All Questions
Tagged with calculus classical-mechanics
61
questions
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Force due to pressure on a curved surface/wall [closed]
Most solutions that I found on the internet concerning the net force due to pressure on a curved wall were using free-body diagrams and I could not find any using a calculus approach
Assuming the ...
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3
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In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?
Here are the equations. ($V$ represents a potential function and $p$ represents momentum.)
$$V(q_1,q_2) = V(aq_1 - bq_2)$$
$$\dot{p}_1 = -aV'(aq_1 - bq_2)$$
$$\dot{p}_2 = +bV'(aq_1 - bq_2)$$
Should ...
Buzz♦
- 16.3k
1
vote
2
answers
131
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On Landaus&Lifshitz's derivation of the lagrangian of a free particle [duplicate]
I'm reading the first pages of Landaus&Lifshitz's Mechanics tome. I'm looking for some clarification on the derivation of the Lagrange function for the mechanical system composed of a single free ...
- 39k
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1
answer
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Landau's derivation of a free particle's kinetic energy- expansion of a function?
I was reading a bit of Landau and Lifshitz's Mechanics the other day and ran into the following part, where the authors are about to derive the kinetic energy of a free particle. They use the fact ...
- 207k
2
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2
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826
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Necessary and Sufficient Conditions for an Equilibrium to be Stable
In the 4th section The condition that convection be absent of the book Fluid Mechanics by Landau and Lifshitz, they give the following statement:
For the (mechanical) equilibrium to be stable, it is ...
CommunityBot
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1
vote
1
answer
64
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Material to Study the Definition, Algebra, and Use of Infinitesimals in Physics [closed]
This is going to be a rather general question about suggestions on best supplementary material to properly explain the use of infinitesimals (or differentials?) for the purposes of integration or ...
- 207k
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1
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76
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Derivation of lagrange equation in classical mechanics
I'm currently working on classical mechanics and I am stuck in a part of the derivation of the lagrange equation with generalized coordinates. I just cant figure it out and don't know if it's just ...
- 194
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2
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Lagrangian total time derivative - continues second-order differential
In the lagrangian, adding total time derivative doesn't change equation of motion.
$$L' = L + \frac{d}{dt}f(q,t).$$
After playing with it, I realize that this is only true if the $f(q,t)$ function has ...
9
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7
answers
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What does it mean to integrate with respect to mass?
I've encountered many integrals that seem to integrate functions of distance with respect to mass, for example, $\int_0^Mr^2dm$ for the moment of inertia of continuous mass distribution.
I'm not sure ...
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2
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129
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Time derivative of unit velocity vector?
Let's say I have some parametric curve describing the evolution of a particle $\mathbf{r}(t)$. The velocity is $\mathbf{v}(t) = d\mathbf{r}/dt$ of course. I am trying to understand what the expression ...
- 5,078
2
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5
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263
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Why does $\vec{r}\cdot\dot{\vec{r}}=r\dot{r}$?
Why is $$\vec{r}\cdot\dot{\vec{r}}=r\dot {r}$$ true? Before saying anything, I have seen the proofs using spherical coordinates for $$\dot{\vec {r}}= \dot{r}\vec{u_r}+r\dot{\theta}\vec{u_\theta}+r\sin\...
- 12.4k
1
vote
1
answer
48
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Lagrangian for 2 inertial frames where only Speed is different by small amount
In Landau & Liftshitz’s book p.5, they go ahead and writes down lagrangians for 2 different inertial frames. They say that Lagrangian is a function of $v^2$.
So in one frame, we got $L(v^2)$.
In ...
- 525
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1
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165
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Recommended physics book(s) that uses calculus and have difficult problems [closed]
What physics book(s) uses calculus and has complex problems (undergrad/olympiad level)?
Context:
I've read "Fundamentals of Physics by Halliday and Resnick" and I found the problems to be ...
- 1
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2
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131
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Does the gradient of potential energy exist independent of coordinates?
Potential energy $U(\vec{r})$ of a conservative force field $\vec{F}$ is defined as a function whose variation between positions $\vec{r}_A$ and $\vec{r}_B$ is the opposite of the work done by the ...
- 12.4k
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2
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63
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Approximation of Small Perturbation [closed]
From Morin's Classical Mechanics, on the chapter of Small Oscillations in Lagrangian Mechanics, he does this approximation on the last equality, I don't understand what happened there.
I get the first ...
- 2,946
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113
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Question on non-holonomic constraints (This is different to the others)
I know there are many posts on non-holonomic constraints and also many on this exact one but I feel that there is still some confusion on it.
"Consider a disk which rolls without slipping across ...
- 207k
1
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1
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33
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Derivatives of the lagrangian of generalized coordinates [closed]
I know that
$$U= \frac{1}{2} \sum_{j,k} A_{jk} q_j q_k \quad \quad T= \frac{1}{2} \sum_{j,k} m_{jk} \dot{q}_j \dot{q}_k $$
and the lagrangian is
$$ \frac{\partial U}{\partial q_k} - \frac{d}{dt} \...
- 11
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1
answer
54
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What does this vertical line notation mean?
Here is the definition of the Noether momentum in my script.
$$I = \left.\frac{\partial L}{\partial \dot{x}} \frac{d x}{d \alpha} \right|_{\alpha=0} = \frac{\partial L}{\partial \dot{x}} = m \dot{x} = ...
- 207k
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1
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163
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The derivative of rotational kinetic energy in terms of period gives me the wrong answer. Why should I use the product rule? [closed]
This is my first question here so I hope I do it correctly. I've tried to solve this, and google it, but I can't find the answer to this particular question. This equation comes from Carroll and ...
5
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2
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Rigorously prove the period of small oscillations by directly integrating
This answer proved that
$$\lim_{E\to E_0}2\int_{x_1}^{x_2}\frac{\mathrm dx}{\sqrt{2\left(E-U\!\left(x\right)\right)}}=\frac{2\pi}{\sqrt{U''\!\left(x_0\right)}},$$
where $E_0:=U\!\left(x_0\right)$ is a ...
- 70.2k
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votes
1
answer
173
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Time to travel a set distance given variable acceleration
Trying to solve a problem for the acceleration of an automated shuttle car at my work, been a while since I studied this stuff so thought I'd reach out for help.
I have a shuttle car that is tasked ...
- 9,508
-1
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1
answer
63
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Derivative of distance [duplicate]
I know that $speed = |\frac{\vec{dr}}{dt}|$
and first derivative of distance with respect time will be $\frac{d\vec{|r|}}{dt}|$
These 2 expressions don't seem to represent the same thing. But when I ...
- 1,775
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37
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Is $n_{cr}=\frac{60}{2\pi}\sqrt{g\frac{\Sigma m_iy_i}{\Sigma m_iy_i^2}} =\frac{60}{2\pi}\sqrt{g\frac{\int y_idx}{\int y_i^2dx}}\quad ?$
I have a question about this formula used to calculate the first critical speed of a drive shaft.
$$
n_{cr}=\frac{60}{2\pi}\sqrt{g\frac{\Sigma m_iy_i}{\Sigma m_iy_i^2}} \tag {1} \quad .$$
It is the ...
- 8,094
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votes
3
answers
432
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Goldstein: derivation of work-energy theorem
I am reading "Classical Mechanics-Third Edition; Herbert Goldstein, Charles P. Poole, John L. Safko" and in the first chapter I came across the work-energy theorem (paraphrased) as follows:
...
- 126k
6
votes
7
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229
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Is every $dm$ piece unequal when using integration of a non-uniformly dense object?
When we want to find the total charge of an object or total mass, usually we start off with a setup such as:
$$ m = \int dm \:\;\:\text{or} \:\;\:q = \int dq$$
in which we then use (and to keep it ...
- 13.1k
2
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2
answers
298
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Solving for the radius of a sphere as a function of time
I have tried to realistically model the famous game Agar.io, which can described as the following: A sphere of initial mass $m_0$ expels part of its mass at a given rate ($\frac{dm_l}{dt}$) for thrust ...
- 19
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1
answer
43
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Doubt in derivation of bending of beam, It's about derivatives and intergration
Radius of curvature of the beam in above picture is given as:
$$ \frac{1}{R} = \frac{d^2 y}{dx^2}$$
Please help me two points used as steps of a derivation in my book:
How was the radius of ...
- 207k
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0
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41
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Minimum seperation of moving objects doubt
Let there be $2$ objects $P_1$(initial velocity $u$ $ms^{-1}$ & acceleration $a$ $ms^{-2}$) & $P_2$ (initial velocity $U$ $ms^{-1}$ & acceleration $A$ $ms^{-2}$) initially separated by ...
1
vote
2
answers
226
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Confused about the solution to the pendulum differential equation
So I’ve learned how to derive the exact solution to the pendulum differential equation (in respect to its period), $\ddot{\theta} + \frac{g}{l}\sin\theta=0$, where $g$ is gravitational acceleration ...
- 113
1
vote
1
answer
114
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Brachistochrone Problem without Trigonometric Substitution
I'm trying to numerically reproduce the cycloid solution for the brachistochrone problem. In doing so, I eventually ended up with the following integral:
$$ x = \int{\sqrt{\frac{y}{2a-y}} dy} $$
...
- 25k