Coulomb's law and gravity

Homework Statement

1. A positively charged object with a charge of +85 nC is being used to balance the downward force of gravity on a 1.8-gram balloon which has a charge of -63 nC. How high above the balloon must the object be held in order to balance the balloon? (NOTE: 1 nC = 1 x 10^-9 C)

Homework Equations

F = k • Q1 • Q2 / d^2

k = 9.0 x 10^9 N • m2 / C2

The Attempt at a Solution

I turned the grams into kilograms, converted the kilograms into weight to equal the force and put all the other numbers back into the equation but no matter what I do, I can't get the answer the website gives as the answer, which is 0.155 meters.

I noticed the "explanation" of the answer didn't include the constant k. Here's the explanation: Use the mass to determine the force of gravity (m • g). The force of gravity on the balloon is 0.0176 N. Thus, the upward electrical force is 0.0176 N. Now that F, Q1, and Q2 are known, Coulomb's law can be used to determine the distance d in the equation. Algebraic rearrangement leads to d = Sqrt [ (Q1• Q2) / F ]. Substitution leads to the answer.

It leaves out k in the formula. But even when I leave out k, I still don't get their answer.
The answer I got was 1.65x10^-5 m. That sounds a little too close.
Thanks.

Doc Al
Mentor
I noticed the "explanation" of the answer didn't include the constant k. Here's the explanation: Use the mass to determine the force of gravity (m • g). The force of gravity on the balloon is 0.0176 N. Thus, the upward electrical force is 0.0176 N. Now that F, Q1, and Q2 are known, Coulomb's law can be used to determine the distance d in the equation. Algebraic rearrangement leads to d = Sqrt [ (Q1• Q2) / F ]. Substitution leads to the answer.
If it's any consolation, you are correct that the book's solution is nonsense. You can't just leave out Coulomb's constant.

(What book are you using?)

Borek
Mentor
I got 5.2 cm, but I solved it using TI-89 that I have not used for several months, so I would not pay too much attention to the result.

$$0.01764 N = \frac {4.8195 \times 10^{-5} J m^3} {d^2}$$

(these units may look strange, but when you solve for d it actually is in meters).

The question is from a website.
I get different answers depending on how I have the number set up before I square it. Do I convert out of scientific notation before I take the square root of the number?

Borek
Mentor
Please elaborate - show how you do the calculations. Notation doesn't change the result.

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Please elaborate - show how you do the calculations. Notation doesn't change the result.

--

Oh, I know what I was doing wrong. I was squaring just the number and not the x10^-3.
I assumed I could add that part afterward.
I got 5.2cm, same as you, for the answer. Or 0.052269903932377759823580627137967m. I assume that's correct.
Stupid website, you'd think after spending so much time on it, they'd have the right answers. But I noticed a couple of mistakes on that same page. Maybe they were drunk when they wrote it. They had a picture of a balloon with the gravity force pointing up.

Thanks for the help.

Well the next problem they have is also driving me nuts. I can't fathom how they got their result.

Here's the question:

Two 1.2-gram balloons are suspended from light strings attached to the ceiling at the same point. The net charge on the balloons is -540 nC. The balloons are distanced 68.2 cm apart when at equilibrium. Determine the length of the string.

Here's how they explain it. I don't get how they concluded the answer was 78.8 cm.

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Borek
Mentor
The only problem I see with the explanation is that they have not shown which angle is theta. Other than that it is simple geometry.

Theta is the angle between vertical and the string.

The only problem I see with the explanation is that they have not shown which angle is theta. Other than that it is simple geometry.

Theta is the angle between vertical and the string.

Well, at the end it tells me: "From a distance triangle, one sees that sin(theta) = 0.5 • d / L Substituting theta and d into this equation leads to the answer."

But when I substitute the values into the equation, I don't get their answer.
Also, I don't understand why I can't use pythagorean's theorem to get the length of the string. Wouldn't the string be the hypotenuse?
Thanks.

Borek
Mentor
Try harder, I got the same result they did.

What are you going to use for a vertical leg length?

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I found out how they got their answer, but I don't quite understand it. I thought I take the sine of 34.1/25.6 after I divided it. But they took the sine of 25.6 before the division, then divided 34.1 by that number. I guess it's been a while since I did some trigonometry.
So if I have the angle and the opposite side, I take the sine of the angle first, then divide the opposite by that number?

Thanks.

Doc Al
Mentor
For any right triangle, the sine of the angle = Opposite Side/Hypotenuse. In this case the hypotenuse is the length of the string, which is what you are trying to find.