Central Force Problem: Nature of Orbits for F=-ar/(r^3) & U=-(a/r)

In summary, the conversation discusses the central force and potential energy in a system and the nature of orbits when the central force is either positive or negative. It is determined that in both cases, the orbit is parabolic, with a positive energy resulting in an elliptic orbit and a negative energy resulting in a hyperbolic trajectory.
  • #1
neelakash
511
1

Homework Statement



Central force F=-ar/(r^3) & Central potential energy,U=-(a/r)
(not U_eff)
Find the nature of orbits if (i)a>0 and (ii)a<0

Homework Equations


The Attempt at a Solution



If we remember the attractive central force E=E(r) diagram,i.e.the one showing the graph of U_eff,we only need to know E_total=K+U.
Where only PE is given.
We see,

U= -integration[F.dr]=integration[dW]=-integration[dK]=-K

Then K=a/r and U=-(a/r)

So,the E=K+U=0

Then in positive and negative both caes we get a parabolic orbit.

Please check if i am correct.
 
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  • #2
Consider a>0.
Then, -(m(v^2)/r)=-(a/r^2)
or,m(v^2)=(a/r)
Hence total energy E=K+U= -(1/2)(a/r)

=>elliptic orbit.

Consider a<0.

Then,F=+b/r where b=-a>0
and U_p=b/r

K is itrinsically positive.So,total energy positive.

From e=sqrt[1+{(2L^2*E)/(m*b^2)}]
e>1.
hence hyperbolic trajectory.
 
  • #3


Your response seems to be mostly correct. However, to fully understand the nature of orbits in this central force problem, we need to consider the energy equation, which is given by E = 1/2mv^2 + U, where v is the velocity of the object and m is its mass.

In the case of a>0, the central potential energy, U, is negative and the kinetic energy, 1/2mv^2, is positive. This means that the total energy, E, is negative. In this scenario, the object will have a bound orbit, which means it will continue to revolve around the central force, never escaping its influence.

In the case of a<0, both U and K are negative, which means that the total energy, E, will also be negative. However, in this case, the object will have an unbound orbit, which means it will escape the central force and continue moving in a hyperbolic or parabolic trajectory.

Therefore, the nature of orbits for this central force problem depends on the sign of the central force, a. If a>0, the orbit will be bound, and if a<0, the orbit will be unbound.
 

1. What is the Central Force problem?

The Central Force problem is a physics problem that involves a particle moving under the influence of a central force, which always acts towards or away from a fixed point called the center. This problem is commonly studied in the field of classical mechanics.

2. What are some examples of central forces?

Some examples of central forces include gravity, electric and magnetic forces, and forces in planetary motion such as the Sun's gravitational force on a planet.

3. How is the Central Force problem solved?

The Central Force problem is solved using Newton's Second Law of Motion, which states that the sum of all forces acting on a particle equals its mass times its acceleration. By setting up the equations of motion and solving for the particle's position and velocity, the problem can be solved.

4. What are some real-world applications of the Central Force problem?

The Central Force problem has many real-world applications, including the motion of planets and satellites in space, the motion of electrons around an atomic nucleus, and the motion of charged particles in a magnetic field.

5. What is the significance of the Central Force problem in physics?

The Central Force problem is significant in physics because it allows us to understand the motion of objects under the influence of a central force, which is a fundamental concept in classical mechanics. It also has many practical applications in fields such as astronomy, engineering, and particle physics.

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