All Questions
61
questions
0
votes
3
answers
80
views
Question regarding gravitational force as external force
So, I was watching a lecture on YouTube for problems on conservation of energy and momentum and I don't quite understand this:
In this question, mass $M$ is released from the peak of the smooth ...
0
votes
1
answer
58
views
Question about Problem $12$ in Chapter $11$ from Kibble & Berkshire's book
I write again the problem for convinience:
A rigid rod of length $2a$ is suspended by two light, inextensible strings of length $l$ joining its ends to supports also a distance $2a$ apart and level ...
1
vote
1
answer
164
views
Why is there a *minimum* energy for a particle to be captured in a $r^{-3}$ potential?
I was stuck in a central force problem from David Morin's Book "Introduction to Classical Mechanics".
The problem states that suppose there is a particle of mass $m$ moving under the ...
- 11
1
vote
2
answers
65
views
Position equation of $U(x)=-U_1[(\frac{x}{x_1})^3-(\frac{x}{x_1})^2 ]$
If $U(x)$ is given by $$U(x)=-U_1\left[\left(\frac{x}{x_1}\right)^3-\left(\frac{x}{x_1}\right)^2 \right]$$ can I find the position equation without harmonic aproximation?
I'm having problem with the ...
- 139
0
votes
2
answers
195
views
Marble "rolling" on the graph of a function
Consider a guide for marbles whose profile locally coincides with a function $f(x)$, for example
$f (x) = - \frac{1}{2} x ^ 4 + x ^ 3 + x ^ 2-2x + 1.$
Suppose that the reference system is chosen in ...
- 11
1
vote
0
answers
50
views
Potentials that prevent the phase flow of the system [closed]
I am trying to solve a question that my professor gave.
When a particle moves in one dimension $x$ in a potential $U(x)$ , the resulting motion over a very short time interval is specified by Newton’...
- 9
0
votes
1
answer
332
views
Doubt from Arnold; Mathematical methods of classical mechanics (page 20)
I am trying to do a problem from Arnold; Mathematical methods of Classical mechanics.
But I didn't get the desired result mentioned by the author.
Let $E_0$ be the value of the potential function at ...
- 23
13
votes
5
answers
4k
views
Why does the incline angle not affect how high a launched object will slide up a frictionless ramp?
I am seeing a problem with the solution given in this book. How did the height of the box have nothing to do with the incline of the ramp? Intuitively it would seem the higher the incline the higher ...
1
vote
0
answers
71
views
Verifying the equation of motion, expressions of kinetic energy and potential energy and how to examine whether motion confined to a plane or not [closed]
A particle is moving in space such that it is attracted towards a fixed point and is proportional to the distance from the fixed point. Derive the Lagrangian and Hamiltonian of the system. Examine ...
- 111
3
votes
1
answer
537
views
Doubt in the expression of Lagrangian of a system [duplicate]
There is a problem given in Goldstein's Classical Mechanics Chapter-1 as
20. A particle of mass $\,m\,$ moves in one dimension such that it has the Lagrangian
\begin{equation}
L\boldsymbol{=}\...
- 436
0
votes
1
answer
178
views
Goldstein equation 1.33
I am trying to read from Goldstein for self-study but I am stuck on equation 1.33. Let me restate some of the lines from Goldstein (with some modification):
If $\textbf{F}_{ij}$ (internal force, ...
- 101
1
vote
1
answer
824
views
Work done for conservative forces is path independent Proof
So I’m looking at the proof for work that is path independent.
There is a line were the integral
Partial derivative V dr from r1 to r2 becomes
Partial derivative V r’ dt from t1 to t2
I’m a bit ...
0
votes
0
answers
40
views
Motion near the local maximum of potential energy
Particle is moving along the $x$ axis in the field with potential energy $U(x)$. $U(x) $has local maximum at $x=0$, and the total energy of particle is equal to $E=U(0)$.
I'm supposed to find how the ...
- 101
0
votes
1
answer
263
views
Taylor Example 4.8. Is my reasoning sound? [closed]
This problem has been giving me all sorts of fits. For one, Taylor states that because the frictional force and normal force are forces of constraint, they produce no work. I'm trying to figure out ...
0
votes
1
answer
148
views
Particle in a cylinder with a spring, sign convention in potential energy (Lagrangian multipliers)
I'm trying to get the force of constraint. The problem I have is when defying the sign of the potential energy using cylindrical coordinates $(\rho,\phi,z)$, what I have is:
$$
V=mgy-\frac{1}{2}k\left(...
- 7
0
votes
1
answer
569
views
Potential energy of a mass bewteen two springs with pendulum hanging [closed]
I need some help with this problem.
A particle of mass $m_1$ hangs from a rod of negligible mass and length $l$, whose support point consists of another particle of mass $m_2$ that moves horizontally ...
- 103
0
votes
2
answers
130
views
Finding total mechanical energy, given the potential [closed]
The average kinetic energy of a particle in a potential of the form
'$V(x, y)=x^{4}+4 x^{2} y^{2}+4 x^{3} y-2 y^{4}$'
is equal to $T$.
How can we find the total energy of the particle?
My attempt:
I ...
- 1,270
0
votes
1
answer
188
views
Elastic potential energy during elastic collisions
While working with problems on elastic collisions, I have come across this observation, that the elastic potential energy of a two-body system is the maximum when the relative velocity equals zero. In ...
- 1,638
-1
votes
1
answer
150
views
Falling Chain help! [closed]
I was going through Example 9.2 in Thornton and Marion's Classical Dynamics, and I am stuck on the Potential Energy part of the Question. How do they get the term at the top of the page on the right? ...
1
vote
2
answers
380
views
Why is the work done by a block into a spring the same from the work done by the spring on the block?
In the following situation:
A 700 g block is released from rest at height h 0 above a vertical
spring with spring constant k = 400 N/m and negligible mass. The block
sticks to the spring and ...
- 493
0
votes
2
answers
815
views
The Theoretical Minimum: Lecture 5, Exercise 3. Finding equations of motion from potential energy [closed]
From Leonard Susskind's book The Theoretical Minimum.
"A particle in two dimensions, x and y, has mass m equal in both directions. It moves in a potential energy $V = \frac{k}{2(x^2+y^2)}$. Work ...
- 141
0
votes
1
answer
42
views
Change in potential energy after infinitesimal variation in position
The a particle with the potential $V(x^2+y^2)$ undergoes an active transformation where
$x\rightarrow x+y\delta$
$y\rightarrow y-x\delta$
The exercise was to prove that the Lagrangian of the system ...
- 62
6
votes
2
answers
2k
views
Bertrand's theorem and nearly-circular motion in a Yukawa potential
The question has arisen as a result of working on part b of problem 3.19 in Goldstein's Classical Mechanics book.
A particle moves in a force field described by the Yukawa potential $$ V(r) = -\frac{...
- 1,031
1
vote
1
answer
2k
views
Force derived from Yukawa potential
This is with regards to problem 3.19 from Goldstein's Classical Mechanics,
A particle moves in a force field described by the Yukowa potential $$ V(r) = -\frac{k}{r} e^{-\frac{r}{a}},
$$ where $k$ ...
- 1,031
0
votes
1
answer
106
views
Feynman Lectures, Chapter 4, Fig 4-3
From the Feynman lectures Chapter 4, Fig 4-3
"We lifted the one-pound weight only three feet and we lowered W pounds by five feet. Therefore W=3/5 of a pound."
If there is a change of 3ft in ...
- 171
2
votes
2
answers
1k
views
Change in Energy when placing an object on the ground
It seems like a simple question but I was wondering where does the energy go when I place an object from a height on the floor.
Initially it's all stored as potential energy, and as I'm moving the ...
- 131
1
vote
1
answer
383
views
Weight and potential energy for a spring pendulum
Consider a spring pendulum like in this figure
suppose the spring is arranged to lie in a straight line and its equilibrium lenght is $l$.
Consider the unit vectors $e_1, e_2$, $e_x, e_{\theta}$ like ...
- 145
0
votes
2
answers
783
views
Classical period of Morse potential [closed]
A particle of mass $m$ and energy $E<0$ moves in a one-dimensional Morse potential:
$$V(x)=V_0(e^{-2ax}-2e^{-ax}),\qquad V_0,a>0,\qquad E>-V_0.$$
From the only other question I have ...
0
votes
1
answer
78
views
Approximating the time it takes for a particle with a potential $-Ax^4$ to approach the origin [closed]
Here's the problem I'm solving:
A particle of mass $m$ can only move along the $x$-axis and is subject to an interaction described by the potential energy function $U\left(x\right) = -Ax^4$, where $A ...
user113773
0
votes
4
answers
102
views
Potential Energy of Conservative Forces [closed]
For a conservative force, its associated potential energy at position $\mathbf{r}$ is
$$U(\mathbf{r}) = - \int_{\mathbf{r}_{0}}^{\mathbf{r}} \mathbf{F}(\mathbf{r'}) \cdot \text{d} \mathbf{r'}$$
...
- 1
0
votes
2
answers
127
views
Work done in circular expansion of a rubberband or an elastic wire
What will be the work done in radially stretching a rubberband it can't be zero as there is potential energy being stored in it
All I came up with it that there would be increase in overall length so ...
4
votes
2
answers
3k
views
Proof of total energy is constant in a central force field
I saw many proofs of total energy is constant in a central force field. But all the proofs end up showing this formula $$m[{\dot r}^2 + r ^2{\dot \theta}^2 ]+ \int f(r)dr = E$$ is constant. But ...
- 1,149
-2
votes
1
answer
576
views
What is the largest radius that the circular track can have such that the car remains in contact with the track? [closed]
A car is travelling at $u$ as it enters a vertical loop of radius $r$.
What is the largest value of $r$ that the track can have such that the car goes up and around the track whilst remaining in ...
- 1,329
58
votes
15
answers
13k
views
When a balloon pops and lets a brick fall, where does the energy come from?
Let's say a scientist attaches a 1 kg brick to a large helium inflated balloon, lets the balloon go, and then it reaches an altitude of 10 000 meters before it pops, dropping the brick.
The brick ...
- 1,298
2
votes
1
answer
125
views
Potential energy and conservation law
I'm preparing for my masters entrance exam on pure mathematics (thought some problems are devoted to classical/lagrangian mechanics). I would be grateful to clarify some basics regarding the ...
- 123
1
vote
0
answers
527
views
Particle in electromagnetic field Lagrangian
Given the two definitions of $\vec E$ and $\vec B$ by scalar potential $\phi$ and vector potential $\vec A$:
$$\vec B=\vec \nabla \times \vec A$$
$$\vec E=-\vec \nabla \phi -\frac 1 c\frac {\partial \...
- 141
0
votes
1
answer
139
views
Final state of a two-block system with friction on a smooth ground
I was working on a modified version of this problem (which I solved):
The modification is as follows:
Consider the following system with two masses, $M$ and $m$, with the length of the body $M$...
- 288
-1
votes
1
answer
143
views
Spring-mass problem [duplicate]
Question: A spring-mass system hangs vertically from a fixed support. The natural length of the spring is l. The block is displaced from the equilibrium position and constrained to move horizontally. ...
2
votes
2
answers
2k
views
Central force - stable/unstable circular orbit for $V = -V_0 \exp(-\lambda^2 r^2)$
This problem is from Introduction to Classical Mechanics With Problems and Solutions by David Morin. The solution is also given in the book for this particular problem.
Problem #6.6.1
A particle ...
- 592
0
votes
1
answer
335
views
Finding height of rope when at an angle? [closed]
The question is that Jeff swings on a 7.6m rope that initially makes an angle of 37° with the vertical. If Jeff starts at rest and has a mass of 78kg, what is the tension in the vine at the lowest ...
- 103
1
vote
0
answers
78
views
Perihelion precession for general potential
I'm trying to show that for the perihelion precession $\Delta\phi$ follows:
$$\Delta\phi=2\int_{r_\textrm{min}}^{r_\textrm{max}}\frac{L}{r^2(E-U_\textrm{eff}(r))^{1/2}}~\mathrm dr$$
where $L$ is the ...
user137707
1
vote
1
answer
1k
views
Force and energy relation: in case of time dependent force
The equivalent problems are also found in Marion problem 7-22, and other formal classical mechanics textbook. Here what i want to know why instructor solution and some websites gives this kinds of ...
- 3,622
0
votes
2
answers
2k
views
How to prove that the potential of a conservative central force depends only on the magnitude of the distance & not on the direction of the vector?
If a conservative central force acts on a body then its potential only depends on the magnitude of the distance between two bodies (does not depend on direction of vector). Is there any proof of it ...
- 41
4
votes
1
answer
442
views
Period on the phase plane (small oscillations)
I have this formula to calculate the period of a motion in the phase space (plan, in this case) along a phase curve.
\begin{equation}
T(E)=\int_{x_1}^{x_2}\frac{dx}{\sqrt{2(E-U(x))}}
\end{equation}
...
- 1,704
0
votes
1
answer
2k
views
Gravitational potential energy lost by an object falling on the earth [closed]
I am stuck on this simple question:
g is the strength of the gravitational field at the surface of the
Earth; R is the radius of the Earth. Show an equation describing the
potential energy lost ...
user63248
5
votes
1
answer
1k
views
How to show period is defined by $T=dS/dE$ (V.I. Arnold Mathemtical Physics) [closed]
I'm looking at a book by VI Arnold on mathematical physics and I've hit a roadblock pretty early on. I'll quote the question:
"Let $S(E)$ be the area enclosed by the closed phase curve ...
- 130
0
votes
1
answer
399
views
Spring Potential Energy [closed]
A spring whose spring constant is $k$, having an initial "free" length is $l$, is being pressed by $2$ hoops on a metal(the parabola) on his both sides. (see image below)
I want to calculate the ...
- 185
0
votes
1
answer
158
views
Potential Energy [closed]
I really need help with this homework question. I thought the function very stranger to be a function of a potential energy. It´s seems to be like Lennard Jonnes pontential energy function. Am I right?...
- 5,257
1
vote
1
answer
335
views
Why does $\omega = \sqrt{V''(x_0) / m}$?
I know that in an equation such that $$\ddot{x} + \omega^2x = 0,$$ the angular frequency $ = \omega$. But why is that ever $ \sqrt{V''(x_0) / m}$? (where $x_0$ is the equilibrium point). I just saw ...
- 171
1
vote
0
answers
163
views
Potential Energy of two masses
If two particles with masses $m_1$ and $m_2$ interact and are located at $\vec{s_1}$ and $\vec{s_2}$ have their potential energy $U$ defined by the modulus of their position vectors, how would I ...
- 11