# Finding theta of a charged pendulum

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1. Aug 29, 2015

### whitejac

1. The problem statement, all variables and given/known data
a small spherical insulator of mass 6.00×10−2 kg and charge +0.400 μC is hung by a thin wire of negligible mass. A charge of −0.220 μC is held 0.290 m away from the sphere and directly to the right of it, so the wire makes an angle theta with the vertical. What is the angle θ (k=1/4πϵ0=8.99×109 N · m2/C2)
2. Relevant equations
F = k q1q2/d^2

3. The attempt at a solution
So I can find the force acting on the mass. That's easy:
Plugging in I get (8.99x10^9)(0.4x10^-12)(-2.2x10^-12) / (0.29 * 0.29)
Which equals: (-7.9112 x 10^-15) / 0.0841
Which equals: (-9.41 x 10^-14)N

I cannot, however, figure out how to correlate this force to an act of motion. I could see it pulling laterally (the y component being negligable?) but how it could do them both with this stage of math because the string should theoretically be fixed in length and thus pull as an arc instea of a straight line, however I believe they are asking for a straight line.

2. Aug 29, 2015

### whitejac

**Edit, I mistook the prefixes. It should be 10^-6, not 10^-12. The resulting force should be -0.941N, much larger.

3. Aug 29, 2015

### TSny

I think the system looks like the figure below. The pendulum charge sits at rest at angle θ with the other point charge located horizontally to the right.

#### Attached Files:

• ###### Pend charge.png
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4. Aug 29, 2015

### whitejac

Yes, i believe it does look like that.
So then, I could find theta by using inverse cosine on the resulting force, which would give me the angle to the lower right of that triangle, and 90-ans would give me theta?

5. Aug 29, 2015

### TSny

I'm not sure. Can you show in more detail what you mean here?

6. Aug 29, 2015

### whitejac

I guess what I mean is:
Theta = 90 - arcos(-0.941)?
The force pulls it away along the x axis, so to the left of the pendulum would create an angle of arccos(-0.941) and with the triangle being a right triangle then the only thing we'd need to conclude the theta would be to subtract that answer from 90 degrees.

7. Aug 29, 2015

### TSny

I don't get 0.941 N for the electric force if I use 0.400 μC and 0.220 μC. It appears to me that your decimal point is in the wrong place in your answer.

More important, what is the justification for taking the inverse cosine of the force? Whenever you take the inverse cosine of a number, the number should be dimensionless.

I recommend that you draw a free body diagram for the pendulum charge and use what you know about the sum of the forces acting on an object that remains at rest.

8. Sep 5, 2015

### whitejac

I cannot recall what my justification for using inverse cosine was, but I simply must thank you for your assistance. I was looking at this problem as more complicated than it had to be. When I did the force body diagram like you said, I saw what the situation really was. the string was an equal and opposite Fx Fy force. Theta is equal to inverse tangent of Fy/Fx. So I was able to solve this problem once I reviewed my vector algebra and got the trigonometry worked out.

9. Sep 5, 2015

### TSny

OK. Good work!