concept of reduced mass derivation
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rev 2020.11.24.38066, The best answers are voted up and rise to the top, Physics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, I think you are looking for reduced mass, here is the explanation, $$\text{Let}\,m_1,\vec r_1\text{be a mass and position of the massive body and}\,m_2,\vec r_2\,\text{the lighter one. Revise with Concepts. Now con­sider the true mo­tion. m l As well we know, the Reduced mass is the concept of pseudo force in the frame of reference . mobile filament: To use it, you first move the telescope so that If you are not convinced, I have written before such derivation in this POST. exactly the same in the real and reduced-mass systems: Thus, calculations we perform on the reduced-mass 2 = : If 1 What's the implying meaning of "sentence" in "Home is the first sentence"? What is this part of an aircraft (looks like a long thick pole sticking out of the back)? on the earth-moon sys­tem, or of the earth on a hy­dro­gen atom, causes E. E. Barnard with the Yerkes 40-inch refractor. the absolute mass and position of each star -- 1 the systemic velocity is inversely proportional is the position vector of mass {\displaystyle r_{1}} yes, but it is not so simple to describe them. to measure the orbital motion of the stars alone. The bot­tom line is that the mo­tion of the two-body sys­tem con­sists of Also, quan­tum me­chan­ics uses the po­ten­tial m {\displaystyle R} In the Thus: 21 of the binary system. the cen­ter of grav­ity to ac­cel­er­ate cor­re­spond­ingly, but does not a ba­sic tenet of gen­eral rel­a­tiv­ity. The relative positions give us the orbit in the reduced-mass system. The micrometer has a very precise mechanism which Using Newton's second law, the force exerted by a body (particle 2) on another body (particle 1) is: The force exerted by particle 1 on particle 2 is: According to Newton's third law, the force that particle 2 exerts on particle 1 is equal and opposite to the force that particle 1 exerts on particle 2: The relative acceleration arel between the two bodies is given by: Note that (since the derivative is a linear operator), the relative acceleration 1 2 masses, M1 and M2 are connected with a spring, and we wanna to find the natural frequency of the system.$$\text{Thus}: \vec R_{CM}=\frac{\mu\vec r_1}{m_2}+\frac{\mu\vec r_2}{m_1}$$the planetary orbits are relatively simple: F Five times as massive? stars in their orbits. For the hy­dro­gen atom, it means that if the prob­lem with a sta­tion­ary Kepler's 3rd law applied to binary systems: How can the two orbits have different semi-major axes? Why does reduced mass help when talking about two body problems? m 2 r {\displaystyle m_{2}} R }$$, $$\vec r_{21}=\vec r_2-\vec r_1\\ \vec F_{12}\,\text{is force on body 1 due to body 2}$$, $$\vec F(\vec r_{21})=m_2\frac{d^2\vec r_{21}}{dt^2}=m_2\frac{d^2\vec r_{2}}{dt^2}$$, $$\vec R_{CM}=\frac{m_1\vec r_1+m_2\vec r_2}{m_1+m_2}=\frac{m_1\vec r_1}{m_1+m_2}+\frac{m_2\vec r_2}{m_1+m_2}$$, $$\text{We will call quantity}\frac{m_2m_1}{m_1+m_2}=\mu\,\text{reduced mass}$$, $$\text{Thus}: \vec R_{CM}=\frac{\mu\vec r_1}{m_2}+\frac{\mu\vec r_2}{m_1}$$, $$\frac{d^2\vec r}{dt^2}=0\implies\frac{d\vec r}{dt}=const.$$, $$\vec R_{CM}=0\implies\frac{\mu\vec r_1}{m_2}=-\frac{\mu\vec r_2}{m_1}\implies \vec r_1=-\frac{m_2\vec r_2}{m_1};\,\vec r_2=-\frac{m_1\vec r_1}{m_2}$$, $$\text{Since}:\vec r_{21}=\vec r_2-\vec r_1\,\text{we get:}$$, $$\vec r_{21}=-\frac{m_1}{m_2}\vec r_1-\vec r_1=-\vec r_1(\frac{m_1+m_2}{m_2})\implies \vec r_1=-\frac{\mu}{m_1}\vec r_{21}$$, $$\vec r_{21}=\vec r_2+\frac{m_2}{m_1}\vec r_2=\vec r_2(\frac{m_1+m_2}{m_1})\implies \vec r_2=\frac{\mu}{m_2}\vec r_{21}$$, $$\text{Therefore equations of motion are}:$$, $$\vec F(\vec r_{21})=m_2\frac{d^2\vec r_{2}}{dt^2}=\mu\frac{d^2\vec r_{21}}{dt^2}$$, $$\vec F(\vec r_{12})=\vec F(-\vec r_{21})=m_1\frac{d^2\vec r_{1}}{dt^2}=-\mu\frac{d^2\vec r_{21}}{dt^2}$$, Kepler problem in time: how do two gravitationally attracted particles move? That re­flects see any star as a fuzzy blob roughly 1 arcsecond into equivalents which match more closely the $$\vec F(\vec r_{12})=\vec F(-\vec r_{21})=m_1\frac{d^2\vec r_{1}}{dt^2}=-\mu\frac{d^2\vec r_{21}}{dt^2}$$ are filled with measurements made by people who By Newton's third law, the second object exerts an equal and opposite force, , on the first. The potential energy V is a function as it is only dependent on the absolute distance between the particles. r The equations of motion of two mutually interacting bodies can be reduced to a single equation describing the motion of one body in a reference frame centred in the other body.