What is the Solution for a Particle Moving Under a Repulsive Central Force?

In summary: I'm sorry, I misunderstood. I was using v_0 at minimum r because that's when U = 0. I should have used mv_0r
  • #1
gcfve
13
0

Homework Statement



1. Homework Statement
A particle of mass m moves under action of a repulsive central force Fr=Cr-3 with constant C greater than 0. At a very large distance from the centre of the force, the partcle has kinetic energy K and its impact parameter is b. Use conservation of energy and angular momentum to show that the closest m comes to the centre of force is rmin=[tex]\sqrt{b^2+ C/2K}[/tex]



2. Homework Equations
L=mbv0
E=.5mvr2+L2/2mr2+V(r)



3. The Attempt at a Solution
So far, I said that since the object is far away its total energy is K
I think V(r) = -3C/r2, but I'm not sure.
And at rmin, E==.5mvrmin2+(mbv)2/2mr2-3C/rmin2

I don't know what to do next.
 
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  • #2
gcfve said:

Homework Statement



1. Homework Statement
A particle of mass m moves under action of a repulsive central force Fr=Cr-3 with constant C greater than 0. At a very large distance from the centre of the force, the partcle has kinetic energy K and its impact parameter is b. Use conservation of energy and angular momentum to show that the closest m comes to the centre of force is rmin=[tex]\sqrt{b^2+ C/2K}[/tex]
2. Homework Equations
L=mbv0
E=.5mvr2+L2/2mr2+V(r)
3. The Attempt at a Solution
So far, I said that since the object is far away its total energy is K
I think V(r) = -3C/r2, but I'm not sure.
And at rmin, E==.5mvrmin2+(mbv)2/2mr2-3C/rmin2

I don't know what to do next.
Your nomenclature is confusing. Do you mean: [itex]F(r) = Cr^{-3}[/itex] ??

AM
 
  • #3
Yeah, that's what i meant
I didnt realize it was like that when i copied it
 
  • #4
gcfve said:

Homework Statement



1. Homework Statement
A particle of mass m moves under action of a repulsive central force Fr=Cr-3 with constant C greater than 0. At a very large distance from the centre of the force, the partcle has kinetic energy K and its impact parameter is b. Use conservation of energy and angular momentum to show that the closest m comes to the centre of force is rmin=[tex]\sqrt{b^2+ C/2K}[/tex]
2. Homework Equations
L=mbv0
E=.5mvr2+L2/2mr2+V(r)
3. The Attempt at a Solution
So far, I said that since the object is far away its total energy is K
I think V(r) = -3C/r2, but I'm not sure.
And at rmin, E==.5mvrmin2+(mbv)2/2mr2-3C/rmin2

I don't know what to do next.
Conservation of angular momentum:

[tex]\vec L = m \vec v \times \vec r[/tex]

so:

(1) [tex]|L| = mv_0rsin\theta = mv_0r(b/r) = mv_0b[/tex]

and:

(2) [tex]U(r) + \frac{1}{2}mv^2 = constant = K[/tex]

So far, you have this correct. [I will use U for potential energy since it is confusing to use v for speed and potential.]

Since F = -dU/dr, -U is the antiderivative of F

[itex]F = Cr^{-3}[/itex] so [itex]U = \frac{1}{2}Cr^{-2}[/itex]Therefore, from (2)

(3) [tex]\frac{1}{2}Cr^{-2} + \frac{1}{2}mv^2 = K[/tex]

Next, you have to find v at minimum r. At that point, what is L (hint: what the angle between v and r?)? You should be able to express v at minimum r in terms of v_0, r and b and then K, r and b. Substitute that into (3) and you should get your answer.AM
 
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  • #5
ok so the angle between r and v at rmin should be 90?
so,
[tex]|L| = mv_0r = mv_0b[/tex]
sp rmin=bv/v0?
 
  • #6
gcfve said:
ok so the angle between r and v at rmin should be 90?
so,
[tex]|L| = mv_0r = mv_0b[/tex]
sp rmin=bv/v0?
The angle is 90. But your expression for L is not correct. Why are you using v_0 at minimum r? [itex]L \ne mv_0r[/itex]. v_0 is the speed at the beginning when U = 0.

AM
 

FAQ: What is the Solution for a Particle Moving Under a Repulsive Central Force?

What is a repulsive central force?

A repulsive central force is a type of force that pushes objects away from a central point. It is an inverse square law force, meaning that as the distance between two objects increases, the force decreases by the square of that distance.

What are some examples of repulsive central forces?

Examples of repulsive central forces include electrostatic repulsion, which occurs between two objects with the same charge, and nuclear repulsion, which occurs between positively charged atomic nuclei.

How is a repulsive central force different from an attractive central force?

While a repulsive central force pushes objects away from a central point, an attractive central force pulls objects towards that point. An example of an attractive central force is the gravitational force.

How is a repulsive central force related to Newton's third law of motion?

Newton's third law states that for every action, there is an equal and opposite reaction. This means that for every repulsive central force acting on an object, there is an equal and opposite force acting on the other object involved in the interaction.

What are some real-world applications of repulsive central forces?

Repulsive central forces play a crucial role in many natural phenomena, such as the repulsion between atoms in a solid object, and the repulsion between positively charged particles in a plasma. They also have practical applications, such as in magnetic levitation and particle accelerators.

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