Initial acceleration of the charge

[sergiokapone]

Homework Statement

Three identical charged balls of mass m and charge q bound in a triangle thread length l. One of the strands break. Calculate the acceleration of the middle ball at the initial moment.

2. Relevant equation

a. Newton's laws

$m\vec a_1 = \vec F_{12} +\vec F_{13} + \vec T_{13}$
$m\vec a_2 = \vec F_{21} +\vec F_{23} + \vec T_{23}$
$m\vec a_3 = \vec F_{31} +\vec F_{32} + \vec T_{31} + \vec T_{32}$b. Constraint Equations

$(\vec r_1 -\vec r_2)^2 = (\vec r_1 -\vec r_3)^2 = const$

c. May be relations between $\vec r$-vectors in CM-system
$\vec r_1 +\vec r_2+\vec r_3=0$

The Attempt at a Solution

The unknown T-forces should to be exluded using Constraint Equations, but what to do with it, I have no idea.

 

[haruspex]

How could you obtain another equation concerning accelerations from you constraint equations (b)?
I believe your equation (c) should be deducible from the other equations.

 

[sergiokapone]

How could you obtain another equation concerning accelerations from you constraint equations (b)?
I believe your equation (c) should be deducible from the other equations.

Yes (c) is the cosequence of the Newton's laws.
I don't know answers. Now I have no idea, how to start to solve this problem.

 

[sergiokapone]

From the (b), I can obtain
First differentiating:
$(\vec r_1 - \vec r_3)(\vec v_1 - \vec v_3)=0$
From the second differentiating:
$(\vec v_1-\vec v_3)^2 + (\vec r_1 - \vec r_3)(\vec a_1 - \vec a_3)=0$

And for the second constraint
$(\vec v_2-\vec v_3)^2 + (\vec r_2- \vec r_3)(\vec a_2 - \vec a_3)=0$

 

[haruspex]

From the (b), I can obtain
First differentiating:
$(\vec r_1 - \vec r_2)(\vec v_1 - \vec v_2)=0$
From the second differentiating:
$(\vec v_1-\vec v_2)^2 + (\vec r_1 - \vec r_2)(\vec a_1 - \vec a_2)=0$

Right. But we are only interested in the initial acceleration. What simplification does that provide?

 

[sergiokapone]

For the initial conditions:
$\vec v_1= \vec v_2 = \vec v_3=0$.
Then
$(\vec r_1 - \vec r_3)(\vec a_1 - \vec a_3)=0$

And
$(\vec r_2 - \vec r_3)(\vec a_2 - \vec a_3)=0$