Question

Miniature black holes. Left over from the big-bang beginning of the universe, tiny black holes might still wander through the universe. If one with a mass of $1 \times {10^{11}}kg$(and a radius of only $1 \times {10^{ - 16}}kgm$) reached Earth, at what distance from your head would its gravitational pull on you match that of Earth’s.

Answer 1

Hint:Here the gravitational pull experienced by you from the black hole would be the same as the gravitational pull experienced by you from the earth. So, in manner of fact we just have to find the distance and to do that apply the general formula for gravitational force.

Complete step by step solution:

The force of gravity between you and the black hole:
${F_{ge}} = G\dfrac{{{M_b}m}}{{{r_b}^2}}$ ;
Now, according to the Newton’s second law F = ma;
${F_{ge}} = mg$;
Equate the above relation with ${F_{ge}} = G\dfrac{{{M_b}m}}{{{r_b}^2}}$:
$mg = G\dfrac{{{M_b}m}}{{{r_b}^2}}$;
Write the above equation in terms of ${r_e}$:
$ \Rightarrow {r_b}^2 = G\dfrac{{{M_b}}}{g}$;
$ \Rightarrow {r_b} = \sqrt {G\dfrac{{{M_b}}}{g}} $;
Put in the given values and solve:
$ \Rightarrow {r_b} = \sqrt {\dfrac{{\left( {6.67 \times {{10}^{ - 11}}} \right)\left( {1 \times {{10}^{11}}} \right)}}{{9.8}}} $
$ \Rightarrow {r_b} = \sqrt {\dfrac{{6.67}}{{9.8}}} $;
Do the necessary calculation:
$ \Rightarrow {r_b} = \sqrt {0.68} $
The distance that is required by the black hole to apply the same pull as earth’s gravity is:
$ \Rightarrow {r_b} \simeq 0.8m$;

The distance from your head would its gravitational pull on you match that of Earth’s would be approximately 0.8m.

Note:Here in this question there is no need for first finding out the gravitational force on the person due to Earth and then equating it with the gravitational force due to black hole. Just apply the formula for gravitational force and write it in terms of the distance and solve for the unknown.