codyg1985’s Think Tank

How small does the earth have to be before I am ripped to shreds?

No, this is not a suicide note.  Hardly.  Let’s suppose that the Earth maintained the same mass, but it was to shrink at a constant rate.   As you may know, as the density of the earth increases, the magnitude of tidal forces increases as well.  In a nutshell, tidal forces result from the force of gravity being stronger at one point on a rigid body (such as a human) than on another point.  The tidal force, or gradient, is not noticeable on the surface of the earth, but close to a black hole the gradient is strong enough to stretch your body like a piece of spaghetti in a process called, amusingly enough, spaghettification.

To perform the calculations, we need some data. First, know the mass of the earth to be 5.97 x 10^24 kilograms and the radius of the earth to be 6.378 x 10^6 meters. The gravitational constant, G, is 6.673 x 10^-11 m^3 kg^-1 s^-2. For the human subject, I am the guinea pi, so we are using my weight (68.039 kg) and height (1.651 m) in the test.  Next, we make a few assumptions.  Firstly, we assume that the cross-sectional area of my ankle bone is 3.32 x 10^-3 m^2 and the cross-sectional area of my thigh bone is 1.91 x 10^-2 m^-2.  We also assume that the ultimate breaking strength of human bone is 1.24 x 10^8 Pa.

Next, we need some equations in which to calculate the appropriate values.  Firstly, the value of the tidal force is calculated by taking the difference in gravitational force at my head and the gravitational force at my feet.  That is:

We also need to calculate the normal stress at my ankles and thighs. The formula for stress, represented by the lowercase Greek letter sigma, is

From those equations it is easy to calculate the forces and stresses at different radii, leaving everything else constant. We have to do a lot of shrinking before we start disconnecting any bones. Once the radius of the earth reaches 5 km, the tidal forces are enough to rip your ankles clean off.  Shrinking even more to where the radius is 125 m, then your leg goes bye-bye.  If we keep going until the Earth is the size of a ping pong ball (radius of 8.87 mm) then we have ourselves a black hole. That value is calculated from the Schwarzschild Radius equation.