Two balls have the same mass of 0.681 kg and identical charges of 18.0 μC.They hang from the ceiling on strings of identical length as shown in the figure. If the angle with respect to the verti- cal of the strings is 20.0°, what is the length of the strings?
To solve this problem, we can use the concept of equilibrium. When the balls are in equilibrium, the net force acting on each ball must be zero.
Considering the forces acting on one of the balls, we have:
1. The tension force acting upward along the string.
2. The gravitational force acting downward due to the weight of the ball.
3. The electrostatic force acting downward due to the repulsion between the two charged balls.
Since the ball is in equilibrium, the upward tension force must be equal in magnitude to the downward gravitational and electrostatic forces.
Let's denote the tension force as T, the gravitational force as mg (where g is the acceleration due to gravity), and the electrostatic force as F_e. We can set up the following equations:
T = mg + F_e (1)
F_e = k * (q^2 / r^2) (2)
where k is the Coulomb's constant, q is the charge of each ball, and r is the length of the string.
Since the balls have the same charges and the same mass, we can simplify equation (1) to:
T = mg + k * (q^2 / r^2) (3)
Now, let's plug in the given values:
m = 0.681 kg
q = 18.0 μC = 18.0 * 10^(-6) C
g = 9.8 m/s^2
k = 8.99 * 10^9 N.m^2/C^2
θ = 20.0°
First, we need to find the gravitational force:
F_g = m * g = (0.681 kg) * (9.8 m/s^2) = 6.6758 N
Next, we need to find the electrostatic force:
F_e = k * (q^2 / r^2)
To find the length of the string, we can use the fact that the angle between the string and the vertical is equal to the angle between the net force and the gravitational force. Therefore:
tan(θ) = F_e / F_g
Rearranging the equation gives:
F_e = F_g * tan(θ)
Substituting the values, we have:
F_e = (6.6758 N) * tan(20.0°) = 2.6008 N
Now, we can substitute the known values into equation (3):
T = mg + k * (q^2 / r^2)
2.6008 N = (0.681 kg) * (9.8 m/s^2) + (8.99 * 10^9 N.m^2/C^2) * ((18.0 * 10^(-6) C)^2 / r^2)
Solving for r, we get:
r^2 = ((8.99 * 10^9 N.m^2/C^2) * ((18.0 * 10^(-6) C)^2)) / (2.6008 N - (0.681 kg) * (9.8 m/s^2))
r^2 = 3.3278926541722595e-7 m^2
Taking the square root of both sides, we find:
r = 0.000576708263703 m
Therefore, the length of the string is approximately 0.0005767 meters, or about 5.77 millimeters.
To find the length of the strings, we can use the concept of equilibrium and the forces acting on the balls.
Let's break down the forces acting on one of the balls:
1. Gravitational Force: This force acts vertically downwards and can be found using the formula F_g = m * g, where m is the mass of the ball and g is the acceleration due to gravity (9.8 m/s^2). Therefore, the gravitational force is F_g = 0.681 kg * 9.8 m/s^2.
2. Electrostatic Force: This force acts along the string and can be found using the formula F_e = k * (q1 * q2) / r^2, where k is the electrostatic constant (8.99 * 10^9 N.m^2/C^2), q1 and q2 are the charges on the balls, and r is the length of each string. As the balls have the same charge, the electrostatic force on each ball is F_e = (8.99 * 10^9 N.m^2/C^2) * (18.0 μC)^2 / (r^2).
In equilibrium, the vertical component of the electrostatic force should balance the gravitational force. Therefore, we can write the equation:
F_e * sin(angle) = F_g
Plugging in the values:
(8.99 * 10^9 N.m^2/C^2) * (18.0 μC)^2 / (r^2) * sin(20°) = 0.681 kg * 9.8 m/s^2
Now, let's solve for r:
r^2 = (8.99 * 10^9 N.m^2/C^2) * (18.0 μC)^2 / (0.681 kg * 9.8 m/s^2 * sin(20°))
Take the square root of both sides to find r:
r = sqrt((8.99 * 10^9 N.m^2/C^2) * (18.0 μC)^2 / (0.681 kg * 9.8 m/s^2 * sin(20°)))
Evaluating this expression will give us the length of the strings.