Two ping pong balls each have a mass of 1.4 grams and carry a net charge of 0.450uC. one ball is held fixed. at what height should the second ball be placed directly above the fixed ball if it is to remain at rest there?
To determine the height at which the second ball should be placed directly above the fixed ball so that it remains at rest, we need to consider the electrostatic force between the two balls. This force can be calculated using Coulomb's Law.
Coulomb's Law states that the electrostatic 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. Mathematically, it can be expressed as:
F = (k * q1 * q2) / r^2
Where:
F is the electrostatic force between the two balls,
k is the electrostatic constant (approximately 9 x 10^9 N m^2/C^2),
q1 and q2 are the charges of the balls, and
r is the distance between the balls.
In this case, we have two identical ping pong balls with a charge of 0.450 µC each. The fixed ball doesn't move, so the net electrostatic force acting on it is zero. Therefore, we need to find the distance at which the electrostatic force acting on the second ball is also zero.
Setting the net electrostatic force to zero (F = 0), we can rearrange Coulomb's Law to solve for the distance (r):
r = sqrt((k * q1 * q2) / F)
Since F = 0, the formula simplifies to:
r = sqrt((k * q1 * q2) / 0)
However, dividing by zero is undefined, so we cannot directly calculate the distance. This implies that if the second ball is placed directly above the fixed ball, it will not remain at rest. The electrostatic force between them will cause the second ball to experience an upward force and move away from the fixed ball.
Therefore, in order for the second ball to remain at rest, it cannot be placed directly above the fixed ball, as the electrostatic force will always be present.
To determine the height at which the second ball should be placed in order to remain at rest, we need to consider the gravitational force and the electrostatic force between the two balls.
Step 1: Calculate the gravitational force:
The gravitational force between two objects can be calculated using the formula:
F_gravity = (G * m1 * m2) / r^2
Where:
F_gravity = gravitational force
G = gravitational constant (6.67430 x 10^-11 N * m^2 / kg^2)
m1 = mass of the first ball (1.4 grams = 0.0014 kg)
m2 = mass of the second ball (1.4 grams = 0.0014 kg)
r = distance between the centers of the two balls (unknown)
Since one ball is fixed, the gravitational force will act on the second ball only. Therefore, the equation becomes:
F_gravity = (G * m_fixed * m2) / r^2
Step 2: Calculate the electrostatic force:
The electrostatic force between the two balls can be calculated using Coulomb's Law:
F_electric = (k * q1 * q2) / r^2
Where:
F_electric = electrostatic force
k = Coulomb's constant (8.99 x 10^9 N * m^2 / C^2)
q1 = charge of the first ball (0.450 uC = 0.450 x 10^-6 C)
q2 = charge of the second ball (0.450 uC = 0.450 x 10^-6 C)
r = distance between the centers of the two balls (unknown)
Step 3: Equate the gravitational and electrostatic forces:
Since the second ball needs to remain at rest, the gravitational force and the electrostatic force must have the same magnitude but act in opposite directions. Therefore, we can set them equal to each other:
F_gravity = F_electric
(G * m_fixed * m2) / r^2 = (k * q1 * q2) / r^2
Step 4: Solve for the height:
Simplifying the equation by canceling out r^2, m_fixed, m2, q1, and q2, we get:
G = k
Since both G and k are constants, the equation implies that the height does not affect the forces acting on the balls. Therefore, the height at which the second ball should be placed directly above the fixed ball to remain at rest is arbitrary.