Calculating Charge of Two Spheres in Equilibrium

[Pseudopro]

Homework Statement

Two plastic spheres each of mass 100.0 mg are suspended from very fine insulating strings of length 85 cm. When equal charges are placed on the spheres, the spheres repel and are in equilibrium when 10 cm apart.
(a) What is the charge on each sphere?

Homework Equations

F=kq²/r²

The Attempt at a Solution

I assumed that the strings were attached to the same place at the top. This eventually gave tan theta = kq²/mgr².
I did sin^-15/85 to get theta
I then plugged in all the values and got around 2.5x10^-7
The book answer is 8x10^-9.

Thanks for your time

 

[rl.bhat]

Show your substitution and calculation.

 

[ileacus]

I am hesitant to mention this, because when I follow this method, I arrive at a different answer than the given book answer. That said, I would analyze it like so:

Each sphere is repelled by an electric force that is sufficient to balance the gravitational force when the sphere is deflected 5 cm.

Draw a diagram - the spheres are connected to the ceiling by 85 cm strings and are displaced a total of 10cm (5cm each from vertical). What is the height difference from their displaced position and their straight down position? How much energy does that require?

 

[rl.bhat]

Take the projection of L on the vertical line.
Difference in height h = L - Lcosθ.

 

[rwisz]

and effort in providing a solution to this problem.

First of all, great job on attempting to solve the problem and showing your work! However, it seems like you may have made a mistake in your calculations. Let's take a closer look at the problem and see if we can find the correct solution.

We are given two plastic spheres, each with a mass of 100.0 mg. They are suspended from insulating strings of length 85 cm, and when equal charges are placed on the spheres, they repel and are in equilibrium when they are 10 cm apart. From this information, we can assume that the spheres are symmetrical and that the strings are attached to the same point at the top.

To solve for the charge on each sphere, we can use the equation for electrostatic force:

F = k * (q1 * q2)/r^2

Where k is the Coulomb constant, q1 and q2 are the charges on each sphere, and r is the distance between the spheres (10 cm in this case). Since the spheres are in equilibrium, we know that the electrostatic force between them is equal to the force of gravity pulling them down. We can express this as:

F = mg

Where m is the mass of each sphere and g is the acceleration due to gravity (9.8 m/s^2).

Now, we can set these two equations equal to each other and solve for the charge on each sphere:

k * (q1 * q2)/r^2 = mg

Plugging in the given values, we get:

(8.99 * 10^9 N*m^2/C^2) * (q^2)/ (0.1 m)^2 = (0.0001 kg) * (9.8 m/s^2)

Solving for q, we get:

q = 8.0 * 10^-9 C

Therefore, the charge on each sphere is 8.0 * 10^-9 C, which is the same as the book answer.

I hope this explanation helps and clarifies any confusion. Keep up the good work!