What is Classical mechanics: Definition and 1000 Discussions
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mechanics, if the present state is known, it is possible to predict how it will move in the future (determinism), and how it has moved in the past (reversibility).
The earliest development of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts based on foundational works of Sir Isaac Newton, and the mathematical methods invented by Gottfried Wilhelm Leibniz, Joseph-Louis Lagrange, Leonhard Euler, and other contemporaries, in the 17th century to describe the motion of bodies under the influence of a system of forces. Later, more abstract methods were developed, leading to the reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. These advances, made predominantly in the 18th and 19th centuries, extend substantially beyond earlier works, particularly through their use of analytical mechanics. They are, with some modification, also used in all areas of modern physics.
Classical mechanics provides extremely accurate results when studying large objects that are not extremely massive and speeds not approaching the speed of light. When the objects being examined have about the size of an atom diameter, it becomes necessary to introduce the other major sub-field of mechanics: quantum mechanics. To describe velocities that are not small compared to the speed of light, special relativity is needed. In cases where objects become extremely massive, general relativity becomes applicable. However, a number of modern sources do include relativistic mechanics in classical physics, which in their view represents classical mechanics in its most developed and accurate form.
I am reading Chapter 9 of Classical Mech by Goldstein.The symplectic condition for a transformation to be canonical is given as MJM' = J, where M' is transpose of M. I understood the derivation given in the book. But my question is : isn't this condition true for any matrix M? That is it doesn't...
Hello everyone. So I wanted to get some opinions on what some of you thought was a better choice, as far taking PDE's or classical mechanics 2 goes. First let me start off by giving a little info; I've already taken calc 1-3 and ordinary differential equations, physics 1 & 2...
Homework Statement
The Observer (me) is at Inertial Frame of Reference:
1)
I am sitting at a car moving east v=30 km/h relative to earth
A bird flying east v = 10 km/h relative to earth
What is the speed of bird for me?
2)
I am sitting at a car moving east v=30 km/h relative to earth
A bird...
Homework Statement
A spherical satellite of radius r is moving with velocity v through a uniform tenuous atmosphere of density ρ. Find the retarding force on the satellite if each particle which strikes it (a) adheres to the surface and (b) bounces off it elastically.
I know the answer should...
Here's what my prof says:
"Define F_{mean} to be the mean force, F_close to be the force on the side of the Earth closer to the moon, and F_far to be the force on the side of the Earth furthest away from the moon.
On the closer side the net force is F_close - F_mean > 0
On the further side the...
Homework Statement
I'm doing a question from a past paper, preparing for an upcoming exam. There is no solutions so I was wondering if my answer is correct for all parts.
Take a star to be a uniform sphere with mass M_{i}=3.0 \times 10^{30} Kg and radius R_{i} = 7.0 \times 10^{8}m that...
Homework Statement
Find the angle Theta so that the system is in equilibrium
Mass of each block: 2 Kg
Mass of ring: 4π Kg
Homework Equations
Static equilibrium:
Rotational equilibrium
The Attempt at a Solution
Static equilibrium:
2g + 2g + 4πg = Normal
N = 4g + 4πg
Taking...
I could've peeked at the solution manual and end with it at that. However, I'm trying to change this nasty habit and solve this problem myself. I might be too ambitious. Here goes.
Homework Statement
Consider the infinite Atwood's machine shown. A string passes over each pulley, with one end...
Homework Statement
(2) Suppose a particle of mass m is subjected to a repulsive force F = +kx.
(a) What is the general solution for the motion of the system?
(b) If the particle begins with a position x(0) = x0 and with velocity v(0) = v0 at t = 0 what are the values of the constants appearing...
This is not a homework question, I'm just curious.
Suppose you had a sailboat sailing on calm water.
Let's say the sailboat is capable of sailing at 10 knots under the current wind conditions using only its sails.
Now say the sailboat had a small propeller motor that was capable of propelling...
Homework Statement
From Fetter and Walecka 5.1:[/B]
Consider the compound pendulum in FIg 28.1 (mass M, moments of inertia Iij relative to the center of mass, which is a distance L from the point of support Q) but with Q attached to the bottom of a vertical spring (force constant k) and...
Homework Statement
Question :- A block of mass ##5 kg## is attached to a spring. The spring is stretched by ##10 cm## under the load of the block. A impulse is provided to the block such that it moves up with a velocity of ##2 m/s##. Find the height it will rise.
Homework Equations
##F_s =...
Homework Statement
Planet X of mass mx = 2.1 × 1024 kg orbits S in uniform circular motion at a distance rx and with a period Px = 2.1 years (=66225600 s). The mass of the star S is MS = 2 × 1031 kg and its radius is RS = 3.2 × 108m.
Homework Equations
T=2pi * sqrt(r3/(GM)
The Attempt at a...
Homework Statement
[/B]
Question :- Find the acceleration of block of mass ##M##. The coefficient of friction between blocks is ##\mu_1## and between block and ground is ##\mu_2##.
free body diagram at the end.
Variables :-
##f_1## - friction between blocks.
##f_2## - friction between block...
Homework Statement
how from this question using Energy principles I can find how frequency depends on g?
Homework EquationsThe Attempt at a Solution
I know that frequency is sqrt(k/m) this is a formula you use in introductory physics. but I am lost on how to do it from energy route or...
Question :-
A car has to move on a path, that is a arc of a circle of radius (##R##). The length of the path is (##L##). Suppose it starts on the highest point of the path, find the highest uniform speed for which, it does not lose contact with the path on any point ?
My attempt :-
I made a...
In my rough understanding Molecular Dynamics using Classical Newtonian mechanics is a 6N dimensional non linear system. 6N dimension because you have 3 position vectors and 3 momentum vectors for each N particles. Nonlinearity because of the terms in force fields. In principle this system can...
I attempt to solve the brachistochrone problem numerically. I am using a direct method which considers the curve ##y(x)## as a Lagrange polynomial evaluated at fixed nodes ##x_i##, and the time functional as a multivariate function of the ##y_i##. The classical statement of the problem requires...
Homework Statement
A solid sphere of density ##ρ## and radius ##R## is centered at the origin. It has a spherical cavity in it that is of radius ##R/4## and which is centered at ##(R/2, 0, 0)##, i.e. a small sphere of material has been removed from the large sphere. What is the the center of...
Homework Statement
Find the equation of motion of the object by setting the derivative of the total energy equal to zero.
Homework Equations
r(theta)=(Rcos(theta), Rsin(theta), q*theta)
v(theta)=dr/dt=(-Rsin(theta)dtheta/dt, Rcos(theta)dtheta/dt, q*dtheta/dt)
derivative of Total Energy =...
Homework Statement
Find the Lagrangian for the double pendulum system given below, where the length of the massless, frictionless and non-extendable wire attaching m_1 is l. m_2 is attached to m_1 through a massless spring of constant k and length r. The spring may only stretch in the m_1-m_2...
Homework Statement
A particle of mass m is subject to force F(x)=kx with K>0. The initial starting position is x0 and the initial speed is zero. Find x(t).
Homework Equations
F(x)=kx
F=ma
The Attempt at a Solution
Honestly, I am totally lost on this. I've written acceleration as v(dv/dx)...
A particle's position is given by $$r_i=r_i(q_1,q_2,...,q_n,t)$$ So velocity: $$v_i=\frac{dr_i}{dt} = \sum_k \frac{\partial r_i}{\partial q_k}\dot q_k + \frac{\partial r_i}{\partial t} $$
In my book it's given $$\frac{\partial v_i}{\partial \dot q_k} = \frac{\partial r_i}{\partial q_k}$$...
Homework Statement
Minimize the functional: ∫01 dx y'2⋅ ∫01 dx(y(x)+1) with y(0)=0, y(1)=aHomework Equations
(1) δI=∫ dx [∂f/∂y δy +∂f/∂y' δy']
(2) δy'=d/dx(δy)
(3) ∫ dx ∂f/∂y' δy' = δy ∂f/∂y' |01 - ∫ dx d/dx(∂f/∂y') δy
where the first term goes to zero since there is no variation at the...
Homework Statement
A uniform cylindrical drum of mass M and radius a is free to rotate about its axis, which i is horizontal. An elastic cable of negligible mass and length l is wrapped around the drum and carries on its free end a mass m. The cable has elastic potential energy \tfrac12...
Homework Statement
I have a block of mass m on a horizontal surface, which is covered in oil. The tell me the viscous resistance force is a function of the velocity, F(v)=-cv1/2, where I am assuming c is a constant of some kind. I need to find the time that it will take for the block to stop...
Homework Statement
Rockets are propelled by the momentum of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that for a Rocket starting...
I was thinking that what is solid reason behind the failure of Classical Mechanics in Black body radiation? Why classical mechanics not explain the curve at lower or higher range of frequncies.
Homework Statement
I have a block of mass M attached to a cord of mass m on an inclined plane of angle θ . A force F is pulling the cord & block up the plane. I have to find the force exerted on the block by the cord. The surface is frictionless
Homework Equations
F = ma
The Attempt at a...
Homework Statement
I have a block on an inclined plane (the angle is 40). The block is projected with an initial speed of 2m/s and μ=0.05. I need to find the time it takes the block to go up the inclined plane and return to the point it started out.
Homework EquationsThe Attempt at a Solution...
This book should introduce me to Lagrangian and Hamiltonian Mechanics and slowly teach me how to do problems. I know about Goldstein's Classical Mechanics, but don't know how do I approach the book.
Homework Statement
A man of mass 100 kg can pull on a rope with a maximum force equal to two fifths of his own weight. [Take g = 10 ms^2] In a competition, he must pull a block of mass 1600 kg across a smooth horizontal floor, the block being initially at rest. He is able to apply his maximum...
Note: All bold and underlined variables in this post are base vectors
I was reading the book 'Introduction To Mechanics' by Kleppner and Kolenkow and came across an example I don't quite understand. The example is this: a bead is moving along the spoke of a wheel at constant speed u m/s. The...
In literature I have read it is said that the Hamiltonian ##H## is the generator of time translations. Why is this the case? Where does this statement derive from?
Does it follow from the observation that, for a given function ##F(q,p)##, $$\frac{dF}{dt}=\lbrace F,H\rbrace +\frac{\partial...
I am not sure if this is the right place to ask it, but this is a question that I thought today, and it gave me some curiosity to understand. Imagine that a car will curve, we can say the turn is a bit tight , what are the factors that can help it to flip? I was wondering about some aspects...
Homework Statement
The three vectors A, B, and C point from the origin O to the three corners of a triangle. Show that the area of the triangle is given by 1/2|(BxC)+(CxA)+(AxB)|.
Homework EquationsThe Attempt at a Solution
I know that the magnitude of the cross product of any two vectors...
You have an infinitesimally small mass in the center of octahedron. Mass is connected with 6 different springs (k_1, k_2, ... k_6) to corners of octahedron.
Equilibrium position is in the center, you don't take into account gravity, only springs.
Find normal modes and frequencies.
Relevant...
Hello, I've got a problem and I have no idea how to start. I'll be happy for any hint. Thanks
Homework Statement
Two beads each of mass m are at the top (Z) of a frictionless hoop of mass M and radius R which lies in the vertical plane. The hoop is supported by a frictionless vertical support...
Homework Statement
Consider the pendulum depicted in the adjacent figure: a mass m
is attached to non stretching chord of length `. Directly below the
pendulum is a circular disc rotating with constant angular velocity
w. We attach to the disk a frame whose x-axis is in the plane of the...
Hello,
I will be enrolling in an undergraduate Classical Mechanics course and I was wondering if the book by Spivak "Physics for Mathematicians: Mechanics" would help me understand the concepts more in depth than usual.
Until the time that I will be taking the course, I will already have...
Homework Statement
A satellite moving in a highly elliptical orbit is given a retarded force concentrated at its perigee. This is modeled as an impulse I. By considering changes in energy and angular momentum, find the changes in a (semi major axis) and l (semi latus rectum). Show that \delta...
Homework Statement
Suppose the asteroid of [other problem] has a mass of 6 \times 10^{20} \textrm{kg} . Find the proportional change in the kinetic energy of the Earth in this encounter. What is the change in the semi-major axis of the Earth's orbit? By how much is its orbital period...
I'm a freshman in Computer Engineering at a university. I have a Classical Mechanics course that will cover these topics:
Newton's law of motion
Vector Algebra
Equilibrium of bodies
Plane Trusses
First moment of area, centroid, etc.
Calculation of virtual displacement and virtual work...
Homework Statement
Two identical charged particles, each of mass m and charge e, are intitialy far apart. One of the particles is at rest at the origin, and the other approaching it with velocity v along the line x=b, y=0 where b=e^2/2 \pi \epsilon_0 mv^2. Find the scattering angle in the CM...
Homework Statement
Assume that the pressure p in a star with spherical symmetry is related to the density \rho by the (distinctly unrealistic) equation of state p= \tfrac12 k\rho^2, where k is a constant. Use the fluid equilibrium equation obtained in Problem 23 to find a relation between \rho...
I really want to know whether this equation is nonholonomic or not.
(As far as I know, Nonholonomic constraint has a term of velocity and do non-integrable. But this formula does not dependent on a path, because it is a total differential form.)