Two identical spheres with mass m are hung from silk threads...

Two identical spheres with mass $m$ are hung from silk threads of length $L$ (Fig. $\mathbf{P} 2 \mathbf{1 . 6 0}$ ). The spheres have the same charge, so $q_{1}=q_{2}=q .$ The radius of each sphere is very small compared to the distance between the spheres, so they may be treated as point charges. Show that if the angle $\theta$ is small, the equilibrium separation $d$ between the spheres is $d=\left(q^{2} L / 2 \pi \epsilon_{0} m g\right)^{1 / 3} .$ (Hint: If $\theta$ is small, then $\tan \theta \cong \sin \theta .)$

Key Concepts

Small-Angle Approximation

The small-angle approximation allows for the simplification of trigonometric functions when the angle is very small, typically approximating sin? and tan? by ? (in radians). This approximation is crucial in simplifying the equations derived from the force balance, making the derivation of the equilibrium distance more tractable.

Pendulum Mechanics

Pendulum mechanics describes how a mass suspended from a fixed point behaves under the influence of gravity and other forces. Here, it provides the framework for analyzing the suspended charged spheres, where the thread’s tension must be resolved into components that balance both the gravitational and the horizontal repulsive forces, leading to an understanding of the system's equilibrium configuration.

Force Equilibrium

Force equilibrium involves setting the net force on an object to zero, implying that all forces acting on the object balance out. In this problem, both the vertical (gravitational and tension components) and horizontal (electrostatic repulsion and tension components) forces must cancel, which leads to equations that determine the equilibrium separation between the spheres.

Gravitational Force

Gravitational force is the force exerted by the Earth on objects with mass. In this context, it represents the downward force acting on each sphere due to gravity, which must be balanced by the vertical components of other forces in the system for equilibrium.

Coulomb's Law

This is the principle that describes the electrostatic force between two point charges. It states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. In problems involving charged spheres assumed to be point charges, this law helps compute the repulsive force that acts to push the spheres apart.

Transcript

00:01 Hello, everyone.

00:02 In this problem, we're asked to show how the separation between two charges that are suspended through threads of length l, both of mass m and both of charge q, is related to the mass of the spheres or the charges and the length of the threads.

00:23 So here's a situation we have.

00:25 So there is some structure onto which you suspend two charged spheres.

00:30 Both have the same charge, they have the same mass, and they have the same length of fred by which they are suspended.

00:38 And so this results after some time in this equilibrium situation where you have the the charges rebel each other and they are subtending the same angle from the bisector of their distance.

00:54 So what's going to happen is you can look at the free body diet.

01:00 For these two objects.

01:02 So on the left, you have the left charge.

01:06 So on the left charge, the tension in the left rope is pointing up in this direction.

01:14 The mass or the weight of this charge is acting down, and the coolum force is repelling this force to the left.

01:22 On the right, you have that the right rope's tension is pointing up to the left.

01:26 The coolum charge is pushing the charge to the right and the weight of the right charge or the right sphere is also acting down.

01:34 So essentially you would be able to just look at one scenario here.

01:39 So it's enough to just consider the left -hand side or just the right -hand side because this is a very symmetric situation.

01:45 So in fact, that's exactly what you'll see.

01:47 So if we analyze the forces and we look at the sum of the forces in the wide direction, then we see that there is no net force, right, because the charges are not moving.

01:56 This is in equilibrium.

01:57 So the sum of the forces in the right direction is zero, and the sum of the forces acting in the wide direction is tl, so the left tension times cosine theta minus the mass or the weight of the left sphere.

02:13 So you can rearrange that to find what tl is.

02:15 And you find that tl is equal to mg over cosine theta, because wl, so the weight of the left sphere is just m times g.

02:24 The same happens on the right -hand side, essentially.

02:26 So if you look at the sum of the forces in the wide direction, you find that the right tensions times cosine theta minus wr is summing to zero, and then rearranging that, you find that the right tension is mg or cosine theta, which of course is exactly equal to what the left -hand side's tension was.

02:44 So if that's the same, then you can also set up the x directions to point in opposite directions, in which case the x forces or the net x force is also going to be the same.

02:56 Actually, you don't have to do that...

Please give Ace some feedback