my mass is 0.5kg, so when a force of 1N in the direction of my movement acts on me for 1s, I accelerate to 1002m/s and in that time I travel 1001m (all relative to Earth).
All this ...
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The Hamilton's equations of a charged particle in electromagnetic field
Here: let's do it a little bit differently that will help make things more clear. For a charged body, with charge $e$, the potential energy $eφ$ has the same relation to the body's energy $E$, as $e𝐀$...
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Why does a higher frequency mechanical wave have more energy?
I’m not really familiar with the mathematics of this, so I’ll propose a simple answer:
First we need to look at the definition of a mechanical wave. A mechanical wave is a vertical transfer of energy ...
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Why does a higher frequency mechanical wave have more energy?
Wave equation in classical mechanics. Sometimes it's possible to recast the governing equation of an elastic medium as a wave equation,
$\dfrac{1}{c^2} \partial_{tt} u - \nabla^2 u = f$,
being $u$ the ...
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What makes a higher frequency sound wave more energetic?
It's an Excellent question;
The energy of a sound wave is determined by its amplitude, not its frequency. Amplitude refers to the extent of the vibration or oscillation in the wave, and it corresponds ...
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How to determine which coordinates to use for calculating the Hamiltonian?
In general, the lagrangian of a simple pendulum of longitude $l$ that moves in radial and angular directions is:
$$\mathcal{L}(r, \theta) = \frac{1}{2}m(\dot{r}^2 + r\dot{\theta}^2) - mgr(1-\cos(\...
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