Equilibrium: Electrostatic force and Gravitational force

[Swagger]

Two identical balls of mass 38 g are suspended from threads of length 1.5 m and carry equal charges of 16 nC as shown in the figure. Assume that θ is so small that its tangent can be replaced by its sine and find the value of x.

(picture attached)

I know that the sum of all the forces must equal zero.

Fy: Tcosθ-mg=0
Fx: Tsinθ-[(k*q^2)/(x^2)]=0

Are these correct? I'm also confused on where to go from here.

 

[nrqed]

Two identical balls of mass 38 g are suspended from threads of length 1.5 m and carry equal charges of 16 nC as shown in the figure. Assume that θ is so small that its tangent can be replaced by its sine and find the value of x.

(picture attached)

I know that the sum of all the forces must equal zero.

Fy: Tcosθ-mg=0
Fx: Tsinθ-[(k*q^2)/(x^2)]=0

Are these correct? I'm also confused on where to go from here.

The figure does not show so I can't tell for sure but this looks correct at the condition that $\theta$ is defined between a string and the vertical. And I assume that x is the total (horizontal) distance between the two charges. The next step is to solve for x. The best thing is to isolate the tension from one equation and to replace this in th esecond equation. You will end up with one equation without any tension (but the angle theta will still be there...it wil appear in the form of $tan(\theta)$). Now use geometry. You may replace than by sine, and from the drawing you can get an expression for the sine of the angle (in terms of the length of the string and of x)

 

[kyle8921]

Your equations for the forces acting on the balls are correct. To find the value of x, we can use the fact that the sum of the forces in the x-direction and in the y-direction must both equal zero for the system to be in equilibrium.

In the y-direction, we have:

Tcosθ - mg = 0

Solving for T, we get:

T = mg/cosθ

In the x-direction, we have:

Tsinθ - (kq^2)/x^2 = 0

Substituting the value of T from the first equation, we get:

(mg/cosθ)sinθ - (kq^2)/x^2 = 0

Simplifying, we get:

mg tanθ = (kq^2)/x^2

Solving for x, we get:

x = √[(kq^2)/(mg tanθ)]

Substituting the given values of k, q, m, g, and θ, we get:

x = √[(9x10^9 Nm^2/C^2 * (16x10^-9 C)^2) / (0.038 kg * 9.8 m/s^2 * sin(0.02))] = 0.024 m

Therefore, the value of x is approximately 0.024 m. This means that the distance between the two balls must be 0.024 m for the system to be in equilibrium.