# Two Charged Masses Suspended From Strings

• Geromy
In summary, the problem involves two similar charged masses hanging from silk threads and forming an angle θ. Using Coulomb's Law and trigonometry, it can be shown that the distance between the two masses at equilibrium is x = ((q2L)/(2∏ε0mg))1/3. The forces acting on the masses are the force due to charges and the force of gravity, which form a right triangle with the force of tension as the hypotenuse. By equating the forces and using trigonometric ratios, the final equation can be derived.
Geromy

## Homework Statement

Two similar tiny balls of mass m are hung from silk threads of length L and carry equal charges q. The angle formed by the two strings is bisected by an imaginary line, forming angle θ. Assume that θ is so small that tan θ can be replaced by its approximate equal, sin θ. The distance between the two charged masses, at equilibrium, is x. Show that, for equilibrium,

x = ((q2L)/(2∏ε0mg))1/3

## Homework Equations

Clearly, Coulomb's Law is a vital piece of this problem, and other sources that I've checked suggest such things as the constancy of the ratio of the sides of a triangle divided by the sine of that side's opposite angle and combining the electric force and gravitational force into the components of the force of tension.

## The Attempt at a Solution

Despite having a couple of ideas about how to start, I haven't made much progress. I noticed that there is an identical problem here (https://www.physicsforums.com/showthread.php?t=305517) that suggested the constant sine ratio rule, but I can't for the life of me get to the right answer. Any help would be appreciated.

Haha, so, I figured it out! I thought I would post here in case anyone wants an explanation.

Consider the forces acting on a single one of the charged masses: the force due to the charges is kq2/x2 - x is the radius between the two charges, since it's the distance they're separated by.

The force due to gravity is also pretty straightforward: it's simply mg.

Since these two forces are perpendicular, they form two sides of a right triangle. The hypotenuse of this triangle is equal to the final force acting on the charged mass, the force of tension. This triangle contains angle θ, since the force of tension is in the direction of the silk thread.

Since now we've reintroduced θ, we can start using trigonometry. Tan θ = Opp/Adj, which in this case is our force due to electric charge (F) over the force due to gravity (W), or F/W.

However, Tan θ = Sin θ, which equals x/2L

when you combine equations, you get 2kq2L = mgx3, and from here it's a simple matter of algebra to get to the final equation.

Well done!

Welcome to PF!

ehild

## 1. How do the charges in the masses affect each other?

The charges in the masses can either attract or repel each other, depending on their polarity. Like charges (positive-positive, negative-negative) will repel each other, while opposite charges (positive-negative) will attract each other.

## 2. What factors determine the strength of the force between the charged masses?

The strength of the force between the charged masses is determined by the magnitude of the charges and the distance between them. The larger the charges and the closer they are, the stronger the force will be.

## 3. Can the masses be suspended at any distance from each other?

No, the masses must be suspended at a specific distance from each other in order for the forces to be balanced. If the distance between the masses is too small, the forces will be too strong and the system will become unstable.

## 4. How does the length of the strings affect the motion of the masses?

The length of the strings can affect the motion of the masses by changing the tension in the strings. If the strings are shorter, the tension will be greater, causing the masses to move faster. If the strings are longer, the tension will be lower, causing the masses to move slower.

## 5. What happens if the charges in the masses are equal?

If the charges in the masses are equal, the forces between them will be balanced and the masses will remain stationary. This is known as electrostatic equilibrium.

• Introductory Physics Homework Help
Replies
19
Views
5K
• Introductory Physics Homework Help
Replies
21
Views
651
• Introductory Physics Homework Help
Replies
1
Views
2K
• Introductory Physics Homework Help
Replies
2
Views
2K
• Introductory Physics Homework Help
Replies
4
Views
3K
• Introductory Physics Homework Help
Replies
1
Views
2K
• Introductory Physics Homework Help
Replies
3
Views
3K
• Introductory Physics Homework Help
Replies
11
Views
1K
• Introductory Physics Homework Help
Replies
13
Views
1K
• Introductory Physics Homework Help
Replies
6
Views
891