A human being of mass 52 kg is in close proximity to the surface of a spherical object with radius 2.9 km and uniform density 2 times that of water (water's density is 1000 kg/m ^ 3).
Given that G = 6.67 * 10^-11 N m^2 / kg^2:
In order to find the force F = G m1 m2 / r^2, we need to know both masses m1 and m2 and the distance r between the centers of mass.
- We know that the density is 2 * 1000 kg/m^3.
- If we knew the volume of the sphere, we could then find its mass.
- The radius of the sphere is 2.9 kilometers = 2900 meters.
- Its volume is therefore (4 `pi /3)r^3 = 1.021606E+11 cubic meters.
- Multiplying this volume by the density 2000 kg/m^3 we obtain the sphere's total mass
- sphere mass m1 = 2.043213E+14 kilograms.
- We can now find the gravitational force.
- We substitute m1 = 52 kg, m2 = 2.043213E+14 kilograms r = 2900 meters into F = G m1 m2 / r ^ 2, and use the universal gravitational constant G = 6.67 * 10^-11 N m ^ 2/kg ^ 2 to obtain force
- F = .084 Newtons.
- If the sphere is compressed to a radius of 290 meters, its mass remains the same, as does the mass of the person.
- So we use the same masses m1 and m2, the same value of G (since it is a universal constant), and the new value 290 meters for r to obtain
- force F = 8.426 Newtons.
- Note that this is 100 times the force experienced at the first radius. The person is 10 times closer to the mass, so r ^ 2 is 1/10 as great, so the gravitational influence of the mass is spread over the surface of an imaginary sphere with 1/10 the radius.
- This imaginary sphere has 1/100 times the area of the first, so the gravitational field will be 100 times as intense. The force on the person will therefore be 100 times as great.
- When the original sphere is compressed to radius 29 meters, which is 1/100 its original radius, a proportionality argument similar to the above tells us that the gravitational field will become 100^2 = 10,000 times as intense.
- This is confirmed by substituting the original masses, with the new radius r = 29 meters, into F = G m1 m2 / r ^ 2.
- We obtain F = 842.649 Newtons, which is in fact 10000 times as great as the force found at the original radius.
The weight of the individual on Earth is 9.8 m/s ^ 2 ( 52 kg) = 509.6 Newtons.
- In order to find the radius at which F = G m1 m2 / r ^ 2 takes this value, we solve this equation for r to obtain r = `sqrt[G m1 m2 / F].
- Substituting the known values of m1 and m2 and the desired value of F, we obtain r = 37.29123 meters.
If all the mass in a sphere of radius R is compressed into a sphere of radius r, then the gravitational influence of the mass will be concentrated in a smaller area and the gravitational field will therefore be greater. The ratio of areas will be the square of the ratio of radii:
The ratio of gravitational fields will therefore be the inversse of the area ratio: