{\displaystyle \scriptstyle {\vec {\text{ F }}}}
is the force, F is a vector valued force function, F is a scalar valued force function, r is the position vector, ||r|| is its length, and
r
^
{\displaystyle \scriptstyle {\hat {\mathbf {r} }}}
= r/||r|| is the corresponding unit vector.
Not all central force fields are conservative or spherically symmetric. However, a central force is conservative if and only if it is spherically symmetric or rotationally invariant.
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}##:
$$T= \frac...
I've already found the potential and force that produce the given orbit. my results were:
##V=-\frac{al^2}{mr^3}##
##\vec{F}=-\frac{-3al^2}{mr^4}\hat{r}##
Now, I've been trying to find the period using the equation
##t=\sqrt{\frac{m}{2}}\int_{r_0}^{r}\frac{dr'}{\sqrt{E-V_{eff}}}##
Using...
I know that the force must be a central force and that under central forces, angular momentum is conserved. But I am unable to mathematically show if the angular and linear momentum are constants.
Radial Momentum
##p=m\dot r = ma\dot \theta=ma\omega##
Angular Momentum
##L=mr^2\dot\theta =...
Here were my assumptions: Energy and angular momentum are both conserved because the only force acting here is a central force. The initial angular momentum of this particle is ##L = mv_0b## and we can treat E as a constant in the homework equation given above. I solved for the KE (1/2 mv^2) in...
This is not a homework. In Chapter 8: Central-Force Motion, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, page 325, Problem 8-19, we are asked to calculate the mass of the planet Saturn. In the instructor's solution manual, the solution for this...
In Mathematical Methods for Physicists, 6th Edition, page 44, Example 1.8.2, the curl of the central force field is zero.
1. Why are central force fields irrotational?
2. Why are central force fields conservative?
Any help is much appreciated...
A particle of mass m in xy plane is attracted toward the origin with the force
$$\begin{align}\vec{f} = - \frac{k^{2} m}{r^{6}}\vec{r}\end{align}$$ where ##\vec r## is position vector of particle measured from origin. If it starts at position ##(a,0)## with speed $$v=\frac{k}{\sqrt{2} a^{2}}$$...
My textbook says that for a central force at the origin, the angular momentum is constant, because the derivative rxF is zero since F points radially outwards so it is in the same direction as r. Ok, but what about the angular momentum about a point other than the origin, or the angular momentum...
Homework Statement
A particle is been acted by a central force exerted by a sphere of mass 'M' at a distance 'xo' initially from the particle (of same mass). At t=0 the particle has velocity Vt perpendicular to the line joining the particle and the center of the sphere. Find the time at which...
Homework Statement
The problem is related with central forces. In the problem I am given the equation of the orbit of a particle subjected to a central force (with an angular momentum "l"), r=a/(phi+1)^2 (where "r" is de distance to the center of forces and "a" a constant). I am asked for the...
Hi,
This is the statement of the problem of AP French's textbook "Newtonian Mechanics".
1. Homework Statement
The commander of a spaceship that has shut down its engines and is
coasting near a strange-appearing gas cloud notes that the ship is
following a circular path that will lead...
I am self studying Kleppner and Kolenkow's an Introduction to mechanics. But i have one doubt about how they got into the equation no 3 of the example problem 9.3 in Central Force Motion.
Please clarify my doubt.
Homework Statement
Consider Newton’s force law for two particles interact through a central force F12(r1',r2',u1,u2), where by Newton’s third law F12 = -F21.
m1(d^2r1/dt^2) = F12(r1,r2,u1,u2)
m2(d^2r2/dt^2) = F21(r1,r2,u1,u2)
A. Show that Newtonian mechanics is form invariant with respect...
PROBLEM:
Show that Special Relativity predicts a precession of π(GMm/cl)2 radians per orbit for any elliptic orbit under a pure inverse-square force.
where G is gravitational constant, M is mass of larger body, m is mass of smaller orbiting body, c is speed of light and l is angular momentum...
Homework Statement
A particle of mass ##m## is placed on a smooth table and attached to a fixed point ##O## on the table by a spring with spring constant ##k## and natural length ##l##.
(i) Show that the particle can execute circular motion about ##O## with angular velocity ##\omega## provided...
I understand that in a two body problem under central force, corresponding to a potential V(r)(assume one body is massive compared to the other so that its motion is negligible), conservation of angular momentum implies the motion of the body to be in a plane spanned by position r and momentum p...
Homework Statement
http://i50.tinypic.com/mmws94.png
Homework Equations
The Attempt at a Solution
1. Angular momentum is constant; J = constant
2. Total energy is constant; E = constant
Not sure why I am ending up with a 2gr instead..as U = ∫ f dr = ∫ mg dr
Homework Statement
1) A satellite having a mass of 5000 kg describes a circular path around the Earth of radius
8000 km. Find (a) its angular momentum and (b) its potential, (c) kinetic and (d) total
energy. Also calculate (e) the orbital period.
2) Assume the satellite in the previous...
Imagine a mass in an elliptical orbit around a central force. If such force is inversely proportional to the distance squared then the mass will accelerate when going nearer the focus. At the same time the angular momentum of the orbit is a constant.
Can one show that the work done by the force...
Homework Statement
Assuming Earth to be a solid sphere, show that if a straight hole were drilled from pole to pole that a particel dropped would undergo simple harmonic motion. Show that the period of the oscillation depends only on the Earth's density and is independent of size. What is...
hi,can anyone solve this two problems??
these are from the "textbook on spherical astronomy" written by W.smart
chapter five,problem number 18 and 19,Euler's theorem and Lambert's theorem
thank you ;-)
Homework Statement
Two particles of mass m are connected by a light inextensible string of length l. One of the particles moves on a smooth horizontal table in which there is a small hole. The string passes through the hole so that the second particle hangs vertically below the hole. Use the...
I'm getting two different radial equations depending on when I plug in the angular momentum piece. Here's the Lagrangian:
L = \frac{1}{2} \mu (\dot{r}^2 + r^2 \dot{\phi}^2) - U(r)
The Euler-Lagrange equation for phi gives angular momentum (conserved), which can be solved for \dot{\phi}...
Homework Statement
A particle of mass m moves under the influence of a central force given by F(r) = -k/r^n. If the particle's orbit is circular and passes through the force center r = 0. Show n = 5. Find the radius of the circular orbit.
Homework Equations
The Attempt at a...
A point mass ‘A’ is kept at the origin. Another B is kept at the x-axis at x = H. Another C is kept at distance H from origin and distance h from B. A, B and C thus form an isosceles triangle with vertex A. Given that A's mass >> B's mass >> C's mass; Newton’s gravitational law governs the...
Hi, I've spent quite a while trying to figure this out but can't quite seem to get there... any ideas?
1. The Question
Two particles of mass m are connected by a light inextensible string of length l. One of the particles moves on a smooth horizontal table in which there is a small hole. The...
Hiya. I'm revising for a classical mechanics first year exam, and the lecture notes on central forces are less then ideal. I'm looking for a fairly simple treatment of the subject with especial regard to circular motion, bound and unbound orbits, binary systems, kepler's laws, distance of...
Hello
Im currently working my way back through a few past papers to ready myself for the new physics year, and was wondering if anybody could help me with a few puzzling questions that have me and my lecture notes stumped!
I have created a image file with my questions and how far I have got...
central forces-please chek answer!
hello...
i have this question in my classical mechanics book and was wondering if some one could help me?
the question says that there is a particle of mass "m" that is moving under an attractive central force F=k/r^5 where k<0
show that the particles...