{\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.
When we throw a ball in a projectile motion, the ball follows a parabolic path due to gravity. And we see that earth moves in an elliptical path around the sun due the same force of gravity. So why two paths are different due to the same force?
Explain using the idea of central force
Goldstein 2nd ed.
In its Appendix is given the derivation of Bertrands Theorem.Here ##x=u-u_0## is the deviation from circularity and ##J(u)=-\frac{m}{l^{2}} \frac{d}{d u} V\left(\frac{1}{u}\right)=-\frac{m}{l^{2} u^{2}} f\left(\frac{1}{u}\right)##
If the R.H.S of A-10 was zero, the solution...
(Goldstein 3rd edition pg 72)
After reducing two body problem to one body problem
>We now restrict ourselves to conservative central forces, where the potential is ##V(r)## function of ##r## only, so that the force is always along ##\mathbf{r}##. By the results of the preceding section, I've...
The total energy of the particle is ##u^2 / 2 - k/R##. When ##u^2 \gg 2k/R##, we take the total energy to be ##u^2/2## only. By the conservation of energy, we have:
$$
\frac{u^2}{2} = \frac{w^2}{2} - \frac{k}{p}
$$
Take the angular momentum expression ##l = bu##, we can replace ##u## with...
A body on a circular orbit in the field of a central force (satellites in gravity field of Earth; a charge in a magnetic field) is subjected to a force which is always perpendicular to its initial velocity v, hence in a time period dt it acquires an additional velocity dv, which is also...
In this chapter, the stability of an object orbiting in a circular orbit of radius r_c in an arbitrary force field f is considered.
The author arrives at the equation of a harmonic oscillator, for small deviations x from the circular orbit:
\ddot{x} + \left[-3\frac{f(r_c)}{r_c} -...
I calculated the potential energy of the particle as follows :
But I am not sure how to calculate the kinetic energy. I know that if it was a satellite orbiting a Earth, I could use ##\frac {GMm} {r^2} = \frac {mv^2} {r}## to calculate the velocity v and they I could calculate kinetic energy...
I want to focus this question on understanding the force ##F(r)## I get (thus, I want to focus on c) ). However, below the dashed line, I included steps on how I derived ##F(r)##.
We are going to work in polar coordinates.
Knowing that the acceleration is:
$$a = \Big( \ddot r - r \dot...
The thought of increasing a satellite's (for example) speed to allow it to transfer from a "higher energy" elliptical orbit to a "lower energy" circular orbit (in reference to the effective potential energy plot that arises after introducing the concept of an effective mass to simplify the...
Hello friends and fellow Physics enthusiasts
At the beginning let me confess that I am more of a sleeping member though I do follow this forum quite closely. Today, I am posting to seek some help from fellow Physics educators. I hope this is the right section of the forum to post, if not, I...
In Chapter 8: Central-Force Motion, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, page 323, Problem 8-5, we are asked to show that the two particles will collide after a time ##\tau/4√2##.
I don't have any problems with the derivations and with...
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...
Homework Statement
Homework Equations
Conservation of energy/ angular momentum
The Attempt at a Solution
I used conservation of momentum to do part d. My answer is VB/VA=sinα /X. i don't know how to do part e. What is the criteria to determine whether the satellite will return to the...
Homework Statement
A model yacht runs on a horizontal frictionless oval track as shown (viewed from above) in the figure. The curved parts of the track are semi-circles of radius ##R = 0.5 m##; the straight sides have length ##L = 1 m##. The mass of the yacht is ##m = 0.5 kg.##
A force of...
Hi There!,
I'm here just to know an answer to a question that is bothering me for a while now. We know that motion of a body under the influence of central force with given center [say A] is planar . I was thinking whether this is possible even when body is allowed to move under the influence of...
Homework Statement
a particle moves in a circle under the influence of an inverse cube law force. Show that the particle can also move with uniform radial velocity,either in or out.
Find theta as a function of r for motion with uniform radial velocity.[/B]Homework Equations
f=-2A/r^3[/B]...
Homework Statement
I just need a hint. So we are given:
F = -kr
We are asked:
Show that:
(a) The orbit is an ellipse with the force center at the center of the ellipse.
Homework Equations
I guess we break it up into its components:
The Attempt at a Solution
m d2x/dt2 = -kx => x...
Homework Statement
From the homework:
In General Relativity it is found that the radial equation of an object orbiting a non-rotating black hole has the form $$\dot r^2 + (1 - 2 \frac {V_o} {r} ) (\frac {l^2} {r^2} + 1) = E^2$$ where ##r## is the radial coordinate, ##l## is the angular...
Hi, I'm studying undergrad mechanics, Central force motion, Marion's book in specific,
Here, the Potential Energy is defined weird way (in my opinion though)
(μ is reduced mass)
So potential Energy becomes
Called "Effective Potential Energy"
But, I can't agree with calling it potential just...
1. The derivation
In a 3-dim space,a particle is acted by a central force(the center of the force fixed in the origin) .we now take the motion entirely in the xy-plane and write the equations of the motion in polar coordinate
how can i derive from these equation that
T(kinetic...
Consider a central force. The central force is radial by definition, so ##\vec{F}=f(r) \hat{r}##. Therefore, by definition, the acceleration caused by the force, in the direction of ##\hat{\theta}## must be zero, ##\vec{a_{\theta}}=0##.
In presence of central force angular momentum is...
Homework Statement
A force field in 2-d F~ = −kr(rˆ) with U(r) = k(r^2)/2 acts on a particle of mass m.
The particle is now in a non-circular orbit. In terms of the particle’s angular momentum L and energy E,
d) What is its closest approach to the origin? e) What is its furthest distance from...
Homework Statement
A particle of mass m moves under the influence of a central force
F(r)=-mk[(3/r^2)-2a/r^3]rhat
Show that if the particle is moving in a circular orbit of radius a, then its angular momentum is L=mh=m√(ka)
Homework Equations
L=mvr for circular orbit
The Attempt at a...
Question: A body of mass m is moving in a repulsive inverse cubed force given by
F = K/r^3 where K > 0
show the path r(θ) of the body given by
1/r = A cos[β(θ-θο)]
Find the values of constants A and β in terms of E, L.
Work done so far:
we did a similar problem with inverse square so I am...
Homework Statement
a satellite is in a circular orbit a distance $h$ above the surface of the Earth with speed $v_0$, booster rockets are fired which double the speed of the satellite without changing the direction. Find the subsequent orbit.
Homework Equations
The Attempt at a Solution
Before...
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...
On page 108 in Goldstein 3rd edition in the paragraph after equation (3.94) he says that ##\psi##` can be obtained from the orbit equation (3.36) using the limits as ##r_0=\infty## ##r=r_m## which the distance of closest approach and ##\theta_0=\pi## which is the initial direction.
So looking...
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.
Hi
we are going through cirkular centralforce and I am complete stuck...
I can't find the derivation to why w1/w2=√r2/r1 is correctw1/w2=√r2/r1sorry I am lost...
best regards
Fred
Good afternoon!
Exercise: Prove that if a body moves under the action of a force F = k⋅u × v, where u is an arbitrary unit vector and v the velocity, the motion is circular with angular velocity ω = k ⋅ u or, in a more general case, a spiral parallel to u.
Source: (Alonso & Finn: Fundamental...
Homework Statement
An object from space (like an asteroid) approaches Earth. A collision will occur if the scattering cross-section is less than π*Re2. If the distance of closest approach is much greater than Re, no collision would occur. Find an implicit expression for the cross-section in...
Homework Statement
Hey, sorry in advance if something I write is unclear but I am not native English speaker.
I have a central force that is defined as F=-f(r) \frac{\vec{r}}{r} where f(r) is some function of distance and \vec{r}=(x,y,z). I have to calculate potential energy when...
I guess this question can apply in all the generality of the 3D Schrodinger eqn. with a central force, the case I'm thinking of however is the the hydrogen atom.
When solving the equation, we derive the quantization of the angular momentum, which has me thinking that before we begin quantizing...
Homework Statement
A particle of mass m moves under an attractive central force of Kr^4 with an angular momentum L. For what energy will the motion be circular? Find the frequency of the radial oscillations if the particle is given a small radial impulse.
Homework Equations...
Homework Statement
I'm having trouble understanding these notes. Can anyone help me understand how Equation 1 is arrived at, and then how Equation 2 is arrived at?
A central force field \underline{F}(\underline{r}) is a field of forces, which are directed from a mass m inwards or...
In a central force problem,
angular momentum is conserved.
we quantized one of the component of L, say Lz.
Also, we quantized the angular momentum, L = √l(l+1)h_bar
If we know Lx and Ly without uncertainty,
then we know the direction of L.
Hence we know the motion of the particle is confined...
Homework Statement
An object of mass 'm' if revolving in a circular path of radius 'R', this is analogous to a gravitational motion except that the force is applied from a point on the circle itself, it is required to find the force law
Homework Equations
from the point of application...
Homework Statement
An object of mass M moves under the influence of an attractive central force F = - A r^4 \hat{r} where \hat{r} is a unit vector in the radial direction.
If the object is in a circular orbit of radius R , find its speed v as a function of M,A, and,R.
Homework...
Could someone please explain how I work with $-mf(r)e_r$ in this question. Usually we get given an equation (like the one for f(r)) and have to work out the orbit by getting a differential equation etc. I'm not too sure how to work it this way around.
"A particle of mass m moves under the...
Homework Statement
To whom it may concern,
I am trying to understand the central force problem of the Dirac equation. In particular, I am following Sakurai's Advanced Quantum Mechanics book. There (section 3.8, p.122), it is shown that there is an operator
K = \beta(\Sigma . L +...
Hi, I'm new to this forum and I've got a question.
In all articles I've found about the two body problem, they first start off by writing the distances between the masses as one function r(t) and one function R(t) (which is just simple the position of the reference frame).
*note: i use initial...
Homework Statement
Consider the central force F=-k/(r^3) u(hat in r direction). This is an attractive force. For various values of k, explain what the orbit of a particle about a force center looks like.
Homework Equations
Included within question.
The Attempt at a Solution
I am not quite...
Hi everyone,
I would like to know if the energy of each body of a two body gravitationnal problem is separately conserved. I know that the individual angular momentum are separately conserved and that the TOTAL energy of the two bodies is conserved. However, I don't know if there could be...
Hi everybody
I have been trying to develop an intuitive general understanding of how the trajectory of an object , that is moving at constant velocity, is changed by the sudden appearance of a central force but I am not sure my reasoning is correct.
(1) Consider an object "A" not subject...
There are many good treatments of the classical central force problem in many undergraduate and graduate textbooks. But I was unable to find a similar treatment of the retarded central force problem. I am looking for the classical treatment of the potentials of type:
\delta(t'-t +...
1. Homework Statement
A particle moves under the influence of a central force, F_r(r) = − α r^−3. At time t = 0, the radius = r0 and the velocity = v0 as shown in the figure.
(A) Determine the potential energy function U(r).
(B) Determine the velocity vc such that the particle will move...
Homework Statement
A particle moves under the influence of a central force, F_r(r) = − α r^−3. At time t = 0, the radius = r0 and the velocity = v0 as shown in the figure.
(A) Determine the potential energy function U(r).
(B) Determine the velocity vc such that the particle will move on...
Homework Statement
A comet is released from rest at a distance r_max from the sun.
Therefore it has no angular momentum.
We can assume the sun is a point mass ( i.e. sun's radius is zero ).
How long does it take for the comet to hit the sun?
Let m = comet mass
let M_s = sun's mass...
Homework Statement
Suppose g:(0, +∞) → ℝ is continuous, and consider F:ℝd\{0} → ℝd, where F(x) = xg(|x|). Prove F is conservative.
Homework Equations
F is conservative iff there exists a C1 function f:ℝd\{0} → ℝd, s.t. F = grad(f). (edit: Or is the codomain of f actually ℝ, so that it's a...