A solid lead sphere of radius 10 m (about 66 ft across!) has a mass of about 57 million kg. If two of these spheres are floating in deep space with their centers 20 m apart, the gravitational attraction between the spheres is only 540 N (about 120 lb). How large would this gravitational force be if the distance between the centers of the two spheres were tripled?
Well, if the distance between the centers of the two spheres were tripled, they would definitely have some serious distance issues. These spheres would be drifting so far apart in deep space that they might be mistaken for distant relatives at a family gathering.
But let's not get too emotional here. To answer your question, the gravitational force between two objects is inversely proportional to the square of the distance between them. So, if we triple the distance between the centers of the two spheres, the force would decrease by a factor of 1/9.
If the original force was 540 N, dividing it by 9 would give us a force of about 60 N. So, the gravitational force between the spheres would be reduced to a mere 60 N, which is like asking for a penny back in change after spending all your money on black market space snacks.
But hey, in the vastness of space, every interaction counts, no matter how small. So, even with a tripled distance, these spheres would still feel a smidgen of gravitational attraction.
To calculate the gravitational force between two spheres, we can use Newton's law of universal gravitation:
F = G * ((m1 * m2) / r^2)
Where:
F is the gravitational force
G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
m1 and m2 are the masses of the two spheres
r is the distance between the centers of the two spheres
Let's solve the problem step by step:
Step 1: Calculate the initial gravitational force between the spheres with a distance of 20 m.
F1 = G * ((m1 * m2) / r1^2)
Step 2: Calculate the new gravitational force when the distance between the centers of the spheres is tripled.
F2 = G * ((m1 * m2) / r2^2)
Step 3: Calculate the ratio of the new gravitational force to the initial gravitational force.
Ratio = F2 / F1
Step 4: Substitute the given values into the formula and calculate the result.
Let's do the math:
Step 1:
F1 = G * ((m1 * m2) / r1^2)
= (6.67430 × 10^-11 m^3 kg^-1 s^-2) * ((57 million kg) * (57 million kg) / (20 m)^2)
≈ 10.3312 N
Step 2:
F2 = G * ((m1 * m2) / r2^2)
= (6.67430 × 10^-11 m^3 kg^-1 s^-2) * ((57 million kg) * (57 million kg) / (3 * 20 m)^2)
≈ 1.148 N
Step 3:
Ratio = F2 / F1
= 1.148 N / 10.3312 N
≈ 0.1110
Step 4:
The gravitational force would be approximately 0.1110 times the initial force when the distance between the centers of the two spheres is tripled.
To calculate the gravitational force between the two spheres, we can use the formula for gravitational force:
F = (G * m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2), m1 and m2 are the masses of the two spheres, and r is the distance between their centers.
In the given scenario, we have:
- Mass of each sphere (m1 and m2) = 57 million kg
- Radius of each sphere = 10 m
- Distance between the centers of the two spheres (r_initial) = 20 m
- Gravitational force between the spheres at the initial distance (F_initial) = 540 N
Now, let's calculate the gravitational force when the distance between the centers of the spheres is tripled (r_final = 3 * r_initial).
Step 1: Calculate the new distance between the centers (r_final):
r_final = 3 * r_initial = 3 * 20 m = 60 m
Step 2: Calculate the new gravitational force (F_final):
F_final = (G * m1 * m2) / r_final^2
Substituting the given values:
F_final = (6.67430 × 10^-11 m^3 kg^-1 s^-2 * 57 million kg * 57 million kg) / (60 m)^2
Calculating the expression inside the brackets first:
(6.67430 × 10^-11 m^3 kg^-1 s^-2 * 57 million kg * 57 million kg) = 2.0404566 x 10^15 m^3 kg^2 s^-2
Now, substitute this value into the formula:
F_final = (2.0404566 x 10^15 m^3 kg^2 s^-2) / (60 m)^2
F_final = 2.0404566 x 10^15 m^3 kg^2 s^-2 / 3600 m^2
Simplifying further:
F_final = 5.667934 x 10^12 N
Therefore, the gravitational force between the spheres would be approximately 5.667934 x 10^12 N if the distance between their centers were tripled.