Two ping-pong balls each have a mass of (1.4x10^0) g and carry a net charge of (3.99x10^0) μC. One ball is held fixed. At what height should the second ball be placed directly above the fixed ball if it is to remain there?
force gravity=force elec
mg=kqq/r^2
solve for r.
To find the height at which the second ball should be placed directly above the fixed ball, we need to consider the gravitational and electrostatic forces acting on the balls.
Let's break down the steps to get the answer:
Step 1: Determine the gravitational force between the balls.
The gravitational force between two objects can be calculated using Newton's law of gravitation:
F_gravity = (G * m1 * m2) / r^2
where F_gravity is the gravitational force, G is the gravitational constant (6.67430 x 10^-11 m³ kg^-1 s^-2), m1 and m2 are the masses of the two objects, and r is the distance between them.
Given:
Mass of each ball (m1 and m2) = 1.4 x 10^0 g = 0.0014 kg
Distance between the balls (r) = height where the second ball should be placed
Step 2: Determine the electrostatic force between the charged balls.
The electrostatic force between two charged objects can be calculated using Coulomb's law:
F_electrostatic = (k * |q1 * q2|) / r^2
where F_electrostatic is the electrostatic force, k is the electrostatic constant (8.988 x 10^9 N m² C^-2), q1 and q2 are the charges of the two objects, and r is the distance between them.
Given:
Charge of each ball (q1 and q2) = 3.99 x 10^0 μC = 3.99 x 10^-6 C
Distance between the balls (r) = height where the second ball should be placed
Step 3: Set the gravitational force equal to the electrostatic force.
To find the height at which the second ball should be placed, we need to equate the gravitational force and the electrostatic force:
F_gravity = F_electrostatic
Step 4: Solve for the height (r).
Combine the equations from Step 1 and Step 2, and solve for r:
(G * m1 * m2) / r^2 = (k * |q1 * q2|) / r^2
Simplifying and canceling out the r^2 term:
G * m1 * m2 = k * |q1 * q2|
Rearranging for r:
r = sqrt((G * m1 * m2) / (k * |q1 * q2|))
Step 5: Calculate the height (r).
Substitute the given values into the equation from Step 4 and calculate r:
r = sqrt((6.67430 x 10^-11 m³ kg^-1 s^-2 * 0.0014 kg * 0.0014 kg) / (8.988 x 10^9 N m² C^-2 * 3.99 x 10^-6 C * 3.99 x 10^-6 C))
Using a calculator, perform the calculation:
r ≈ 8.91 x 10^-4 meters
Therefore, the second ball should be placed at a height of approximately 8.91 x 10^-4 meters directly above the fixed ball to remain there.