• Support PF! Buy your school textbooks, materials and every day products Here!

Seperation between two charged ball

  • Thread starter Lady M
  • Start date
  • #1
9
0

Homework Statement


Two identical insulating balls of mass m hang from massless strings of length l and carry identical electric charges, q. you may assume that the angle of separation θ is so small that tanθ ≅sinθ≅θ.
1.png

What is the separation distance between the two masses x?

The Attempt at a Solution


I made two attempts at a solution, though it has been so long since I worked a problem like this that I don't know if either attempt is correct (I do know that both methods give different values for x, so at least one is incorrect).

1st using energy conservation:
2.png


I believe this method gives the incorrect answer as there is some initial energy between the two particles I have not accounted for, when the potential energy is zero but the energy from the E-field is at its greatest. Because I do not know the diameter of the balls, or their starting position, I assume this method is a bad one?

2nd method using statics:
3.png

Based off the free body diagram...
4.png

thus
5.png

so then
6.png

where
7.png

as such
8.png

and finally
9.png

Is this correct? Can someone please point me in the right direction?
 

Attachments

Last edited:

Answers and Replies

  • #2
34,060
9,934
I believe this method gives the incorrect answer as there is some initial energy between the two particles I have not accounted for, when the potential energy is zero but the energy from the E-field is at its greatest.
Right. You don't know the total energy in the system, and the particles being at the same place is an unphysical situation.
2nd method using statics:
Shouldn't there be some equations in this part?
 
  • #3
9
0
Right. You don't know the total energy in the system, and the particles being at the same place is an unphysical situation.Shouldn't there be some equations in this part?
My images did not upload. Give me a second... Sorry.

Edit: I've fixed it now. Not sure why it didn't work the first time...
 
  • #4
Chandra Prayaga
Science Advisor
649
148
Was there any reson for you to doubt your second method?
 
  • #5
9
0
Was there any reson for you to doubt your second method?
The problem mentioned that I should be aware of the small angle approximation, which I did not use. Also, I was uncertain if I would have to account for the second ball in this problem. I feel like I had already done so, but I wanted to be certain of that.
 
  • #6
34,060
9,934
You expressed x as function of the angle (and other constants). The answer should not depend on the unknown angle. If you could use the angle in the answer, simple geometry would be much faster.
 
  • #7
9
0
You expressed x as function of the angle (and other constants). The answer should not depend on the unknown angle. If you could use the angle in the answer, simple geometry would be much faster.
Yes, you could just take the sin of that angle, and multiply it by twice the length of the string.

So is there a better way to do this? Is there a way to do this without knowing the final angle of separation or the distances between the particles?
 
  • #8
Chandra Prayaga
Science Advisor
649
148
Notice that the final result you got contains both x and θ. You can eliminate one of them simply in the small angle approximation. You can use:

tanθ≈sinθ=x/l

and then solve for x in terms of just the mass and charge of each ball.
 
  • #9
9
0
Notice that the final result you got contains both x and θ. You can eliminate one of them simply in the small angle approximation. You can use:

tanθ≈sinθ=x/l

and then solve for x in terms of just the mass and charge of each ball.
Ah, so then
10.png
would be the final answer in terms of everything we "know"?
 

Attachments

  • #10
Chandra Prayaga
Science Advisor
649
148
Absolutely!
 
  • #11
ehild
Homework Helper
15,427
1,827
Notice that the final result you got contains both x and θ. You can eliminate one of them simply in the small angle approximation. You can use:

tanθ≈sinθ=x/l
No, sinθ=x/(2l)
 
  • #12
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
32,760
5,036
  • #13
Chandra Prayaga
Science Advisor
649
148
Oops, sorry. As seen in the diagram, ehild above is right. I overlooked that.
 
  • #14
9
0
No, sinθ=x/(2l)
Yes, you are correct, as we want twice the length of x (the sin part of two triangles). As such the final solution is
10.png
.

Thanks to everyone who has helped me with this problem.
 

Attachments

Related Threads on Seperation between two charged ball

  • Last Post
Replies
2
Views
1K
Replies
2
Views
1K
  • Last Post
Replies
4
Views
7K
Replies
1
Views
3K
Replies
1
Views
3K
Replies
1
Views
5K
Replies
5
Views
6K
Replies
16
Views
3K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
2K
Top