Missing something obvious in this derivation

[TheCanadian]

Hi,

I've attached a file displaying a derivation to make the kinetic energy of a two-body problem into a kinetic energy only involving the reduced mass. When plugging 8.3 into 8.1, I just don't quite see how this derivation makes sense. Shouldn't there be a $$ \mu^2$$ term? Since when squaring the absolute value of r1, aren't the mass terms also squared? If I'm not mistaken, it simply looks like the mass terms are not squared when plugging in 8.3 into 8.1.

I feel like I'm missing something painfully obvious here, so any help is appreciated!

 

[sophiecentaur]

I feel like I'm missing something painfully obvious here,

Isn't it just a matter of dimensions? Why would you expect mass² to come into the energy formula?

 

[fzero]

There's a factor of $m_1+m_2$ that cancels against one of the factors in the denominator.

 

[andrewkirk]

Since when squaring the absolute value of r1, aren't the mass terms also squared?

Yes, they are squared. But if you make the substitutions and work through the algebra, you just end up with $\frac{m_1m_2}{m_1+m_2}$

 

[Let'sthink]

I agree with fzero.

 

[Chandra Prayaga]

Right. Just put it down on paper, instead of imagining what will happen, and it comes out very easily.