A small ball with a mass of 30.0 g and a charge of −0.200 μC is suspended from the ceiling by a string. The ball hangs at a distance of 5.00 cm above an insulating floor. If a second small ball with a mass of 50.0 g and a charge of 0.400 μC is rolled directly beneath the first ball, will the second ball leave the floor? What is the tension in the string when the second ball is directly beneath the first ball?
To determine whether the second ball will leave the floor and calculate the tension in the string, we need to consider the gravitational and electrostatic forces acting on the balls.
First, let's calculate the gravitational force acting on both balls using the formula:
F_gravity = m * g
Where,
F_gravity is the gravitational force,
m is the mass of the ball,
g is the acceleration due to gravity (approximately 9.8 m/s^2).
For the first ball:
m1 = 30.0 g = 0.030 kg
F_gravity1 = 0.030 kg * 9.8 m/s^2 = 0.294 N
For the second ball:
m2 = 50.0 g = 0.050 kg
F_gravity2 = 0.050 kg * 9.8 m/s^2 = 0.490 N
Next, let's calculate the electrostatic force between the balls using Coulomb's law:
F_electrostatic = k * (|q1| * |q2|) / r^2
Where,
F_electrostatic is the electrostatic force,
k is the electrostatic constant (approximately 9.0 x 10^9 N·m^2/C^2),
|q1| and |q2| are the magnitudes of the charges,
r is the distance between the charges.
|q1| = 0.200 μC = 0.200 x 10^-6 C
|q2| = 0.400 μC = 0.400 x 10^-6 C
r = 5.00 cm = 0.050 m
F_electrostatic = (9.0 x 10^9 N·m^2/C^2) * (0.200 x 10^-6 C) * (0.400 x 10^-6 C) / (0.050 m)^2
Evaluating this expression gives F_electrostatic ≈ 576 N.
Now let's analyze the situation:
- If the electrostatic force is greater than the gravitational force for the second ball (F_electrostatic > F_gravity2), the second ball will leave the floor.
- If the electrostatic force is less than or equal to the gravitational force for the second ball (F_electrostatic ≤ F_gravity2), the second ball will remain on the floor.
Comparing the values:
F_electrostatic ≈ 576 N
F_gravity2 = 0.490 N
Since F_electrostatic is much greater than F_gravity2, the second ball will leave the floor.
To calculate the tension in the string when the second ball is directly beneath the first ball, we need to consider the net force acting on the first ball. The net force is the difference between the gravitational force and the electrostatic force:
Net force on the first ball = F_gravity1 - F_electrostatic
Net force on the first ball = 0.294 N - 576 N
Net force on the first ball ≈ -575.706 N
Since the net force is negative, the tension in the string will be in the opposite direction, pulling the first ball upward.