Two spherical objects with a mass of 2.88 kg each are placed at a distance of 2.12 m apart. How many electrons need to leave each object so that the net force between them becomes zero?
G•m²/R =k•q²/R,
q=m•sqrt(G/k)=
=2.88•sqrt(6.67•10^-11 /9•10^9) =2.48•10^-10 C,
N =q/e=2.48•10^-10/1.6•10^-19 =1.55•10^9
To find the number of electrons needed to leave each object so that the net force between them becomes zero, you can follow these steps:
Step 1: Calculate the gravitational force between the two objects.
The gravitational force between two objects can be calculated using the equation: F = G * (m1 * m2) / r^2, where F is the gravitational force, G is the gravitational constant (approximately 6.674 × 10^-11 N m^2 / kg^2), m1 and m2 are the masses of the objects, and r is the distance between their centers.
Given:
Mass of each object (m1 and m2) = 2.88 kg
Distance between the objects (r) = 2.12 m
Plugging in the values, we get:
F = (6.674 × 10^-11 N m^2 / kg^2) * (2.88 kg * 2.88 kg) / (2.12 m)^2
Calculating this gives us the magnitude of the gravitational force between the objects.
Step 2: Calculate the net electric force required to balance the gravitational force.
To make the net force between the objects zero, we need to balance the gravitational force with the electric force. The electric force between two charged objects can be calculated using the equation: F = k * (q1 * q2) / r^2, where F is the electric force, k is the electrostatic constant (approximately 8.99 × 10^9 N m^2 / C^2), q1 and q2 are the charges of the objects, and r is the distance between their centers.
Since the net force should be zero, the electric force should have the same magnitude as the gravitational force.
Plugging in the values for the electric force equation, we get:
(6.674 × 10^-11 N m^2 / kg^2) * (2.88 kg * 2.88 kg) / (2.12 m)^2 = (8.99 × 10^9 N m^2 / C^2) * (q1 * q2) / (2.12 m)^2
Step 3: Solve for the charge (q1 and q2) of the objects.
Rearranging the equation, we have:
q1 * q2 = [(6.674 × 10^-11 N m^2 / kg^2) * (2.88 kg * 2.88 kg) / (2.12 m)^2] * [(2.12 m)^2 / (8.99 × 10^9 N m^2 / C^2)]
Simplifying this equation will give us the product of the charges of the two objects (q1 * q2).
Step 4: Calculate the number of electrons needed to achieve the required charge.
The charge of a single electron is approximately -1.602 × 10^-19 C.
To find the number of electrons needed, divide the total charge (q1 * q2) by the charge of a single electron.
Number of electrons = (q1 * q2) / (-1.602 × 10^-19 C)
Performing the calculations at each step will yield the final answer.
To find out how many electrons need to leave each object so that the net force between them becomes zero, we need to use the concepts of the electric force and Coulomb's law.
Coulomb's law states that the 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 force between the objects, k is Coulomb's constant (8.99 x 10^9 N.m^2/C^2), q1 and q2 are the charges on the objects, and r is the distance between them.
In this case, we want to find the charge (number of electrons) needed on each object to make the net force between them zero. Since we know the mass of each object, we can use the mass to calculate the total charge.
1. Calculate the weight of each object:
Weight = mass * gravitational acceleration
We will assume a standard gravitational acceleration of 9.8 m/s^2.
Weight = 2.88 kg * 9.8 m/s^2 = 28.224 N
2. Convert the weight into the force of gravity acting on each object:
Force = mass * gravitational acceleration
3. Find the charge (number of electrons) needed on each object to make the gravity force equal to the electric force:
F = (k * q1 * q2) / r^2
F = Force
k * q1 * q2 = Force * r^2
q1 * q2 = (Force * r^2) / k
4. Since we want the net force to be zero, q1 and q2 should have the same magnitude but opposite signs:
|q1| * |q2| = (Force * r^2) / k
5. Now we can find the magnitude of the charge on each object:
|q1| = |q2| = sqrt((Force * r^2) / k)
6. Convert the magnitude of the charge into the number of electrons:
1 elementary charge = 1.6 x 10^-19 C
Number of electrons = |q1| / (1.6 x 10^-19)
Plug in the values and calculate the number of electrons needed to leave each object to make the net force between them zero.