Two spheres are mounted on identical horizontal springs and rest on a frictionless table, as in the drawing. When the spheres are uncharged, the spacing between them is 0.05 m, and the springs are unstrained. When each sphere has a charge of +1.6 £gC, the spacing doubles. Assuming that the spheres have a negligible diameter, determine the spring constant of the springs?
3. Three charges are fixed to an x, y coordinate system. A charge of +18 £gC is on the y axis at y = +3.0 m. A charge of -12 £gC is at the origin. Last, a charge of +45 £gC is on the x axis at x = +3.0 m. Determine the magnitude and direction of the net electrostatic force on the charge at x = +3.0 m. Specify the direction relative to the ¡Vx axis.
4. A particle has a charge of +1.5 £gC and moves from point A to point B, a distance of 0.20 m. The particle experiences a constant electric force, and its motion is along the line of action of the force. The difference between the particle¡¦s electric potential energy at A and B is EPEA - EPEB = +9.0ƒù10-4 J.
a. Find
To determine the spring constant of the springs in question 1, we can use Hooke's law which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.
Let's consider the situation when the spheres are uncharged. In this case, the spacing between them is 0.05 m, and the springs are unstrained. This means that the force exerted by each spring on the spheres is zero, as they are in equilibrium.
Now, when each sphere has a charge of +1.6 μC, the spacing between them doubles. This change in spacing can be attributed to the repulsive electrostatic force between the charged spheres. Let's denote the spring constant as k.
At the new equilibrium position, the repulsive force between the spheres is balanced by the restoring force exerted by the springs. We can equate these forces to find the value of k.
The electrostatic force between two charges can be calculated using Coulomb's Law, which states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
From the given information, we can calculate the electrostatic force between the spheres when they are charged:
F = k_e * (q1 * q2) / r^2
Here, k_e is the electrostatic constant (which has a value of 8.99 * 10^9 N m^2/C^2), q1 and q2 are the charges (+1.6 μC each), and r is the distance between them when charged (0.1 m, double the original spacing).
Now, we can set this electrostatic force equal to the spring force:
k_e * (q1 * q2) / r^2 = k * (d - r)
Here, d is the original spacing between the spheres (0.05 m).
Simplifying the equation, we get:
k = k_e * (q1 * q2) / (d - r)
Plugging in the values, we get:
k = (8.99 * 10^9 N m^2/C^2) * ((1.6 * 10^-6 C)^2) / (0.05 m - 0.1 m)
k = 2.304 * 10^5 N/m
Therefore, the spring constant of the springs is approximately 2.304 * 10^5 N/m.
For question 3 and 4, additional information is required to calculate the net electrostatic force and the difference in electric potential energy.