Two spheres are attached to two identical springs and
separated by 8.0 cm, as in Figure 14. When a charge of
2.5 106 C is placed on each sphere, the distance
between the spheres doubles. Calculate the force constant
k of the springs.
The forces being used in this scenario are the:
-electric force (Fe)
-spring force (Fs)
In each sphere, since the other is pushing away from it (due to the like charge), and since the spring is trying to restore equilibrium by pushing the sphere back, the equation will be Fnet = Fe - Fs.
Fnet = Fe - Fs
Fnet = 0 (the sphere's are stationary)
Fs=Fe
-kx = K(q1)(q2) / r^2
-kx = K(q)^2 / r^2
x is negative because it is being compressed (as stated by Hooke's law)
x= 0.04m (because this is how many meters each spring retracts back)
q1=q2= 2.5 * 10^-6 C
K= 9 * 10^9
r= 0.16 (the new distance between the two sphere)
-k(0.04) = 9 * 10^9 (2.5 * 10^-6 C)^2 / (0.16)^2
k = 54.9316 N/m
k= 55 N/m
from the center each sphere moves 8 - 4 = 4 cm = .04 m
that is how far each spring is compressed.
F = k x = 0.04 k
now what is the electrostatic force when the spheres are 16 cm = 0.16 meters apart?
F = [ 9*10^9 ] Q1Q2/(0.16)^2
so
0.04 k = [ 9*10^9] [ 2.5*10^6] [2.5*10^6] / [ 1.6^2*10^-2 ]
Those charges are huge. Typo?
Why did the spheres go to couples therapy?
Because they wanted to work on their spring-tuality!
Now, let's calculate the force constant.
Given that the initial separation is 8.0 cm and it doubles when a charge of 2.5 x 10^6 C is placed on each sphere, we can say that the final separation is 2 times the initial separation, which is 16.0 cm.
The force constant (k) is related to the separation distance by the formula:
F = k * x
where F is the force and x is the separation distance.
Initially, the force between the spheres is given by:
F1 = k * x1
After the charges are placed on the spheres, the force becomes:
F2 = k * x2
where x2 is the final separation distance.
Since the force is directly proportional to the distance, we can say:
F2 = 2 * F1
Substituting the values, we get:
k * x2 = 2 * (k * x1)
k * 16.0 cm = 2 * (k * 8.0 cm)
16.0 k = 16.0 k
Solving for k, we find:
k = 1
So, the force constant of the springs is 1 N/m. However, keep in mind that this is a simplified calculation and not reflective of real-life scenarios.
To calculate the force constant, we need to use Coulomb's law to relate the electric force between the spheres to the force exerted by the springs.
Coulomb's law states that the electric force between two charged objects is given by:
F = k * (q1 * q2) / r^2
Where F is the electric force, q1 and q2 are the charges on the two spheres, r is the distance between them, and k is the electrostatic constant.
In this case, when the charge on each sphere is 2.5 * 10^(-6) C, the distance between the spheres doubles. Therefore, the new distance is 2 * 8.0 cm = 16.0 cm = 0.16 m.
The electric force between the spheres is equal to the force exerted by the springs:
F = k * (2.5 * 10^(-6) C)^2 / (0.16 m)^2
To find the force constant k, we need to rearrange the equation to solve for k:
k = F * (0.16 m)^2 / (2.5 * 10^(-6) C)^2
Now we can substitute the values and calculate the force constant:
k = F * (0.16 m)^2 / (2.5 * 10^(-6) C)^2
To solve this problem, we need to use Coulomb's law, which states that the force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
Let's break down the problem step by step:
1. Given that the charge placed on each sphere is 2.5 * 10^6 C.
2. When a charge is placed on each sphere, the distance between them doubles. This means that the initial separation of 8.0 cm becomes 16.0 cm.
3. We can calculate the force before and after the separation doubles using Coulomb's law formula:
F = (k * q1 * q2) / r^2
Where:
F is the force between the charged objects,
k is the force constant of the springs,
q1 and q2 are the charges on the spheres,
and r is the distance between the spheres.
4. Before the separation doubles, the force between the spheres can be calculated as:
F1 = (k * (2.5 * 10^6 C) * (2.5 * 10^6 C)) / (8.0 cm)^2
5. After the separation doubles, the force between the spheres becomes:
F2 = (k * (2.5 * 10^6 C) * (2.5 * 10^6 C)) / (16.0 cm)^2
We know that the force decreases because the separation increases.
6. The problem states that the force constant of the springs is expected to be identical for both scenarios. Therefore, F1 should be equal to F2.
(k * (2.5 * 10^6 C) * (2.5 * 10^6 C)) / (8.0 cm)^2 = (k * (2.5 * 10^6 C) * (2.5 * 10^6 C)) / (16.0 cm)^2
7. We can cancel out the common terms:
(k * (2.5 * 10^6 C) * (2.5 * 10^6 C)) / (8.0 cm)^2 = (k * (2.5 * 10^6 C) * (2.5 * 10^6 C)) / (16.0 cm)^2
k / (8.0 cm)^2 = k / (16.0 cm)^2
8. Since the k values are identical, we can cancel them out as well:
(2.5 * 10^6 C) / (8.0 cm)^2 = (2.5 * 10^6 C) / (16.0 cm)^2
(2.5 * 10^6 C) * (16.0 cm)^2 = (2.5 * 10^6 C) * (8.0 cm)^2
9. Now, we can solve for k:
k = [(2.5 * 10^6 C) * (16.0 cm)^2] / [(2.5 * 10^6 C) * (8.0 cm)^2]
k = (16.0 cm)^2 / (8.0 cm)^2
k = 4.0
Therefore, the force constant of the springs, k, is 4.0.