Reduced mass

The reduced mass (かんさんしつりょう) is effective inertial mass used in the issue of of the Newton dynamics two. It is quantity with a dimension of the mass and enables it to handle one issue of two as if it is a problem. Reduced mass is Greek characters well${\displaystyle \mu \!\,}$I use を and am shown.

It is half of the harmonic average of two mass, and reduced mass becomes always less than mass of each 2 objects. But I cannot consider that gravitational mass in itself to decide gravitational size decreases. The calculation top can express gravity definitely if I replace the other with the sum of mass of 2 objects when I replace one mass with reduced mass.

Reduced mass in the issue of two

Mass${\displaystyle m_{1}\!\,}$ と${\displaystyle m_{2}\!\,}$ There are two objects having を and turns around a center of gravity. This problem and issue of equivalent one object 2 of the position is based; is reduced mass at mass of object 1

${\displaystyle m_{\text{red}}=\mu ={\cfrac {1}{{\cfrac {1}{m_{1}}}+{\cfrac {1}{m_{2}}}}}={\cfrac {m_{1}m_{2}}{m_{1}+m_{2}}}\!\,}$ 

It is said that it is the same that に substitution, the gravity act between two original objects and is the problem for the position of object 1.

I can easily prove this. The power given to object 1 by object 2 is as follows when I use the second law of Newton.

${\displaystyle F_{12}=m_{1}a_{1}\!\,}$ .

As for the power given to object 2 by object 1 likewise

${\displaystyle F_{21}=m_{2}a_{2}\!\,}$ 

である.

Because there is reaction for equal reverse of the size for all action when I obey the third law of Newton,

${\displaystyle F_{12}=-F_{21}.\!\,}$ 

である. Therefore

${\displaystyle m_{1}a_{1}=-m_{2}a_{2}\!\,}$ 

であり

${\displaystyle a_{2}=-{m_{1} \over m_{2}}a_{1}\!\,}$ 

である.

The relative acceleration between two objects is given as follows.

${\displaystyle a=a_{1}-a_{2}=\left({1+{m_{1} \over m_{2}}}\right)a_{1}={{m_{2}+m_{1}} \over {m_{1}m_{2}}}m_{1}a_{1}={F_{12} \over m_{\text{red}}}}$ .

Therefore, object 1 is based on a position of object 2 and works in the same way as an object with the mass same as reduced mass.

The position of object 1 for object 2 knows that I am ruled by a differential equation same as the position of the very small object revolving around an object having mass same as the sum of mass of 2 objects when I use the gravitational formula. Because

${\displaystyle {m_{1}m_{2} \over m_{\text{red}}}=m_{1}+m_{2}.\!\,}$ 

Therefore.

More general reduced mass

The word "reduced mass" may more generally hint at item algebra of the following form.

${\displaystyle x_{\text{red}}={1 \over {1 \over x_{1}}+{1 \over x_{2}}}={x_{1}x_{2} \over x_{1}+x_{2}}\!\,}$ .

I can simplify an expression of the following form by expanding this.

${\displaystyle {1 yen \over x_{\text{eq}}}=\sum _{i=1}^{n}{1 \over x_{i}}={1 \over x_{1}}+{1 \over x_{2}}+\cdots +{1 \over x_{n}}\!\,}$ .

Such a reduced mass is used for the expression of relations of elements such as the plural resistance that is electrical, and thermal, is hydraulic or is dynamic joined together in parallel. Those expressions of relations are decided like consecutive equations to join a factor together by the physical characteristic of the element.

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