Two small metallic spheres, each of mass 0.20 g, are suspended as pendulums by light strings fr

Two small metallic spheres, each of mass 0.20 g, are...

Key Concepts

Trigonometry in Force Analysis

Trigonometry is used to relate the angles in the physical setup to the components of forces and distances. In pendulum problems, the angle of the string with respect to the vertical helps determine the horizontal separation between objects and the components of the tension force. This approach is essential for setting up the equations that govern the system's equilibrium.

Vector Decomposition

Vector decomposition involves breaking down a force into its components, typically along perpendicular axes. In problems involving pendulums, the tension force in the string is often resolved into horizontal and vertical components. This method is crucial for separately analyzing the forces due to gravity and electrostatic repulsion.

Static Equilibrium

Static equilibrium occurs when all the forces acting on an object balance each other out, resulting in no net force and no acceleration. In the context of the charged pendulums, the condition of equilibrium implies that the combined effects of gravitational forces, tension in the strings, and the electrostatic repulsion force are perfectly balanced.

Electrostatic Force

The electrostatic force is the force of attraction or repulsion between electrically charged objects. In problems involving charged pendulums, this force arises due to the like charges on the objects, causing them to repel each other. Analyzing the magnitude and direction of this force using Coulomb's Law is key to determining the behavior of the system.

Coulomb's Law

Coulomb's Law is a fundamental principle in electrostatics that quantifies the electrostatic force between two point charges. It states that the magnitude of the force between the charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. This concept is essential for understanding the repulsive or attractive interactions in charged systems.

Transcript

00:01 The electrical force is going to come, the expression for it is going to come from kulum's law.

00:10 And that gives a magnitude as well as a direction.

00:15 The magnitude of an electrical force is the electrical constant, we'll call it k, times the absolute value of two charges interacting, and then divided by their separation squared.

00:29 The direction of the force comes from looking at the signs and telling whether there's repulsion or traction.

00:40 And that force always acts along a line joining the two charges.

00:45 So as an example of this, here's a situation where two small objects, we'll call them spheres, have been charged up to the same amount.

00:56 We don't know whether they're positive or negative, but they are rebrand.

01:00 Repelling each other, and we would like to find the charge on each one.

01:06 So what we can do is, of course, write that the magnitude of the force is k, little q squared, and the separation we can get from the picture is going to be related to the angle at which they're hanging.

01:27 So we could kind of look at a small triangle in there and get our, over 2, r over 2 divided by, yeah, r over 2 is equal to .3 meters times the sign of the hanging angle, the length of the string, times the sign of 5 degrees, and therefore we can get their separation in terms of meters.

02:07 That is a fairly easy thing to do.

02:09 Let's see here.

02:28 And we get 0 .03, 523 meters.

02:40 Okay, so we do have an r, which we can put into this for r squared.

02:49 But in order to solve for the charge, we must figure out the electrical force.

02:54 And to do that, we are going to use a little bit of equilibrium.

02:59 In equilibrium, newton's second law says that the sum of forces, is equal to zero.

03:08 And so we'll draw a little force diagram on one of the charges.

03:13 It does not matter which one.

03:15 They will have very similar pictures.

03:19 So we'll choose the one on the right.

03:21 What we have is the weight of the small sphere, acting downwards.