# Initial acceleration of the charge

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1. Nov 13, 2015

### sergiokapone

1. The problem statement, all variables and given/known data
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
$\vec r_1 +\vec r_2+\vec r_3=0$

3. 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.

2. Nov 13, 2015

### 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.

3. Nov 13, 2015

### sergiokapone

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.

4. Nov 14, 2015

### 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
$(\vec v_2-\vec v_3)^2 + (\vec r_2- \vec r_3)(\vec a_2 - \vec a_3)=0$

Last edited: Nov 14, 2015
5. Nov 14, 2015

### haruspex

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

6. Nov 14, 2015

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

Last edited: Nov 14, 2015