# What is Central force: Definition and 95 Discussions

In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force.

F

=

F

(

r

)
=

|

F
(

r

)

|

r

^

{\displaystyle {\vec {F}}=\mathbf {F} (\mathbf {r} )=\left\vert F(\mathbf {r} )\right\vert {\hat {\mathbf {r} }}}
where

F

{\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.

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1. ### The Difference in Paths: Exploring Central Force & Gravity

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
2. ### A Doubt in a step while deriving Bertrand theorem

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3. ### A Cyclic coordinates in a two body central force problem

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4. ### Central force field derivation

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5. ### Constant velocity in a central force field

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6. ### I Stability of circular orbits in an arbitrary central force field

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7. ### Torque and angular momentum with a central force

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8. ### Energy of particle when under a central force

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20. ### Effective Potential Energy from Central Force Motion

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21. ### Proof -- motion under a central force in text Symon Mechanics

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22. ### Central force and acceleration in the polar direction

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23. ### Find the min/max r of particle with a central force

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24. ### Angular momentum and central force

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...
25. ### Solve Inverse Cubed Force: Find A & B in r(θ) for E & L

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...
26. ### Central Force problems using radial motion equation

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...
27. ### Finding out the type of central force given the orbit

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...
28. ### Goldstein Central Force Repulsive Scattering

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...
29. ### Perturbed circular orbit under central force motion?

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.
30. ### Is w1/w2 = √r2/r1 the Correct Derivation for Circular Central Force?

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
31. ### Proving F is a Central Force: Alonso & Finn's Physics

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...
32. ### Scattering Cross-Section from a Central Force

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...
33. ### Proving that central force is conservative and getting the energy

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...
34. ### Angular momentum in the 3D Schrodinger eqn with a central force

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...
35. ### Classical Mechanics: Central Force Problem

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...
36. ### Potential for a Central Force Field

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...
37. ### Conservation of angular momentum (central force)

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...
38. ### Central Force with application at circle

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...
39. ### Finding Speed of Object Under Central Force: Equations & Solution

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...
40. ### MHB Solving for Central Force in $r = c\theta^2$ Orbit

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...
41. ### K^2 = J^2 + 1/4 for the central force problem of the Dirac equation

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 +...
42. ### Central force to 2-body problem: which values?

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...
43. ### Explaining Central Force of inverse r^3

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...
44. ### Is the energy conserved FOR EACH BODY in a two-body central force problem?

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...
45. ### Motion of object under central force

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...
46. ### Retarded Central Force Problem in Classical Physics

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 +...
47. ### Circular motion from a central force

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...
48. ### Particle under central force in circular motion

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...
49. ### 2-body central force ( gravity ) problem

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...
50. ### Prove a rotationally symmetric central force field is conservative

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...