CP Two small spheres with mass m=15.0 g are hung by silk threads of length L=1.20 m from a commo

CP Two small spheres with mass m=15.0 g are hung by silk...

Key Concepts

Coulomb's Law

Coulomb's law describes the force between two point charges. According to this law, the magnitude of the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This electrostatic force is repulsive if the charges are of the same sign and attractive if they differ, and it is essential for quantifying interactions in charged-particle systems.

Gravitational Force

Gravitational force is the attractive force that acts between any two masses, with the strength proportional to their masses and the universal gravitational constant. In problems involving suspended objects, gravitational force is typically represented as the weight, which is the product of the mass and the acceleration due to gravity, acting vertically downward.

Tension in the Suspension Strings

The tension in a string or thread is the force exerted along its length, balancing other forces acting on a suspended object. In systems where objects are hanging, the tension must counteract the gravitational force and any additional forces such as those arising from electrostatic interactions, with its components determined by the angle the string makes with the vertical.

Static Equilibrium

Static equilibrium occurs when all forces and torques acting on a system are balanced, resulting in no net acceleration. For a suspended charged sphere, this means that the downward gravitational force, the horizontal electrostatic repulsion, and the tension in the string combine such that the net force in both the horizontal and vertical directions is zero. This condition is crucial for setting up the equations that determine quantities like the charge or the equilibrium angle.

Trigonometric Decomposition

Trigonometric decomposition involves breaking vectors into their horizontal and vertical components using sine, cosine, and other trigonometric functions. This is essential in analyzing forces acting at angles, as it allows the separation of forces into components that can be independently balanced using the equilibrium conditions.

Geometric Relations in Pendulum Systems

The geometry in pendulum configurations, such as the length of the strings and the angle they make with the vertical, directly affects the distance between suspended objects and the components of the forces acting on them. Understanding these geometric relationships is vital for relating the physical dimensions of the setup to the forces and angles determined by static equilibrium conditions.

Transcript

00:01 In this question, we have two small spheres, each with the mass of 15 grams, hung from silk threads of length 1 .2 meters from a common point.

00:11 The spheres are given an equal quantity of negative charge, and the thread hangs at a 25 -degree angle from the vertical.

00:19 So in part a, we're asked to draw a diagram showing the force on each sphere.

00:26 In part b, we're asked to find the magnitude of the charge, and then in part c, we're going to shorten the length of the strings and calculate the consequential angle from there.

00:44 So let's go ahead and start with part a.

00:46 We'll draw a diagram of this.

00:49 So we've got the common points that the threads are coming from, and then we've got two charges.

00:58 One on each string and each string makes an angle of 25 degrees, hard to see, 25 degrees with the vertical.

01:16 And let's go ahead and draw the forces in on these charges.

01:20 So each one is going to have a tension force along the rope and obviously a gravitational force acting straight down as well.

01:35 And then because they have the same type of charge, so they both have a negative charge, we're told, there's going to be a repulsive force between them.

01:47 So that's going to look something like this.

01:50 So it's going to go to the right on the right hand charge and then to the left, on the left hand charge.

01:56 So they essentially have the same forces on acting on both, just a mirror image of each other.

02:03 So we just need to analyze the forces on one of the charges, in order to properly complete this calculation.

02:13 So to start in on trying to calculate the charge, the magnitude of the charge, which is the question in part b, we're going to do an analysis of forces.

02:28 So i'm going to use, let's see, i'm going to use up as my positive y direction and to the right as my positive x direction.

02:42 And i'm going to analyze the forces on the right -hand charge.

02:46 So this object is in equilibrium.

02:51 It's not moving.

02:53 And so we know that the net force in both the x and y directions is equal to zero.

02:59 I'm going to start by analyzing the x direction.

03:03 So in the x direction, we have f .e.

03:07 And in the negative extraction, we have the x component of ft, and that's going to equal zero.

03:17 So what that gives us right off the bat is f .e is equal to ftx.

03:27 And analyzing the triangle here, we can see that ftx is going to be equal to ft sine 25.

03:45 Okay, so that's our first equation.

03:47 And then we can analyze the y direction.

03:51 So f -nat -y is going to be equal to zero as well.

03:56 So that's going to yield f -ty minus f -g is equal to zero.

04:02 And f -t -y is going to be cos -25.

04:10 And taking the f -g over to the other side, we get f -t -cose -25 is equal to f -g.

04:18 So that actually allows us to calculate what ft is because we know fg is just going to be mg.

04:27 So mg over cos 25.

04:31 Make sure that we're converting the 15 to 15 grams for the mass into kilograms.

04:36 So it's going to be 0 .01 kilograms times 9 .81 meters per second squared divided by coase 25.

04:47 And so ft here is equal to 0 .162 newtons.

05:04 Now that's great news because we can use that in the first equation that we derived to find f .e.

05:12 So 0 .162 times sine of 25 degrees will give us f .e.

05:20 So f .e itself is 0 .062.

05:26 8 to 6 newtons.

05:29 So now that we have f .e., we can use that to calculate the charge.

05:35 So we know that the electric force between two charges is going to be k, q1, q2.

05:41 Since the cues are the same, in this case, we can just replace that with q squared.

05:47 And that's going to be divided by the separation between the two charges.

05:54 So we can rearrange this for q.

05:57 We're going to get f .e times k.

06:03 Sorry, f .e.

06:06 Times r squared, divided by k.

06:11 And then we can take the square root of each side.

06:15 That's going to give us q is equal to the square root of all of this.

06:20 Now, before we go ahead and calculate that, let's just take a look at what the separation between the charges is.

06:29 So if we know that the length of one of these strings is 1 .2 meters, then the distance from the center to one charge is going to be 1 .25, sorry, 1 .20 times the sign of 25.

06:54 So to get the total distance between the two charges, we would just multiply that by two.

07:03 So the distance between the two charges is going to be 1 .029 meters.

07:22 Sorry, apologies...

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