Both threads are now lengthened such that L=1.00m, while the charges q1 and q2 remain unchanged. What new angle will each thread make with the vertical? Hint: you must solve for the angle numerically

I assume these are hanging from a "ceiling" from the same point.

The force of the charge keeps them apart, and part of gravity tends them together, and those forces at equilibrium have to be equal.

Frepulsion=kqq/d^2

force gravity=mg sinTheta
mgsinTheta=kqq/d^2

Now, looking at the string arrangement, sinTheta=(d/2)/1m

mg (d/2)=kqq/d^2

d^3=2kqq/mg

because it is a small angle,
sinTheta=Theta(inRadians)
then Theta= (d/2)=cubrt(2kqq/mg)

so to get the answer I do the inverse sine of d?

No, you compute cubrt(2kqq/mg) and that is the angle (in Radians)

oh ok thanks

To find the new angle each thread makes with the vertical, we can use trigonometry. Here are the steps to follow:

1. First, let's determine the forces acting on each thread. Each thread will experience two forces: the tension in the thread and the electrostatic force between the charges.

2. The tension in each thread can be determined using the equation T = mg, where T is the tension, m is the mass of the charges, and g is the acceleration due to gravity. However, since the length of the thread has been lengthened, m will also change. The new mass of the charges can be calculated using the equation m = q / g, where q is the charge of the particle and g is the acceleration due to gravity.

3. Now that we have the tension in each thread, we can analyze the forces acting on each particle. The electrostatic force can be calculated using the equation F = k * (q1 * q2) / r^2, where F is the electrostatic force, k is the Coulomb's constant, q1 and q2 are the charges, and r is the distance between the charges.

4. Since q1 and q2 remain unchanged, the electrostatic force will also remain the same.

5. Using these forces, we can now find the components of the tension in the thread along the vertical and horizontal directions. The vertical component will balance the weight of the charges, while the horizontal component will provide the centripetal force required to keep the charges in circular motion.

6. The vertical component of the tension can be calculated using T * sin(θ), where T is the tension and θ is the angle the thread makes with the vertical.

7. For the vertical component to balance the weight of the charges, it must be equal to mg. Therefore, we can set up an equation T * sin(θ) = mg and solve for the new angle θ.

8. To do this numerically, you can plug in values for the variables (m, g, T), and use trial and error or numerical methods (such as the Newton-Raphson method) to find the angle that satisfies the equation.

By following these steps and performing the necessary calculations numerically, you can find the new angle each thread makes with the vertical.