Three equal charges held by equal length strings

[mr_sparxx]

Homework Statement

Three particles with the same charge q are hold together by three strings of the same length l forming an equilateral triangle. The tension in each string is:

a) $\frac {K q^2 cos 60º}{l^2}$
b)$\frac {K q^2 cos 30º}{l^2}$
c)$\frac {K q^2 sin 60º}{l^2}$ or
d)$\frac {K q^2 sin 30º}{l^2}$

It is a multiple choice question.

Homework Equations

Coulomb's law for electrostatic force: $F = \frac {K q_1 q_2}{d_{12}^2}$

The Attempt at a Solution

Well, numbering the charges from 1 to 3, we get that the total force acting on the third charge is the sum of the tensions plus the sum of the elecrostatic forces:

$$ \vec F = \vec T_{13} + \vec T_{23} + \vec F_{13} + \vec F_{23} = 0 $$

Furthermore, the symmetry of the problem implies that:
$$ T_{13} = T_{23} = T_{12} $$
$$ F_{13} = F_{23} = F_{12} $$

So, in the two directions where forces lie
$$ \vec T_{13} = - \vec F_{13} $$
$$ \vec T_{23} = - \vec F_{23} $$

And, therefore
$$ T_{13} = F_{13} = \frac {K q^2}{l^2} $$

The tension in the other strings is exactly the same.

I am unable to see my mistake, simple as it may be...

 

[Spinnor]

Part 1 has me a little confused, there are three strings and 4 components are given but not associated with a particular string.

What exactly is the question? Parts 2 and 3 look correct.

 

[mr_sparxx]

Thanks for your answer.

I am sorry, I should have said it is a multiple choice question: a, b, c, or d are the answer options. I'll update the post.

Could it be possible that the question is wrongly formulated or the answers wrong? This is the exact text of the question...

 

[Spinnor]

Thanks for your answer.

I am sorry, I should have said it is a multiple choice question: a, b, c, or d are the answer options. I'll update the post.

Could it be possible that the question is wrongly formulated or the answers wrong? This is the exact text of the question...

I think you are headed in the right direction. Please state the exact question you are to answer

 

[mr_sparxx]

The exact question is the one stated in part 1. I have to write a, b, c or d in the answer sheet and my result I am getting is different to all four choices: that's why I am so confused.

 

[Spinnor]

The exact question is the one stated in part 1. I have to write a, b, c or d in the answer sheet and my result I am getting is different to all four choices: that's why I am so confused.

Sorry, I am confused as well, it looks to me like you have the correct answer in part 3 above. I did the problem the same way you did.

 

[Spinnor]

Maybe it is like this with the three strings attached at a common point,

I think we both read it the wrong way? If like above you can figure it out. I thought the strings formed the sides of an equilateral triangle and my mind was locked into that as the only possibility. A Google image search allowed me to see the other possibility.

https://www.google.com/search?q=ten...bAhUFk1kKHZ0FDt44ChD8BQgLKAI&biw=1164&bih=537

 

[TSny]

Maybe it is like this with the three strings attached at a common point,

Good thought. But it doesn't appear that this interpretation would give one of the choices either (unless I'm messing up somewhere).

Edit: Note that answers (a) and (d) are the same. Also, (b) and (c) are the same. @mr_sparxx please check that you have written the choices correctly.

 

[mr_sparxx]

Thank you for your answers and thoughts. I posted it here because I thought it was the most relevant place but, in fact, this problem comes from a past access exam paper one of my students brought to me. I was unable to give him an answer. Tomorrow I'll take a picture of it to be sure .

@Spinnor I am quite sure there was a diagram... but that turn you proposed was quite inspiring. However I think there is a $\sqrt 2$ factor discrepancy with the b/c choices.

Edit: Note that answers (a) and (d) are the same. Also, (b) and (c) are the same.

@TSny I didn't notice!

Anyway, I was not confident enough to think the question was wrong (it is a quite prestigeous university access exam), but I am beginning to think this is the case.

 

[TSny]

If the picture in @Spinnor 's post #7 is correct, then I believe any of the answers (a) through (d) would be correct if the trig function factor is replaced by $\tan 30^0$.

 

[mr_sparxx]

Actually what was wrong was my handwritten copy of the answer choices! I am so embarrased!
Choices were:
a) $\frac {K q^2 sin 60º}{l^2}$
b)$\frac {K q^2 cos 60º}{l^2}$
c)$\frac {2 K q^2 sin 30º}{l^2}$
d)$\frac {2 K q^2 cos 30º}{l^2}$
And it is obviously c)...
Sorry for wasting your time!

Edit: in the end strings were the sides of the triangle, each holding two charges together...

 

[TSny]

OK. No problem. Glad it got cleared up.