Circling particle and force

Two particles of mass m and M undergo uniform circular motion about each other at a separation R under the influence of an attractive constant force F. The angular velocity is ω radians per second. Show that R = (F/ω2)(1/m + 1/M)..

#Mechanics

Easy Math Editor

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Comments

Each particle is in a circular orbit around the center of mass. The distance of the less massive particle to the center of mass is $r_m$ , and the distance of the more massive particle to the center of mass is $r_M$ .

$r_m + r_M = R$

Newton's Second Law equations with centripetal force:

$\frac{G m M}{R^2} = m \, r_m \, \omega^2 \\ \frac{G m M}{R^2} = M \, r_M \, \omega^2$

Combining $r_m$ and $r_M$ :

$R = r_m + r_M = \frac{G M}{R^2 \, \omega^2} + \frac{G m}{R^2 \, \omega^2} \\ = \frac{G (m + M) }{R^2} \frac{1}{\omega^2} \\ = F \frac{m + M}{m M} \frac{1}{\omega^2} \\ = \frac{F}{\omega^2} \Big (\frac{1}{m} + \frac{1}{M} \Big)$

Steven Chase - 2 years, 10 months ago

@Steven Chase That's great .Can you add some visualizations or animations if possible but i got it.While attempting it from the textbook I got confused since it is said they are circling about each other and i thought how it is possible but now it is clear that they are rotating about center of mass. Pls help me with this

Om Tiwari - 2 years, 10 months ago

@Steven Chase Some extra help pls can u provide recommendations regarding books,problems,videos,links and all for preparation of IPHO

Om Tiwari - 2 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$