Classical mechanics Definition and 1000 Threads

Now, I can calculate the velocity of the block using the concept of pseudo force (which here is the centrifugal force), and newton's second law i.e F=ma but, I don't understand why should this block move I mean in the non-inertial reference frame the reason can be given by the centrifugal force...

I tried taking an element of length dx and tried calculating the force of gravity acting on it so that I could just integrate over the whole chain, but I couldn't figure out what to do of that displacement part since the dx element is not moving as is just at rest (hanging). So, how should I...

In my textbook, a hint is the formula above, which can be used when we have a homogenous body. ##M## is the body's mass, but what does the remnant mean?

The book Classical Mechanics by Alexei Deriglazov defines as canonical a transformation Z=Z(z,t) that preserves the Hamiltonian form of the equation of motion for any H. After taking the divergence of the vector equation relating the components of the time derivative of Z in the two coordinate...

After a year of thinking about instantaneous velocity. I think that this notion is no more than a mathematic coincidence when mathematician tried to find the tangent of curve. The only definition of velocity that make sense is ##\frac{\Delta x}{\Delta t}##, this proportion is a quantity that...

If the work done by a force is the consequence of the dot product of the force and the displacement of the particle, then it should be a scalar quantity which is true as well. But then why does question of the work done being positive or negative (depending on the angle between the force vector...

A uniform circular motion of a point always yield an equation u=cos(wt)i +sin(wt)j of position vector. Which we deduce the acceleration and velocity vector with constant magnitude and they are perpendicular at each instant. Can I use the information of them at one instant to predict the position...

TL;DR Summary: What book should i read before arnolds classical mechanics? I have good math background but little physics background. A little while ago, i was in a summer camp for teens who are exceptional at math. Around that time i started getting into physics. A person from there...

I know that I can draw an FBD and apply Newton's 2nd law to find the relevant equations. But my question here is why is the mg vector or the weight of the entire rope same at every point on it I mean to say that if the mass of the entire rope is say M then how can a small point on the rope also...

I think that time derivative of the energy of the oscillator is F times the derivative of q, which means it's the power of the external force. So it's like it is suggesting that F·q is the work done by external force, which makes no sense at all. As far as I concerned, the input energy has no...

Hello, I often get confused when it comes to some physical concepts in classical mechanics. That's mostly because I like to ask a ton of questions and because that I dig myself into a hole that I can't come out of. So, I'm wondering if anyone knows a really good classical mechanics book that...

I express the total kinetic energy of the body, via König theorem, as $$T=\frac{1}{2}mv_p^2+\frac{1}{2}mI{\omega}^2$$ where $$v_p=(v_x,v_y)=(\dot{r}\cos\varphi-r\dot{\varphi}\sin\varphi-\frac{l}{2}(\dot\varphi-\dot\psi)\sin(\varphi-\psi),\dot r \sin\varphi+r\dot\varphi...

For the past few months, I've been on a look out for the best physics community on the Internet and I've just come across this one. My primary goal is to gain as much knowledge as possible in the area of classical mechanics and electromagnetism in a year. I'm fairly new to magnetism, but I can't...

(a). Getting the potential is straight forward, just take the negative derivative of F(x) and then enforce the condition that the function should approach 0 at x = infinity. ##- \int -a e^{-bx} (1 - e^{-bx})## = ##\int a e^{-bx} - a e^(-2bx)## = ## \frac{a}{b} (\frac{1}{2}e^{-2bx} - e^{-bx}) +...

So I’m a high school student and I am planing to participate in higher-category physics competitions in my country. However, I think that my theoretical understanding of the basics isn’t clear enough yet - by basics I mean classical / Newtonian mechanics. I am the type of person that learns by...

My set-up is the following: i have an iron bolt suspended on a string next to an electromagnet, of which I steadily increase the voltage and thereby the magnetic field. Supposing the force is linear with the magnetic field and dependent on the distance between the bolt and magnet. The exact...

Suppose somehow an object is moving upwards with a speed ##v##, at this point I start applying a force ##F## that is equal to its weight, so the net force on the object is zero. So it will continue moving upwards with its initial speed. Suppose after the height difference is ##h##, I stop...

I'm using rigid body dynamics/kinematics in robotics stuff but I don't have a background in mechanics, I'm interested in understanding the kinematics of frame transformations for rigid bodies. Suppose we have two reference frames fixed on a rigid body, F_1 and F_2 and a transformation T which...

(This is a homework assignment that my sister (younger than me) didn't manage to solve, but I am not sure about the attempt I thought of either. Especially in the last point. So I ask you to correct where I am wrong). I solved ##a)## with the following: 1. Since ##B## must be stationary, I...

[Rewriting this as per the suggestions. Thanks once again.] I won't be using the Lagrangian because it was never explicitly stated that I have to so I'll just use conservation of energy. $$ T = \frac{1}{2}mv^2 = \frac{1}{2}m(R\dot{\theta})^2 = \frac{1}{2}mR^2\dot{\theta}^2 $$ $$ V = mgy =...

I use ##l-1## lagrangian coordinates ##\alpha_1,...,\alpha_{l-1}## . ##\alpha_i## is the angle between ##OP_{i-1}## and ##OP_{i}##. As the length of a chord between two rays with angle ##\alpha## is ##d=2Rsin(\alpha/2)##, I write the potential energy of the system as...

Hi, in the Hamiltonian formulation of classical mechanics, the phase space is a symplectic manifold. Namely there is a closed non-degenerate 2-form ##\omega## that assign a symplectic structure to the ##2m## even dimensional manifold (the phase space). As explained here Darboux's theorem since...

We start with something like If we suppose the wire is the green line, we are to figure out what the green line looks like to the right of the red bead. What I first thought of was simply $$\vec{r}'(t)=x'(t)\hat{i}+y'(t)\hat{j}=v_0\hat{i}+gt\hat{j}\tag{1}$$...

My answer to the question is: build a two-stage rocket. Or a ##k##-stage rocket. Then I thought: what happens if we try to make ##k=\infty##? To cut to the chase, my question is how to calculate the infinite series $$\lim\limits_{k\to\infty} \sum\limits_{i=1}^k \ln{\left (...

I have found the Hamiltonian to be ##H = L - 6 (q_1)^2## using the method below: 1. Find momenta using δL/δ\dot{q_i} 2. Apply Hamiltonian equation: H = sum over i (p_i \dot{q_i}) - L 3(q_1)^2. Simplifying result by combining terms 4. Comparing the given Lagrangian to the resulting Hamiltonian I...

I've started reading the Principia and have been trying to follow along with the examples. Unfortunately, I got stuck almost immediately. This example is from 'Axioms, or laws of motion', Law III, Corollary II. It is based on the following picture (everything in red is my addition): The text...

For part (d), there is the formula a = v^2/r I can use. Note that Mg = mv^2/r, we have two unknowns, v and r. I can solve this if conservation of angular momentum is true, i.e. mvr = constant. I am not convinced I can use this however, because is increasing M torque? My idea is that it is an...

I've heard that classical mechanics and electromagnetics are not applicable at small sizes in particle physics. 1) At what size and energy levels are they no longer considered to be applicable at all? 2) What range of size and energy levels could be considered a "transition" area where both...

We know the wave function: $$ \frac {\partial^2\psi}{\partial t^2}=\frac {\partial^2\psi}{\partial x^2}v^2,$$ where the function ##\psi(x,t)=A\ e^{i(kx-\omega t)}## satisfies the wave function and is used to describe plane waves, which can be written as: $$ \psi(x,t)=A\ [\cos(kx-\omega...

vA = 3u/4 and vB = u/4, and 1/8 KE is lost. I can't get to these answers however: for the first part, I got to u = vA + 3vB using conservation of momentum, and the fact that particle B is at an angle, hence I would think its momentum should be 10mvBsin(arcsin(3/5)). Doing the same for A with...

What is the correct statement of Varignon's theorem? On the net I find some discrepancies between the various statements: in some cases the vectors of the system referred to by the theorem must be applied at the same point or such that their lines of action pass through the same point, in other...

Can somebody explain why the kinetic term for the fluctuations was already diagonal and why to normalize it, the sqrt(m) is added? Any why here Z_ij = delta_ij? Quite confused about understanding this paragraph, can anybody explain it more easily?

I have read some text about defining the cross product. It can be defined by both a x b = epsilon_(ijk) a^j b^k e-hat^i and a x b = epsilon^(ijk) a_i b_j e-hat^k why the a and b have opposite indice positions with the epsilon? How to understand that physically?

Consider the above system, where both the wedge and the mass can move without friction. We want to get the equations of motion for the both of them using Lagrangian formalism, where the constraints in the solution sheet are given as: $$y_2=0$$ $$\tan \alpha=\frac {y_1}{x_1-x_2}$$ However...

Assuming that a universe operates according to Aristotle's mechanics, that is to say, the second law of Aristotle in this universe F=mV (V is the velocity of the object's motion)True. (Note that 'm' here does not have a dimension of mass.) In order to obtain a logically consistent Aristotle's...

For the central force ##F=-\nabla U(r_r)## where ##\vec r_r=\vec r_1-\vec r_2##, and ##\vec r_1## and ##\vec r_2## denote the positions of the masses, we get the following kinetic energy using the definition of center of mass ##\vec r_{cm}= \frac{m_1\vec r_1+m_2\vec r_2}{m_1+m_2}##: $$T= \frac...

One of the constraints is given as ##r=R##, which is very obvious. The second constraint is however given as $$\phi - \frac {2\pi} h z=0$$ where ##h## is the increase of ##z## in one turn of the helix. Physically, I can't see where this constraint comes from and how ##\phi=\frac {2\pi}h z##.

Suppose we have mass ##m## attached to spring with constant ##k##, and some coefficient of kinetic friction ##\mu## between the mass and the surface. Its displaced from equilibrium by some distance ##x## at ## t = 0 ##. I've come up with the following ODE to described ##x(t)## using the...

Consider the following setup where the bead can glide along the rod without friction, and the rod rotates with a constant angular velocity ##\omega##, and we want to find the constraint force using Lagrange multipliers. I chose the generalized coordinates ##q=\{r,\varphi\}## and the...

I'm just getting started on Lagrangian mechanics and what I can't understand is, how did Lagrange discover the Lagrangian? Did he just randomly decide to see what would happen if we calculate KE - PE or T - V and then discovered that the quantity is actually mathematically and physically...

My question is about the general relationship between the constraint functions and the constraint forces, but I found it easier to explain my problem over the example of a double pendulum: Consider a double pendulum with the generalized coordinates ##q=\{l_1,\theta_1,l_2,\theta_2\}##,: The...

For a rigid body with three principal axis with distinct moments of inertia, would the principal axis with the intermediate moment of inertia still be unstable in ideal conditions, e.g. no gravity, no friction etc.? From the mathematical derivation I deduce that it should be unstable, since we...

I learned some computer science basics from the book SICP ( Structure and Interpretation of Computer Programs, Authors: Gerald Jay Sussman, Hal Abelson, Julie Sussman ) and I've witnessed a book about mechanics from the same author called Structure and Interpretation of Classical Mechanics...

In classical mechanics, it seems like solving force equations are a question of finding a solvable system of equations that accounts for all existing forces and masses in question. Therefore, I'm curious if this can be mixed with reinforcement learning to create a game and reward function...

The invariance of this volume element is shown by writing the infinitesimal volume elements $$d\eta$$ and $$d\rho$$ $$d\eta=dq_1.....dq_ndp_1......dp_n$$ $$d\rho=dQ_1.......dQ_ndP_1....dP_n$$ and we know that both of them are related to each other by the absolute value of the determinant of...

The homework statement isn't exactly as is mentioned above. The actual problem statement is as follows: This is problem 3.8 from John R. Taylor's Classical Mechanics; however, my question is not related to the main problem itself but one particular aspect of it. Now, in the same textbook (John...

Goldstein's Classical Mechanics makes the claim (pages 382 to 383) that given coordinates ##q,p##, Hamiltonian ##H##, and new coordinates ##Q(q,p),P(q,p)##, there exists a transformed Hamiltonian ##K## such that ##\dot Q_i = \frac{\partial K}{ \partial P_i}## and ##\dot P_i = -\frac{\partial...

I think product of inertia always zero because rolling motion of disk is fully balanced

I am attempting problem number 2.38 from John R. Taylor's Classical Mechanics and I am not getting the correct answer. My procedure is as follows: Equation of motion (taking up as the positive direction): $$m\dot{v}=-mg-cv^2$$ Now to find ##v_\mathrm{ter}##, the terminal velocity, we consider...