Angle between 2 charged spheres hanging from string

[zachem62]

Homework Statement

Two positively charged metal spheres are suspended from the same hook by light strings of equal length, making an angle of 10.0◦ with each other. The charges carried by the spheres are as shown in the diagram. After that, the spheres are brought in contact briefly, then released. If the mass of each sphere is 4.00 g, calculate:

a. the length of each string.
b. the new angle θ between the two strings.

Diagram is attached.

Homework Equations

The Attempt at a Solution

I've solved part (a) of the question by using the net forces in the horizontal and vertical directions. I found L to be 27.88m. I am completely lost on part (b). How can I approach this question?

 

[andrewkirk]

Bringing them into contact will equalise the charge between them*, so that each has $5\mu C$. Taking that into account, can you solve part (b)?

* This is only true if the spheres are identical, or at least have identical capacitance. The problem should have stated this, but instead only said that they have the same mass, and drew a picture that made it look as though they might be identical. However I think it's reasonable for us to assume the spheres are identical, as the problem cannot be solved otherwise.

 

[zachem62]

Bringing them into contact will equalise the charge between them*, so that each has $5\mu C$. Taking that into account, can you solve part (b)?

* This is only true if the spheres are identical, or at least have identical capacitance. The problem should have stated this, but instead only said that they have the same mass, and drew a picture that made it look as though they might be identical. However I think it's reasonable for us to assume the spheres are identical, as the problem cannot be solved otherwise.

Ok so with that info, I can get the electric force on each sphere just after they come into contact. But I'm still lost on how I can go from there to knowing how far the spheres separated to determine that angle. Would the change in energy approach work here?

 

[andrewkirk]

Just use the same approach as you used for part (a). In that case you would have derived an equation relating the two charges, the angle and the string length, with the string length L being unknown. Now that you know L, can't you just use the same equation with the charges each being $5\mu C$ and the angle being the unknown?

 

[zachem62]

Just use the same approach as you used for part (a). In that case you would have derived an equation relating the two charges, the angle and the string length, with the string length L being unknown. Now that you know L, can't you just use the same equation with the charges each being $5\mu C$ and the angle being the unknown?

But in that equation I was able to determine the separation distance between the 2 spheres when they were separate and in equilibrium. Once they come into contact, have equal charges and are let go, how can I determine what that new separation distance will be? That's the only variable that I am missing in the equation I used to find L in part (a).

 

[andrewkirk]

But in that equation I was able to determine the separation distance between the 2 spheres when they were separate and in equilibrium. Once they come into contact, have equal charges and are let go, how can I determine what that new separation distance will be? That's the only variable that I am missing in the equation I used to find L in part (a).

It sounds like you are concerned about the dynamics of what happens after the balls are released, which is fair enough. It's another weakness in the way the question is asked that they appear to want to know what the static situation will be once equilibrium is reached, but they don't explicitly state that.

When the balls are released they will exercise a damped harmonic motion that, because of the damping of air resistance and string friction, will gradually tend towards a static equilibrium. It's the angle in that static equilibrium that is sought.

I suggest you set out on here the calculation you used for (a). Suggestions can then be made about how to adapt it to solve (b).

 

[zachem62]

It sounds like you are concerned about the dynamics of what happens after the balls are released, which is fair enough. It's another weakness in the way the question is asked that they appear to want to know what the static situation will be once equilibrium is reached, but they don't explicitly state that.

When the balls are released they will exercise a damped harmonic motion that, because of the damping of air resistance and string friction, will gradually tend towards a static equilibrium. It's the angle in that static equilibrium that is sought.

I suggest you set out on here the calculation you used for (a). Suggestions can then be made about how to adapt it to solve (b).

Ok so for part (a) here's what I did:

For the left sphere, Fnetx = Ftsinθ - Fe ⇒ Fe=Ftsinθ
Fnety = Ftcosθ - Fg ⇒ Ft = Fg/cosθ
From these 2 equations, derive that Fe = Fgtanθ

Fe= kq1q2/r^2 = Fgtanθ
isolate r⇒ r = √(kq1q2/Fe)
In the triangle of spheres hanging, θ is the angle that the string on one side has with the vertical. r is separation distance of spheres. L = r/(2sinθ)
Therefore, L = √(kq1q2/Fe)/(2sinθ)

That is how I did part (a).

 

[andrewkirk]

Therefore, L = √(kq1q2/Fe)/(2sinθ)

That equation relates the variables $L, q_1,q_2, F_e,\theta$. In (b) you know the values of all those variables except $\theta$. So make $\theta$ the subject of the equation, substitute the values in (b) for the other variables and you can calculate $\theta$. Then you need to double it to get the answer to (b) because the $\theta$ in the question is the whole separation angle, whereas yours is half that angle (it would have been better to use a different letter for the half angle, to avoid this confusion).

By the way, another flaw in the way the question is asked is that it appears to expect you to use Coulomb's Law, but that law is only applicable to point charges. It should have said that the spheres were very small and asked you to calculate the answer approximately, which would then allow you to use Coulomb's Law as you have. The problem becomes much more complex if we are not allowed to make that simplification, because the charge distribution over the spheres will not be uniform.

 

[zachem62]

That equation relates the variables $L, q_1,q_2, F_e,\theta$. In (b) you know the values of all those variables except $\theta$. So make $\theta$ the subject of the equation, substitute the values in (b) for the other variables and you can calculate $\theta$. Then you need to double it to get the answer to (b) because the $\theta$ in the question is the whole separation angle, whereas yours is half that angle (it would have been better to use a different letter for the half angle, to avoid this confusion).

By the way, another flaw in the way the question is asked is that it appears to expect you to use Coulomb's Law, but that law is only applicable to point charges. It should have said that the spheres were very small and asked you to calculate the answer approximately, which would then allow you to use Coulomb's Law as you have. The problem becomes much more complex if we are not allowed to make that simplification, because the charge distribution over the spheres will not be uniform.

So in that situation, I don't know what Fe (electric force) is when the 2 spheres are separated by that given angle, and I cannot determine that without knowing the separation distance. How can I work around this?

 

[haruspex]

So in that situation, I don't know what Fe (electric force) is when the 2 spheres are separated by that given angle, and I cannot determine that without knowing the separation distance. How can I work around this?

So take the angle and distance to be unknowns and use the geometry to write an equation relating them.

 

[zachem62]

So take the angle and distance to be unknowns and use the geometry to write an equation relating them.

I already did that in a previous step, in part (a) when I determined string length where L = r/(2sinθ). This is an equation that relates the angle and separation distance, and using this, I was able to get to the equation L = √(kq1q2/Fe)/(2sinθ). So I still have unknowns other than the angle, preventing me from being able to determine the angle. What could be another way to do this?

 

[andrewkirk]

I don't know what Fe (electric force) is when the 2 spheres are separated by that given angle

Yes you do. In post 7 you wrote a formula for Fe in terms of $\theta$ and Fg. So substitute that formula for Fe into your equation and then you'll have an equation in which the only unknown is $\theta$.

 

[zachem62]

Yes you do. In post 7 you wrote a formula for Fe in terms of $\theta$ and Fg. So substitute that formula for Fe into your equation and then you'll have an equation in which the only unknown is $\theta$.

But in that case the spheres are in equilibrium so we know that Fnet = 0, which let me arrive at that equation that relates Fe and Fg. The question just says they're released after being in contact, and doesn't say the balls have reached static equilibrium. So how's it possible to use that equation again here?

 

[andrewkirk]

But in that case the spheres are in equilibrium so we know that Fnet = 0, which let me arrive at that equation that relates Fe and Fg. The question just says they're released after being in contact, and doesn't say the balls have reached static equilibrium. So how's it possible to use that equation again here?

See post #6.

 

[zachem62]

See post #6.

Okay so you're suggesting I should just assume its gone back to being in static equilibrium and just apply the same equations from part (a)?

 

[haruspex]

Okay so you're suggesting I should just assume its gone back to being in static equilibrium and just apply the same equations from part (a)?

Yes.

 

[zachem62]

Yes you do. In post 7 you wrote a formula for Fe in terms of $\theta$ and Fg. So substitute that formula for Fe into your equation and then you'll have an equation in which the only unknown is $\theta$.

Okay so in part (a) I had these equations: Fe=Fgtanθ=mgtanθ and L = r/(2sinθ). If I plugged in r = √(kq1q2/Fe) in L = r/(2sinθ) I would have L = √(kq1q2/Fe)/(2sinθ). Plugging in Fe=Fgtanθ=mgtanθ into that equation would yield L = √(kq1q2/(mgtanθ))/(2sinθ). So in this equation I have everything needed but when I isolate for θ I get 2 trig quantities so I ended up with ((sinθ)^2)tanθ = kq1q2/(4L^2 *mg). How can I solve for θ here when I have both sinθ and tanθ in the equation?

 

[andrewkirk]

There's no easily apparent route to an analytic solution, but we have numeric values for all the other variables, so we can just use a numeric equation solver, like Wolfram Alpha, to find the value of theta.

 

[zachem62]

There's no easily apparent route to an analytic solution, but we have numeric values for all the other variables, so we can just use a numeric equation solver, like Wolfram Alpha, to find the value of theta.

That sucks. I'm literally at the last step, having just the angle to solve for and its a dead end. I just wish there was a way to solve this because I don't think I can reference wolfram alpha when I submit my assignment.

 

[haruspex]

That sucks. I'm literally at the last step, having just the angle to solve for and its a dead end. I just wish there was a way to solve this because I don't think I can reference wolfram alpha when I submit my assignment.

You can probably get a decent approximation by replacing both sin(θ) and tan(θ) with θ (in radians).