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Old Sep 27th 2008, 02:52 AM   #1
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Question Motion of point mass under two central forces -- source of one stationary one moving.

PROBLEM:
A point mass ‘A’ is kept at the origin. Another B is kept at the x axis at x = H. Another C is kept at distance H from origin and distance h from B. A, B and C thus form an isosceles triangle with vertex A. Given that A's mass >> B's mass >> C's mass; Newton’s gravitational law governs the bodies. C will have motion because of both A and B while B will have motion because of A only.
Find:
The equation of path of C in any coordinate system given that B and C have zero initial velocities.

Attempted solution:

I write, $\displaystyle a_B=\frac{k_a}{r_b^2} $ and solve this by writing, $\displaystyle v\,dv=a\,dr_b$.
We get an equation in $\displaystyle v$ and $\displaystyle r_b$. In this equation, we write: $\displaystyle v=dr_b/dt$ and solve for $\displaystyle r_b $ and get it in terms of $\displaystyle t$. We finally arrive at:
$\displaystyle

r_b=\left(H^{3/2}-k_at\right)^{2/3}
$
Now, follow the link:

A seeker of silences am I...: 27-Sep-2007

You will see two images posted. Click on them to enlarge them. I have carried fwd the solution there.

Method applied to reach till the equations 3 and 4 (given below):

I used polar coordinate systems with the point A as origin. I found out radial and angular components of the accleration of C and compard them with the standard expressions with the same as given here:
Polar coordinate system - Wikipedia, the free encyclopedia

The following equations we finally arrive at:

$\displaystyle

r_{b/c}=\sqrt{r_b^2\,+\,r_c^2\,-2bc\cos \theta}\quad(1)$

$\displaystyle \frac{r_{b/c}}{\sin \theta}\,=\,\frac{r_b}{\sin \phi}\quad(2)
$
These two equations define the angle $\displaystyle \phi$ and $\displaystyle r_b/c$ for us.

Proceeding...

$\displaystyle
-\frac{k_b}{r_{b/c}^2}\,\left(1-\,\frac{r_b^2}{\sin^2 \theta r_{b/c}^2}\right)\,=\,\ddot{r_c}\,-\,r_c\dot{\theta^2}\quad(3)
$
$\displaystyle

-\left(\frac{k_b}{r_{b/c}^2}\frac{r_b}{\sin \theta r_{b/c}}\,+\,\frac{k_a}{r_c^2}\right) \,= \,r_c\ddot{\theta}\,+\,2\dot{r_c}\dot{\theta}\quad (4)
$

$\displaystyle r_{b/c}
$ is defined by first two equations. Eqations 3 and 4 then need be solved for $\displaystyle r_c$ and $\displaystyle \theta$.

Plz help.
Thanks.

Last edited by taureau20; Sep 27th 2008 at 03:16 AM. Reason: Error in latex coding
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central, dfq, forces, kinematics, mass, motion, moving, point, source, stationary



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