Two small metallic spheres, each with a mass of 0.20 g, are suspended as pendulums by light stri

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

Key Concepts

Coulomb's Law

Coulomb's law describes the electrostatic interaction between two charges. It states that the force between them is directly proportional to the product of the charges and inversely proportional to the square of the distance between their centers. This fundamental principle is essential in calculating the repulsive force between the charged spheres.

Equilibrium of Forces

The concept of equilibrium is central to the analysis of the system, where the net force on each sphere is zero. This means that the gravitational force is balanced by the horizontal component of the tension in the string (which in this case is provided by the electrostatic repulsion) and the vertical component balances the weight of the spheres. Setting up these equilibrium conditions allows for solving unknown quantities.

Decomposition of Forces

Decomposition of forces involves breaking a force vector into its components, typically along perpendicular axes. For a pendulum, the tension in the string is resolved into vertical and horizontal components using trigonometric functions. This decomposition is crucial for analyzing how different forces act to maintain equilibrium.

Pendulum Mechanics

Pendulum mechanics involves understanding the motion and forces acting on a pendulum system. In static equilibrium, the analysis focuses on how gravitational force, tension in the string, and any additional forces such as electrostatic repulsion interact. This framework is essential for solving problems involving swinging masses under various forces.

Trigonometric Relationships

Trigonometry is used to relate the angles of the system to the force components. In this context, sine and cosine functions help calculate the contributions of the tension force in the horizontal and vertical directions relative to the pendulum’s angle, facilitating the setup of equations representing the equilibrium condition.

Transcript

00:01 So here's a problem where we have two objects that are being suspended at the same point on the ceiling.

00:14 They have a mass of 0 .2 grams, which is 0 .0 .002 kilograms.

00:25 And the length of the strings is 30 centimeters and the angle that they make with the vertical is 5 degrees and they're they're both charged we just don't know what the charge is but we're trying to figure out what is the charge ultimately so what is the charge on each sphere so there's a way that we can do this it does take a little bit of trigonometry but we can go through it together so the first thing is you're going to want to start off by looking at the force diagram.

01:01 So if i just pick one of those, let's pick this left one here.

01:04 It's going to feel a force of mg going down or the weight of it.

01:09 It's going to feel a force of tension going up.

01:12 And it's going to feel a repelling force from electricity or from the coolum force.

01:19 And it's in equilibrium.

01:22 So here's the key here is that.

01:24 This force is going to equal the electrostatic force and then this force here is going to equal the weight.

01:33 So what i'm going to do is to make things easier, i'm just going to focus on this right triangle.

01:39 Because those are equal, i'm going to redraw it like this.

01:42 The tension, the electrostatic force, even though it is going in the opposite direction, they're equal to each other, so i'm just going to label it with that.

01:51 And then this is mg and of course the same thing is true.

01:54 It's going in the opposite direction, but because they're equal, i'm just going to call it that because it's the same magnitude.

01:59 And i know this angle here is 5 degrees.

02:03 So what we can do is, ultimately what we're looking for is fe, because then that fe will lead us to the charge.

02:12 We can figure out what the weight is, and by using trig, we can figure out what fe is.

02:17 So let's start with mg.

02:19 So fg is equal to mg, and the mass is 0 .2 times 10 to negative 3.

02:26 That's converting from grams to kilograms times 9 .8.

02:32 So when i put that in the calculator, 0 .2 times 10 and negative 3 times 9 .8, i get a force of 0 .00196 newton.

02:45 So that's what this value is going up there.

02:51 So now we have a side and we have an angle.

02:54 By doing both, by using these, we can actually find the other side.

02:57 So if i have tan theta equals the opposite over the adjacent, i just scroll down a little bit, i can figure this out.

03:08 So tangent of the angle 5 degrees, opposite is going to be fe, and the adjacent is mg, which we just found out was .00196.

03:21 So now let's go ahead and plug this into the calculator.

03:24 So that times tangent of 5 degrees.

03:28 Is going to give us an electrostatic force of 1 .715 times 10 to the negative 4 newton's...

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