Two spherical balls are placed so their centers are 2.6 m apart. The force between the two balls is 2.75 x 10-12 N. What is the mass of each ball if one ball is twice the mass of the other ball?
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To calculate the mass of each ball, we can use the formula for the gravitational force between two objects:
F = G * m1 * m2 / r^2
Where:
F = force between the two balls (2.75 x 10^-12 N)
G = gravitational constant (6.67430 x 10^-11 m^3 kg^-1 s^-2)
m1 = mass of the first ball
m2 = mass of the second ball
r = distance between the centers of the balls (2.6 m)
Given that one ball is twice the mass of the other ball, we can denote the mass of the smaller ball as m and the mass of the larger ball as 2m.
Substituting the values into the formula, we get:
2.75 x 10^-12 N = (6.67430 x 10^-11 m^3 kg^-1 s^-2) * m * 2m / (2.6 m)^2
Simplifying further, we have:
2.75 x 10^-12 N = (13.34860 x 10^-11) * 2m^2 / (2.6)^2
2.75 = 13.34860 * (2m^2) / 6.76
Multiplying both sides by 6.76 to isolate (2m^2), we get:
2m^2 = 2.75 * 6.76 / 13.34860
2m^2 ≈ 1.3976
Dividing both sides by 2, we have:
m^2 ≈ 0.6988
Taking the square root of both sides:
m ≈ √0.6988
m ≈ 0.8358 kg
Therefore, the mass of the smaller ball is approximately 0.8358 kg, and the mass of the larger ball is approximately 2 times that, which is approximately 1.6716 kg.
To solve this problem, we can use the equation for the gravitational force between two objects:
F = (G * m1 * m2) / r^2
Where F is the force between the two objects, G is the gravitational constant (approximately 6.67430 × 10^-11 N m^2/kg^2), m1 and m2 are the masses of the objects, and r is the distance between their centers.
In this given problem, we are told that the force (F) between the two balls is 2.75 x 10^-12 N, and the distance (r) between their centers is 2.6 m.
We are also given the information that one ball is twice the mass of the other ball. Let's assume the mass of the smaller ball is m, so the mass of the larger ball would be 2m.
Now, we can substitute all the given values into the equation and solve for m.
2.75 x 10^-12 N = (6.67430 × 10^-11 N m^2/kg^2) * (m) * (2m) / (2.6 m)^2
Simplifying this equation, we get:
2.75 x 10^-12 N = (3.33486 × 10^-11 N m^2/kg^2) * (2m^2) / (2.6 m)^2
Cross multiply to further simplify:
(2.75 x 10^-12 N) * (2.6 m)^2 = (3.33486 × 10^-11 N m^2/kg^2) * (2m^2)
Rearrange the equation and isolate m:
m^2 = [(2.75 x 10^-12 N) * (2.6 m)^2] / [(3.33486 × 10^-11 N m^2/kg^2) * 2]
m^2 = 3.1814 x 10^-23 kg
Taking the square root of both sides, we find:
m = 1.7826 x 10^-12 kg
Since we assumed the mass of the smaller ball to be m, and we calculated the value of m, the mass of the smaller ball is approximately 1.7826 x 10^-12 kg.
Finally, to find the mass of the larger ball, we know that it is twice the mass of the smaller ball. Thus, the mass of the larger ball is approximately 2 * (1.7826 x 10^-12 kg) = 3.5652 x 10^-12 kg.
Use Newton's universal law of gravity.
F = G M1 M2/R^2
In this case, M2 = 2 M1. Therefore
F = 2 M1^2/R^2.
M1 is the mass of the smaller sphere. R is the separation. Look up the universal constand G; it must be in your notes, or textbook.
Solve for the smaller mass, M1. Double that for M2. Show your work if further help is needed.