Coulomb's Law and Electric Force

[Gazaueli]

Homework Statement

An object with a small mass (m) causes two balloons to float (neither ascending nor descending) in the air at a fixed distance from each other, as shown in the diagram. The balloons, with a volume (V) each carry a positive charge, Q. The balloons are said to be point charges, and their mass is not considered.

1. Create a diagram of all the forces at work.

2. Give a literal expression for the angle θ according to m, g, p, and V (study the condition of flotation, and do not consider Archimede's Principle exercised on the mass (m)).

3. By supposing that m=10g, L=1.5m, and d=30cm, calculate Q.

Homework Equations

Coulomb's Law: F=$\frac{k|qQ|}{r^{2}}$

Gravitational Force: F=$\frac{GmM}{r^{2}}$

Tension Force: F=Fgrav-ma

The Attempt at a Solution

First off, my knowledge of physics is very elementary. Before this class, I had never taken a physics course and the last math course I had was very basic. As I'm studying in Switzerland, I unfortunately don't have the option of taking an introductory course at university, and was therefore placed directly in this class, which I think is rather advanced physics. It's a lecture style course without step-by-step problem solving, and little to no explanation. Students are expected to already have a solid background and be able to follow along. As you can imagine, I'm rather lost. I need very clear, precise and elementary explications: i.e as you would give to a child, because this is honestly not my strong suit. I don't have a strong mathematical background either, so again, easy, clear explanations (when possible) are best. I apologize if the question is worded rather strangely; it was translated from my homework which is in French, so if something seems unclear, I will do my best to explain it.

1. For the drawing of the diagram, and the figure from the problem, please see the attached photo.

3. I attempted to solve #3 first as I'm really unsure of how to go about solving #2. Considering the diagram, I tried to calculate the gravitational force, and the tension forces. I believe that because the balloons are at equilibrium, one can assume that these two forces cancel each other out, but I'm not sure about that given that the string creates an angle and is not perpendicular to the gravitational force. From that I would assume then that the only force acting upon the sphere is the electric force. But with the angles coming into play, I'm really thrown off. I attempted to calculate the electrical force acting on the balloon and got 1.0x10$^{9}$ N/C$^{2}$, but I'm pretty sure this is a huge number and therefore doesn't seem possible.

I apologize if there's really no sense in my calculations. I'm trying my best to comprehend this, but I seem to be missing quite a lot.

Any help is greatly appreciated! Thank you!

 

[mfb]

Is the diagram upside-down? If the balloons allow the mass to float, I would expect that the baloons are above the mass. It's like a hot-air balloon - you don't put the gondola on top.

I attempted to solve #3 first as I'm really unsure of how to go about solving #2.

You need #2 (or something close to the solution) to solve #3.

Considering the diagram, I tried to calculate the gravitational force, and the tension forces. I believe that because the balloons are at equilibrium, one can assume that these two forces cancel each other out

Cancelling forces are a good approach, but the balloons do not have a mass, therefore they have no gravitational force. How do balloons work (what is pushing them up)?
You can consider horizontal and vertical forces separately, both have to cancel. Can you split the tension into a horizontal and vertical component with trigonometry?

Which forces act on the mass?

I attempted to calculate the electrical force acting on the balloon and got 1.0x10⁹ N/C², but I'm pretty sure this is a huge number and therefore doesn't seem possible.

A Coulomb is a huge amount of charge, that is fine. Realistic charges of objects are somewhere in the Nanocoulomb range.

 

[Gazaueli]

Hello mfb,

Thank you for taking the time to respond!

Is the diagram upside-down? If the balloons allow the mass to float, I would expect that the baloons are above the mass. It's like a hot-air balloon - you don't put the gondola on top.

No, the diagram is not upside down. That's exactly as it appears in the problem. From what I understand it's the mass that causes the two balloons to float up. It's lifted upwards from the bottom towards the two balloons thus causing them to lift and float.

Cancelling forces are a good approach, but the balloons do not have a mass, therefore they have no gravitational force. How do balloons work (what is pushing them up)?

I believe they're being pushed up by the charge on the object which is lifted towards them, no?

You can consider horizontal and vertical forces separately, both have to cancel. Can you split the tension into a horizontal and vertical component with trigonometry?

Yes, I thought this might be what I need to do, but I'm unsure of how to go about doing this. From what I've learned (from trying to understand the book explanations), I think this is where the angle θ comes into play. So I need to find the x and y components of F$_{Tens}$, right? From that, I understand that I first need to find the angle θ. Given that the length (L) of the string is 1.5m, and the distance (d) of 30cm (=0.3m), I can use the inverse sin fonction to find θ which gives me: θ=11.5.

The formula to find the x component (from my book) is: A$_{x}$=Acosθ$_{A}$
But this is where I get a bit confused. As I don't have the tension force (which I think is represented by A in the formula), I'm not sure what I'm supposed to plug into find the x component...

You need #2 (or something close to the solution) to solve #3.

I'm not sure what it means to "give a literal expression for the angle θ according to m, g, p, and V"

 

[LeonhardEu]

When it says literal i believe it means an approximation you should make. This approximation is that for small θ, sinθ = θ

 

[mfb]

From what I understand it's the mass that causes the two balloons to float up. It's lifted upwards from the bottom towards the two balloons thus causing them to lift and float.

No, that does not make sense.

I believe they're being pushed up by the charge on the object which is lifted towards them, no?

Only the balloons are charged, and their Coulomb force is purely horizontal.
Did you consider buoyancy?

Yes, I thought this might be what I need to do, but I'm unsure of how to go about doing this. From what I've learned (from trying to understand the book explanations), I think this is where the angle θ comes into play. So I need to find the x and y components of F$_{Tens}$, right?

Right

From that, I understand that I first need to find the angle θ. Given that the length (L) of the string is 1.5m, and the distance (d) of 30cm (=0.3m), I can use the inverse sin fonction to find θ which gives me: θ=11.5.

Try to solve (2) before you use the values of (3), it is really easier that way. I guess you mean θ=11.5°. Usually, angles are expressed in radians and not in degrees.

The formula to find the x component (from my book) is: A$_{x}$=Acosθ$_{A}$

Be careful how the book uses the angle to get this formula. Is it the same way your angle is defined?

But this is where I get a bit confused. As I don't have the tension force (which I think is represented by A in the formula), I'm not sure what I'm supposed to plug into find the x component...

Just plug in a symbol for your tension. You cannot calculate a numerical value at that point, but that does not matter.

I'm not sure what it means to "give a literal expression for the angle θ according to m, g, p, and V"

"Find some formula to calculate θ if you know m, g, p and V".


@LeonhardEu: No, that approximation is not good for sin(θ)=1/5.

 

[Gazaueli]

No, that does not make sense.

Hmm, I don't understand why that wouldn't make sense. It clearly states in the questions that the small object causes the two balloons to float. Why isn't it considered in this problem? I thought it was only the masses of the balloons that are ignored here.

Did you consider buoyancy?

No, I didn't. Would I then need to use Archimedes' Principle? If so, how would I apply it to this problem? I really don't understand the connection between electrostatics and buoyancy.

I guess you mean θ=11.5°. Usually, angles are expressed in radians and not in degrees.

Yes, I meant θ=11.5°, but I don't understand the difference between radians and degrees. I always thought angles were expressed in degrees. ?

I took a look at the corrections, to see if I might make some connections, but I honestly feel more lost than before.

It states that the forces at work are 1) the electrostatic force of the first balloon exercised on the second balloon, 2) the electrostatic force of the second balloon exercised on the first balloon. These two forces I understand. Because of the like charges on the two balloons, the forces repel each other. I also understand that they are horizontal forces.
It also states that the other two forces at work are 3) $\vec{P}$ of the carried mass, but that we ignore the mass and gas inside them. Is this a pressure force?
And finally, 4) $\vec{P}$$_{i}$, archimedes' principle that each balloon undergoes, upwards.

Again the last two Forces have totally thrown me off. I'm even more confused than before now that I've taken a look at the correction paper. I really don't know where to begin with this.

 

[gneill]

The problem only makes sense if the mass is suspended below the balloons. Nowhere does the problem state that there is a charge on the mass, or that it is lifted to some position between the balloons. If that were the case the actual mass of the object would be irrelevant. Without a load to balance the buoyancy of the balloons there could be no equilibrium established.

So. The most likely setup implied by the problem description would be:

Another issue is that the problem doesn't adequately explain all the variables. In particular, when it says "Give a literal expression for the angle θ according to m, g, p, and V", what is p? The other variables m, g, and V are pretty standard designators of mass, gravitational acceleration, and Volume. But lower-case p is not so well established as a particular item. Could it be that it was meant to be $\rho$ for density of the balloons?

 

[mfb]

Hmm, I don't understand why that wouldn't make sense. It clearly states in the questions that the small object causes the two balloons to float. Why isn't it considered in this problem? I thought it was only the masses of the balloons that are ignored here.

Without the mass, the balloons would rise upwards, as they have no mass on their own, but displace air.

No, I didn't. Would I then need to use Archimedes' Principle? If so, how would I apply it to this problem? I really don't understand the connection between electrostatics and buoyancy.

Right.
There is no direct connection between those concepts, they are just both sources of forces on the balloons.

Yes, I meant θ=11.5°, but I don't understand the difference between radians and degrees. I always thought angles were expressed in degrees. ?

They are like two different unit systems.
You can give express a given length in meters or feet. Both work, but feet are more common in the US and meters are more common everywhere else. In the same way, radians are more common in physics and trigonometry and degrees are more common if no trigonometry is needed.