Deriving Pith Ball Mass from Coulomb's Law: A Scientific Exploration

[kashika1212]

Homework Statement

One pith ball is mounted on an insulating stick and the other is hung on a string from the ceiling. The two balls are brought together such that they are barely touching, the initial tension is only vertical. Upon introducing a charged object to the stationary pith ball, charge is equally distributed to both of the balls. The hanging ball swings out from the mounted ball and eventually comes to rest. Your task is to determine the magnitude of charge on the balls given the following measurements:
-A pith ball has a diameter of 40mm
-The hanging ball is hung from the ceiling with a 147.3cm long string. Note: this measurement is to the top of the pith ball, not to the ball's center.
-After charging, the two pith balls have a center-center displacement of 6.40cm

Relevant Equations

$F_{c} = \frac{\left ( k\left ( q_{1} \right )\left ( q_{2} \right ) \right )}{r^{2}}$
$F_{g} = m*g$

I actually found most of it out I'm just struggling with how to find the mass of the balls. I'm not sure how you would do that since could only derive two equations from the information given or are we assuming the mass is so small that it doesn't matter?

Q = charge of one pith ball
d = distance between the COM of the two balls
L = length of the string plus the radius of the ball
h = change in height after the swing (though I suppose this is probably negligible)
m = mass of pith ball

$\frac{F_{c}}{F_{g}} = \frac{d}{2} * \frac{1}{L-h}$
$F_{c} = \frac{d}{2} * \frac{1}{L-h} * F_{g}$
$\frac{d*m*g}{2(L-h))} = \frac{k*Q^{2}}{d^{2}}$
√$\frac{d^{3} * m * g}{2(L-h) * k)}$ = Q

Any help would be appreciated. Thanks in advance!

 

[haruspex]

Do you think that it could be solvable without knowing the mass?

 

[kashika1212]

I tried going about solving it without having mass in my equations but I always ended up having to find it to get Q. If there is a way to solve it without knowing the mass than I've never been taught it. If you do know a way to do it without mass could you provide some hints/guidance?

 

[haruspex]

I tried going about solving it without having mass in my equations but I always ended up having to find it to get Q. If there is a way to solve it without knowing the mass than I've never been taught it. If you do know a way to do it without mass could you provide some hints/guidance?

Seems to me you don't even need to try to solve it. It should be evident that the greater the mass (for a given charge) the less the angle, so knowing the mass is essential.

 

[hutchphd]

But you could go ahead and find the charge to mass ratio for the pith ball. That will teach you the physics. You should also feel good that you knew something was missing. Perhaps you can look up the density of pith since you know the ball size.

 

[haruspex]

find the charge to mass ratio for the pith ball

Not quite. That could be done if the stationary ball had a known fixed charge, but here they both have charge Q, meaning m will be proportional to Q², as in the equation obtained.

 

[kashika1212]

Thank you all so much, I thought I was going crazy cause I couldn't figure out how to do it without mass. So is there no way to derive the mass either since density of the pith ball is not known, meaning the problem did not give enough information?

 

[hutchphd]

That would be my assessment, too. Whether we are all crazy is a different matter...