Why Does the Reduced Mass Concept Yield a Positive Total Energy?

[AdityaDev]

According to the concept: when a planet revolves around a star, and when both the bodies move in circular orbits due to the interaction between each other, both the bodies can be replaced by a single body of mass $\mu$ revolving in a circular orbit of radius equal to the distance between both the bodies.
so the total energy of the system becomes $\frac{1}{2}\mu v^2$
But it is positive. The total energy has to be be negetive
$-\frac{GMm}{r}+E_1+E_2$ where e1, e2 are kinetic energies of the bodies comes out to be negetive after solving.
Why do I get positive sign using concept of reduced mass?

 

[Orodruin]

Please provide references for this statement:

so the total energy of the system becomes $\frac{1}{2}\mu v^2$

Did you try actually writing down the energy of the system in the CoM system?

 

[AdityaDev]

Only the statement is given.
The energy of the body in circular motion should be ½μν² right? Since both the bodies can be replaced by a single body of reduced mass $\mu$
I got the total energy using $-\frac{GMm}{r}+\frac{1}{2}I_1\omega^2+\frac{1}{2}I_2\omega^2=-\frac{GMm}{2r}$ which is also given in my textbook.
On using ½μν², i get $\frac{GMm}{2r}$.

 

[AdityaDev]

I understand now. I tried using the relation $$mv^2/r=GMm/R^2$$ which is wrong since there is no force of gravitation when you just have a single body.

 

[Orodruin]

No no no, you replace the two bodies with a single body of mass $\mu$ moving in a central gravitational potential. You still need to provide the reference, what book are you using?

 

[AdityaDev]

DC Pandey, Understanding Physics for JEE main and advanced.
I did the 1/2μν2 part myself. I just posted what i understood so that you could correct me. This is taken from the textbook (exactly the same including errors):

$$m_1r_1=m_2r_2$$
$$m_1r_1\omega^2=m_2r_2\omega^2=\frac{Gm_1m_2}{r^2}$$
$$L=(I_1+I_2)\omega^2=\mu r^2\omega^2$$
Kinetic Energy of the system, $K=\frac{1}{2}\mu r^2\omega^2$
"Thus, the two bodies can be replaced by a single body whose mass is equal to the reduced mass. The single body revolve in a circular orbit whose radius is equal to the distance between the bodies and force of circular motion is equal to force of interaction between the two bodies for actual separation"