I think it's a conservation of energy problem

In summary, an alpha particle with a kinetic energy of 10.5 MeV collides with a lead nucleus at an angle and with an initial nonzero angular momentum of magnitude L=p_0 b. The lead nucleus is stationary and can be treated as a point charge, with an atomic number of 82. The alpha particle, a helium nucleus with atomic number 2, causes the path to curve but at closest approach, it will be moving perpendicular to the line joining the two particles. By using the conservation of momentum and energy equations, the closest distance between the two particles can be found by setting the initial potential energy to 0 and solving for Rmin. The distance b is essential in the angular momentum equation but can be ignored in
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Homework Statement

An alpha particle with kinetic energy 10.5 MeV makes a collision with lead nucleus, but it is not "aimed" at the center of the lead nucleus, and has an initial nonzero angular momentum (with respect to the stationary lead nucleus) of magnitude L = p_0 b, where p_0 is the magnitude of the initial momentum of the alpha particle and b=1.00×10−12 m. (Assume that the lead nucleus remains stationary and that it may be treated as a point charge. The atomic number of lead is 82. The alpha particle is a helium nucleus, with atomic number 2.)

a) find the distance of closest approach

Homework Equations



conservation of momentum and energy, I guess; formula for electric potential energy

The Attempt at a Solution

[Edit] Scroll to last post to see my attempt

I definitely need some hints on this one. I don't think we've said anything about angular moment of particles during the physics course I'm in. I did a google search to refresh my memory on angular moment, and I recall that it's basically rotation.

Well, in this problem, I can't figure what's rotating around what. Or how this brings on a collision. I appreciate any help. However, I have solved a problem kind of like this that involved conservation of linear momentum and energy to find the closest distance two particles of the same charge got before jutting off in the opposite directions due to their opposing force on one another. But how can I apply the concept of angular momentum?
 
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  • #2
Seraph404 said:
Well, in this problem, I can't figure what's rotating around what.

Hi Seraph404! :smile:

Nothing's rotating …

any moving object has angular momentum about any point not on its line of travel …

and its angular momentum is conserved …

angular momentum = speed times perpendicular distance. :wink:
 
  • #3
Wait... does the alpha particle cause the lead nucleus to rotate?
 
  • #4
Seraph404 said:
Wait... does the alpha particle cause the lead nucleus to rotate?

You can assume not …
Seraph404 said:
Assume that the lead nucleus remains stationary and that it may be treated as a point charge.
 
  • #5
So then it's just approaching at an angle then? That's the only difference between this and what I described?
 
  • #6
They're both charged, so the path will curve, but at closest approach, it will be moving perpendicular to the line joining them, which is all you need to know about the path.
 
  • #7
So far is this the correct setup?

L1 = L2
P0b = p(Rmin) [Rmin is the minimum distance the two particles get to each other]
v0b=vf(Rmin) [mass doesn't change so I canceled it]

K0+U0 = Kf+Uf

K0 = .5mV0^2 = 1 mil(1.602E-19) J [ I can use this to find initial velocity, assuming that I can use atomic mass given from the periodic table and divide it by avogadro's number to find the mass of a single atom (one source on the Internet said I could ignore the mass of the electrons - that all the mass is really contained in the nucleus)]
U0 = kq1q2/b [would the charge on on the alpha particle be the atomic # times the charge of one proton? and similar for the second particle?]
Kf = .5mVf^2 [ I think I can change my expression for angular momentum and substitute in for this Vf so that I only have one unknown variable]
Uf = kq1q2/ Rmin

Then I think I can plug in "v0b/Rmin" for Vf. And then I get my terms containing Rmin on one side and everything else on the other. The plug in numbers and take the reciprocal of each term on each side of the equation so that I have a quadratic allowing me to solve for Rmin.
Will this work? I'm working on it now to see, but if this set up is wrong, it sure would save a lot of time to know beforehand.

[Edit] Well, I tried it and I ended up with a negative distance of much greater magnitude than what I started out with. So now that I've expended my best guess, any other ideas on how to approach this problem?
 
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  • #8
Just got up … :zzz:

Everything looks ok :smile:, except …
Seraph404 said:
K0 = .5mV0^2 = 1 mil(1.602E-19) J

U0 = kq1q2/b

where does 1 mil(1.602E-19) come from?

and U0 = 0 (you can take it that the alpha particle starts so far away that it's effectively at infinity … the physical significance of b is only that it's the distance by which it would miss the nucleus if there was no repulsion). :wink:
 
  • #9
I looked up on google what MeV stood for and I saw that it was 1 electron volt times a million. Maybe my source was wrong *shrug*.

So is b significant at all in finding the closest distance that the alpha particle comes to the lead nucleus?
 
  • #10
Seraph404 said:
So is b significant at all in finding the closest distance that the alpha particle comes to the lead nucleus?

b is essential to the angular momentum equation, but you can ignore it for the energy equation. :smile:
 

1. What is the conservation of energy principle?

The conservation of energy principle states that energy cannot be created or destroyed, but it can be transformed from one form to another.

2. How is the conservation of energy applied in scientific problems?

The conservation of energy is applied in scientific problems by using it as a fundamental principle to analyze and understand physical phenomena, such as motion and changes in energy forms.

3. What are some examples of conservation of energy problems?

Some examples of conservation of energy problems include a pendulum swinging back and forth, a ball rolling down a hill, and an object being launched into the air.

4. How is the conservation of energy related to the laws of thermodynamics?

The conservation of energy is related to the first law of thermodynamics, which states that energy cannot be created or destroyed, but it can be transferred or converted from one form to another.

5. What are the implications of violating the conservation of energy principle?

If the conservation of energy principle is violated, it would mean that energy is being created or destroyed, which goes against the fundamental laws of physics. This could lead to incorrect conclusions and predictions in scientific experiments and observations.

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