Question

After the big-bang, tiny black holes may have formed. If one with a mass of `1 x 10^11 kg` (and a radius of only `1 x 10^-16 m`) reached Earth, at what distance from your head would its gravitational pull on you match that of the Earth's?

Answer 1

Your head would need to be about 0.82 metres from the black hole to experience 1g.

Explanation:

The acceleration you would experience is:

`a=(GM)/r^2`

Where `M=10^11kg` is the mass of the black hole, `G=6.674m^2kg^(-1)s^(-2)` is the gravitational constant and `r` is the distance from the black hole.

The acceleration due to gravity on Earth is `a=9.81ms^(-2)`.

So, to experience 1g of acceleration, thee distance is:

`r^2=(GM)/a=6.67/9.81`

This gives `r=0.82m`.

Being so close to a black hole puts you in the region where tidal effects can occur. At `r=0.6m`, `a=18.5ms^(-2)`. At `r=0.4m`, `a=41.7ms^(-2)`.

Incidentally, the Schwarzschild radius for a black hole is given by:

`r=(GM)/c^2`

Where `c` is the speed of light. A black hole with a mass of `10^11kg` has a radius of `7.4*10^(-17)m`, which is slightly smaller than `10^-16m`.