Two metallic spheres, each with a mass of 0.46 g are suspended as pendulums by light strings from a common point. They are given the same electric charge, and the two come to equilibrium when each string is at an angle of 5.2° with the vertical. If the string is 21.0 cm long, what is the magnitude of the charge on each sphere?
To find the magnitude of the charge on each sphere, we can use Coulomb's law and the principles of electrostatic equilibrium.
Let's first analyze the forces acting on one of the spheres:
1. The gravitational force acting on the sphere is given by the equation: F_grav = m * g, where m is the mass of the sphere and g is the acceleration due to gravity. In this case, m = 0.46 g = 0.46 × 10^(-3) kg, and g ≈ 9.8 m/s^2.
2. The electrostatic force between the two charged spheres is given by the equation: F_electrostatic = (k * q^2) / r^2, where k is the electrostatic constant (approximately 9 × 10^9 N·m^2/C^2), q is the magnitude of the charge on each sphere, and r is the distance between the centers of the spheres.
3. The tension force in the string can be resolved into two components: a vertical component (T_vert) balancing the gravitational force, and a horizontal component (T_horiz) balancing the electrostatic force.
Since the two spheres come to equilibrium when each string makes an angle of 5.2° with the vertical, we can consider the following forces:
- The vertical component of tension (T_vert) counteracts the gravitational force (F_grav).
- The horizontal component of tension (T_horiz) counteracts the electrostatic force (F_electrostatic).
Now, let's analyze the forces in more detail:
1. The vertical component of tension:
T_vert = T * cos(theta), where T is the tension in the string and theta is the angle between the string and the vertical.
In this case, theta = 5.2° = 5.2° * (pi/180) rad. We have the value of T_vert, which is equal to F_grav:
T * cos(theta) = m * g
T * cos(5.2° * (pi/180)) = 0.46 * 10^(-3) * 9.8
2. The horizontal component of tension:
T_horiz = T * sin(theta), where T is the tension in the string and theta is the angle between the string and the vertical.
In this case, T_horiz balances the electrostatic force between the two spheres:
T * sin(theta) = (k * q^2) / r^2
Now, we can combine the two equations and solve for q:
(T * sin(theta)) = (k * q^2) / r^2
(T * sin(theta)) * r^2 = k * q^2
q^2 = [(T * sin(theta)) * r^2] / k
q = sqrt{ [(T * sin(theta)) * r^2] / k }
By substituting the known values:
- T * sin(theta) = (0.46 * 10^(-3) * 9.8) * sin(5.2° * (pi/180))
- r = 21.0 cm = 21.0 × 10^(-2) m
- k = 9 × 10^9 N·m^2/C^2
We can now calculate the magnitude of the charge on each sphere using the formula:
q = sqrt{ [(T * sin(theta)) * r^2] / k }