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
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