Deriving Equation of Reduced Mass W/ Angular Momentum

[Dgray101]

Homework Statement

Derive m*= mM/(m+M) *hint* total angular momentum Iω=m*r²ω equals the sum of the individual angular momenta, where r = r_e + r_n; re and rn are the distances of the electron and nucleus respectively from the center of mass.

Homework Equations

Angular Momentum = mvr
Conservation of Angular momenta
m=mass of electron
M=mass of nucleus

The Attempt at a Solution

m*r²ω = m(re)²ω + M(rn)²ω

I write out this equation but algebraically I can't seem to solve for m* properly so I must be missing something :/ sorry about the notation in the equation I am still learning how to properly use the tools in the message figure.

 

[gneill]

Write expressions for r_e and r_n in terms of r, given that they are determined by the location of the center of mass.

 

[Dgray101]

So like m*r2ω = m(r-r_n)^2 ω + M(r-r_e)^2 ω ?

 

[gneill]

Forget the angular velocity for the moment. The distances r_e and r_n from the center of mass can be determined entirely by the masses and the total separation r.

Given m and M and r, what are those distances?

 

[Dgray101]

I have no idea :/ my guess is it has something to do with the gravitation equations. But I never took anything with this in first year because I was in engineering :/ not the normal physics stream.

 

[gneill]

I think that if you've done engineering you can find the center of mass given two objects and their separation. What's the distance of each from that center of mass?

 

[Dgray101]

I didn't do that well in engineering mechanics because it was my first year, and wasn't expecting it to be so brutal :/ but if I recall

Center of Mass = (1/M+m) *( (mr_e) + (Mr_n) ) ?

 

[gneill]

I didn't do that well in engineering mechanics because it was my first year, and wasn't expecting it to be so brutal :/ but if I recall

Center of Mass = (1/M+m) *( (mr_e) + (Mr_n) ) ?

You've got the right idea, but there needs to be some reference point from which r_e and r_n are measured for the expression to make sense. Since it's the center of mass itself that we want as the reference point, things would need adjusting.

I'll suggest another way to approach the problem. You want to get expressions for r_e and r_n in terms of r and the masses m and M. So look at it as if it were two objects of mass M and m on a massless rod in a uniform gravitational field. You want to find the lengths r_e and r_n from the pivot point where the system will be balanced:

Write an expression for the moments about the pivot point. That's one equation. For the other, use the fact that r is the sum of r_e and r_n. Solve for r_e and r_n.

 

[Dgray101]

so... I had said that the sum of the moments about the center of mass would be...

r_emg - r_nmg = r(m+M)g (I think anyways)

which becomes r(m+M) = Mr_n + mr_e

Would it be helpful to perhaps take out a factor of mM in this case on the right hand side?

 

[gneill]

so... I had said that the sum of the moments about the center of mass would be...

r_emg - r_nmg = r(m+M)g (I think anyways)

No, the moments need to balance for equilibrium. The right hand side should be zero (no net torque).

Then use the fact that the sum of the distances is r to write a second equation. Solve for r_e and r_n.

 

[Dgray101]

Oh okay so r_emg - r_nMg = 0

r_e = (r_nM) / m

Then pluging that into the sum of re and rn --> r = (M+m)r_n / m

So now could we sub r into the original equation in the problem statement and solve for m*?

 

[gneill]

Oh okay so r_emg - r_nMg = 0

r_e = (r_nM) / m

Then pluging that into the sum of re and rn --> r = (M+m)r_n / m

So now could we sub r into the original equation in the problem statement and solve for m*?

You don't want to replace r in the angular momentum equation, you want to replace both $r_e$ and $r_n$ in that expression.

So rearrange your equation above to isolate $r_n$. Then solve for $r_e$ in a similar fashion. You should end up with two expressions:

r_n = <something involving r,M,n only>

r_e = <something involving r,M,n only>

Those expressions will replace r_e and r_n in your angular momentum equation.