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

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

Two small spheres with mass $m=15.0 \mathrm{~g}$ are hung by silk threads of length $L=1.20 \mathrm{~m}$ from a common point (Fig. $\mathrm{P} 21.60$ ). When the spheres are given equal quantities of negative charge, so that $q_{1}=q_{2}=q,$ each thread hangs at $\theta=25.0^{\circ}$ from the vertical. (a) Draw a diagram showing the forces on each sphere. Treat the spheres as point charges. (b) Find the magnitude of $q$. (c) Both threads are now shortened to length $L=0.600 \mathrm{~m},$ while the charges $q_{1}$ and $q_{2}$ remain unchanged. What new angle will each thread make with the vertical? (Hint: This part of the problem can be solved numerically by using trial values for $\theta$ and adjusting the values of $\theta$ until a selfconsistent answer is obtained.)

Key Concepts

Geometric Analysis in Static Equilibrium

The geometry of the system, including angles and distances, plays a key role in linking the forces through trigonometric relationships. In the pendulum problem, the angle of the thread with the vertical and the length of the thread determine the separation between the spheres, which is then used in Coulomb's law to compute the electrostatic force.

Coulomb's Law

This concept describes the electrostatic force between two point charges. According to Coulomb's law, the force is directly proportional to the magnitude of the charges and inversely proportional to the square of the distance between them. It provides the basis for calculating the repulsive force that acts between the charged spheres in the system.

Force Equilibrium

In a static system, the sum of all forces acting on an object must equal zero. In this context, the balance involves gravitational forces, tension in the supporting threads, and the electrostatic repulsive forces. Understanding how these forces balance is crucial to determining the system's configuration when the spheres are in equilibrium.

Resolution of Forces

Resolving forces into their horizontal and vertical components is essential when dealing with systems at an angle. For the pendulum-like setup, the tension in the thread must be separated into components that counteract both gravitational force (vertical) and the electrostatic force (horizontal). This technique simplifies the analysis of equilibrium by applying Newton's laws to each direction independently.

Numerical Methods for Solving Nonlinear Equations

Often, complex equilibrium problems lead to nonlinear equations that do not have closed-form solutions. In these cases, numerical methods such as the trial-and-error technique or iterative methods are employed to estimate the unknown quantities—in this problem, the angle resulting from a change in the length of the threads.

Transcript

0:00 Hello everyone.

00:01 In this problem we are asked to find the charge, to draw the diagram and find the charge on two spheres that are hung from a given point on a ceiling or something, an experimental flat surface, and they're hung by threats of equal length, and they're charged to equal charges of hue.

00:27 They both have the same mass, 15 grams or 15 times the minute's 3 kilograms, and initially the length of the thread is 1 .20 meters.

00:38 In this setup, the charges make a 25 degree angle with the vertical, and we're told that we were told to find or draw the free -bated diagrams for the object.

00:52 So on the left, i draw the free -bated diagram for the left charge and then the right for the right charge.

00:57 So on the left, we have that the left -hand charge is feeling a cool and force repelling it to the left, and it's got a tension that is acting along the direction of the rope or the thread, which is making a 25 -degree angle with the vertical.

01:13 And it's got the weight, its weight pulling down.

01:16 Now, weight, we don't have to distinguish between the left and the right values or the left -and -the -right charged three years because they have the same mass.

01:24 So on the right, instead we have that the tension in the right is pulling along the second thread or the right thread.

01:33 The coolant force is again repelling.

01:35 So now the second to the right and the mass or the weight is also pulling down just like in the first case.

01:42 I notice that because they have the same mass and the same charge and the same length of the thread, this is a highly symmetric situation.

01:50 And what this allows us to do is to essentially just use one.

01:54 Of the equations for only one of the charges to find all the values that we need.

01:58 So let's first find the separation between the two charges as a function of all the other variables.

02:06 So given that we know the length and we know the angle, we can actually work out what half of the separation is by drawing this right -handed triangle, where the opposite to the 25 -degree angle is d over 2 and the hypotenuse is l.

02:20 So then the sine theta is equal to opposite over hypotenuse.

02:23 So it's due d over 2 over l.

02:26 And that simplifies the d over 2l, and that allows us to find that d is equal to 2l times sine theta.

02:32 Now we can combine this with the fact that the coulom force is always going to be given by q1 times q2 over 4 pi epsilon noth times r squared, where r is the separation between the charges, and q1 and q2 are the charges of the object.

02:44 We can rearrange this, or we can plug in our pan value for d.

02:49 So applying this to our situation where q1 is equal to q2 and r is equal to d, we find that the coulomb force in both of these, or the magnitude of the coolum force on both of these charges, is going to be q squared over 4 pi epsilon not times d squared.

03:02 This substituting in the value or this expression for d that we have found is equal to q squared over 16 pi epsilon knot l squared times sine squared theta.

03:12 Okay, so now that we have the freeway diagrams and we are there of all the variables in the problem, we're asked to find what the value of q is.

03:21 So first, we'll look at the equations of motion for the left -hand charge, let's say.

03:27 It doesn't matter which when you choose.

03:29 The equations are going to be the same...

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