Two spheres are placed so that their centers are 2.6m apart. the force bvetween the two spheres is 2.75 x 10 to the power of negative 12 Newton.What is the mass of each sphere if one sphere is twice the mass of the other sphere?
2.7E-12=G(2m)(m)/2.6^2
solve for m
90
To solve this problem, we can use Newton's Law of Universal Gravitation, which states that the force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Let's assign the following variables:
- F: Force between the spheres (2.75 x 10^-12 N in this case)
- m1: Mass of the first sphere
- m2: Mass of the second sphere (twice the mass of the first sphere)
- r: Distance between the centers of the spheres (2.6 m in this case)
- G: Universal gravitational constant (6.67430 x 10^-11 N(m/kg)^2)
Now, let's set up the equation using these variables:
F = (G * m1 * m2) / r^2
Since one sphere is twice the mass of the other, we can express m2 in terms of m1:
m2 = 2 * m1
Replacing m2 in the equation:
F = (G * m1 * (2 * m1)) / r^2
Simplifying:
2.75 x 10^-12 N = (G * 2 * m1^2) / (2.6 m)^2
Now, let's calculate the mass of each sphere:
1. Rearrange the equation:
2 * m1^2 = (2.75 x 10^-12 N * (2.6 m)^2) / G
2. Calculate the right side of the equation using the given values:
(2.75 x 10^-12 N * (2.6 m)^2) / G = 1.447506126 x 10^-6 kg^2
3. Divide both sides of the equation by 2:
m1^2 = (1.447506126 x 10^-6 kg^2) / 2 = 7.23753063 x 10^-7 kg^2
4. Take the square root of both sides to solve for m1:
m1 = √(7.23753063 x 10^-7 kg^2) = 0.0008517 kg (rounded to four decimal places)
Since m2 is twice the mass of m1:
m2 = 2 * 0.0008517 kg = 0.0017034 kg (rounded to four decimal places)
Therefore, the mass of the first sphere is approximately 0.0009 kg, and the mass of the second sphere is approximately 0.0017 kg.