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

AI Thread Summary
A particle of mass m is influenced by a repulsive central force Fr=Cr^-3, and the discussion focuses on determining the closest approach to the center of the force, rmin, using conservation of energy and angular momentum. The potential energy is identified as U(r) = 1/2 Cr^-2, and the total energy at a large distance is K. The conservation of angular momentum is applied, leading to the relationship between the impact parameter b and the velocity at the closest approach. The angle between the velocity and the radius vector at rmin is confirmed to be 90 degrees, which is crucial for deriving the expression for rmin. The discussion emphasizes the need for careful application of these principles to arrive at the final result.
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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=\sqrt{b^2+ C/2K}



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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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=\sqrt{b^2+ C/2K}
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: F(r) = Cr^{-3} ??

AM
 
Yeah, that's what i meant
I didnt realize it was like that when i copied it
 
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=\sqrt{b^2+ C/2K}
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:

\vec L = m \vec v \times \vec r

so:

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

and:

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

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

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

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

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
 
Last edited:
ok so the angle between r and v at rmin should be 90?
so,
|L| = mv_0r = mv_0b
sp rmin=bv/v0?
 
gcfve said:
ok so the angle between r and v at rmin should be 90?
so,
|L| = mv_0r = mv_0b
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? L \ne mv_0r. v_0 is the speed at the beginning when U = 0.

AM
 
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