Two charged spheres repel (attached to strings)

[FrogPad]

It's been awhile since I've had physics I, so this problem is giving me a headache.

Q) Two very small conducting spheres, each of a mass $1.0 \times 10^{-4}\,\,(kg)$, are suspended at a common point by very thin nonconducting threads of a length $0.2 \,\,(m)$. A charge $Q$ is placed on each sphere. The electric force of repulsion separates the spheres, and an equilibrium is reached when the suspending threads make an angle of $10 \,\, (deg)$. Assuming a gravitational force of $9.80 \,\, (N/kg)$ and a negligible mass for the threads, find $Q$.

My Work)
We first will deal with:
$$\vec F_{12} = \frac{\hat R_{12} k q_1 q_2}{R^2_{12}}$$
$$\sum \vec F_i = m \vec a$$
Since equilibrium is reached, $\vec a = \vec 0$. Thus,

$$\sum F_i = \vec T_2 + \vec F_G + \vec F_{12} = \vec 0$$

We now find the forces.

$$\vec F_G = -\hat y (9.8 \times 10^{-4})$$

Setting up the coordinate system we assume the orgin as at the point of interesection of the two threads. Thus, a vector that points to sphere-two is:
$$\vec S_2 = \hat x(0.2 \sin 5^{\circ}) - \hat y(0.2 \cos 5^{\circ})$$

The vector quantities for coloumbs law are as follows:
$$\vec R_{12} = \hat x (2(0.2\sin 5^{\circ}))$$
$$R = 0.4 \sin 5^{\circ}$$
$$\hat R = \hat x$$

Thus, since the spheres have an equal charge
$$\vec F_{12}=\frac{\hat x k Q^2}{0.16 \sin^2 5^{\circ}}$$

Now I know the tension has to exert a force that holds the sphere in place, so gravity and the electric repulsion keep it from moving away. So do I just say that $\vec T_2 = -\vec S_2$?

I'm not really sure what to do. Is what I'm doing even correct?

thanks in advance

 

[FrogPad]

I figured it out

 

[kyle8921]

Hello,

It seems like you are on the right track with your calculations. However, I would recommend double checking your equations and units as there are a few errors. For example, the gravitational force should be in Newtons (N) not Newtons per kilogram (N/kg). Also, the distance between the spheres should be 0.4 meters, not 0.2 meters. Once you correct these errors, you can solve for the unknown charge Q using the equation you have set up for equilibrium.

Additionally, I would suggest using the vector notation for the electric force instead of just the magnitude. This will give you a more accurate representation of the forces acting on the spheres. And yes, you are correct in saying that the tension in the strings will be equal and opposite to the vector sum of the electric and gravitational forces.

I hope this helps and good luck with your problem!