The masses of the earth and moon are 5.98x1024 and 7.35x1022 kg, respectively. Identical amounts of charge are placed on each body, such that the net force (gravitational plus electrical) on each is zero. What is the magnitude of the charge placed on each body?
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To find the magnitude of the charge placed on each body, we need to equate the gravitational force and the electrical force on each body.
The gravitational force between two bodies is given by the equation:
Fg = (G * m1 * m2) / r^2
where Fg is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the bodies, and r is the distance between them.
The electrical force between two charged bodies is given by the equation:
Fe = (k * q1 * q2) / r^2
where Fe is the electrical force, k is the Coulomb's constant, q1 and q2 are the charges on the bodies, and r is the distance between them.
Since the net force on each body is zero, we can equate the gravitational force and the electrical force:
(G * m1 * m2) / r^2 = (k * q1 * q2) / r^2
Given that the masses of the earth (m1) and the moon (m2) are 5.98x10^24 kg and 7.35x10^22 kg, respectively, we can substitute these values into the equation:
(G * 5.98x10^24 kg * 7.35x10^22 kg) / r^2 = (k * q1 * q2) / r^2
To solve for the charges q1 and q2, we need to know the values of the universal gravitational constant (G) and the Coulomb's constant (k). The values are:
G = 6.67430 x 10^-11 Nm^2/kg^2
k = 8.98755 x 10^9 Nm^2/C^2
Substituting these values into the equation, we have:
(6.67430 x 10^-11 Nm^2/kg^2 * 5.98x10^24 kg * 7.35x10^22 kg) / r^2 = (8.98755 x 10^9 Nm^2/C^2 * q1 * q2) / r^2
Since the right side of the equation depends on the product of q1 and q2, we can assume that the charges q1 and q2 are equal, and let q be the magnitude of the charge on each body.
Simplifying the equation further, we get:
(6.67430 x 10^-11 Nm^2/kg^2 * 5.98x10^24 kg * 7.35x10^22 kg) / r^2 = (8.98755 x 10^9 Nm^2/C^2 * q^2) / r^2
Now, we can cancel out the r^2 term from both sides of the equation:
(6.67430 x 10^-11 Nm^2/kg^2 * 5.98x10^24 kg * 7.35x10^22 kg) = (8.98755 x 10^9 Nm^2/C^2 * q^2)
To solve for the charge q, we isolate q on one side of the equation:
q^2 = (6.67430 x 10^-11 Nm^2/kg^2 * 5.98x10^24 kg * 7.35x10^22 kg) / (8.98755 x 10^9 Nm^2/C^2)
Taking the square root of both sides, we find:
q ≈ √[(6.67430 x 10^-11 Nm^2/kg^2 * 5.98x10^24 kg * 7.35x10^22 kg) / (8.98755 x 10^9 Nm^2/C^2)]
Evaluating this expression, we can find the approximate magnitude of the charge placed on each body.
To find the magnitude of the charge placed on each body, we need to equate the gravitational force between the Earth and the Moon to the electrical force between them.
Step 1: Calculate the gravitational force between the Earth and the Moon
The gravitational force between two objects is given by the equation:
F_grav = (G * m1 * m2) / r^2
where F_grav is the gravitational force, G is the universal gravitational constant (6.67430 × 10^-11 N m^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between them.
In this case, m1 is the mass of the Earth (5.98x10^24 kg), m2 is the mass of the Moon (7.35x10^22 kg), and r is the average distance between the Earth and the Moon (which is approximately 3.844x10^8 meters).
So, we can calculate F_grav:
F_grav = (G * m1 * m2) / r^2
= (6.67430 × 10^-11 N m^2/kg^2) * (5.98x10^24 kg) * (7.35x10^22 kg) / (3.844x10^8 meters)^2
Step 2: Set the gravitational force equal to the electrical force
Since we want the net force (gravitational plus electrical) on each body to be zero, we can set the gravitational force equal to the electrical force:
F_grav = F_elec
Step 3: Calculate the electrical force
The electrical force between two objects with charges q1 and q2 is given by Coulomb's Law:
F_elec = (k * q1 * q2) / r^2
where F_elec is the electrical force, k is the Coulomb constant (8.9876 × 10^9 N m^2/C^2), q1 and q2 are the charges of the two objects, and r is the distance between them.
In this case, we want F_elec to be equal to F_grav, so we can rewrite Coulomb's Law as:
(G * m1 * m2) / r^2 = (k * q1 * q2) / r^2
Step 4: Calculate the magnitude of the charge on each body
Since the distances and masses are the same for both bodies, we can simplify the equation:
(G * m1 * m2) = (k * q1 * q2)
Now, we can solve for q1 and q2:
q1 = (G * m1 * m2) / (k * q2)
q2 = (G * m1 * m2) / (k * q1)
For the charges to be equal, q1 must be equal to q2. So, we can set q1 equal to q2 and solve for the magnitude of the charge:
q = (G * m1 * m2) / (k * q)
Simplifying the equation, we get:
q^2 = (G * m1 * m2) / k
q = sqrt((G * m1 * m2) / k)
Plugging in the values, we can calculate the magnitude of the charge placed on each body:
q = sqrt((6.67430 × 10^-11 N m^2/kg^2) * (5.98x10^24 kg) * (7.35x10^22 kg) / (8.9876 × 10^9 N m^2/C^2))
Calculating the value will give you the magnitude of the charge placed on each body.