Mass of a black hole - given only the diameter

[PirateFan308]

Homework Statement

Cosmologists have speculated that black holes the size of a proton could have formed during the early days of the Big Bang when the universe began. If we take the diameter of a proton to be $1.0*10^{-15}$, what would be the mass of a mini black hole?

Homework Equations

$v=\sqrt{\frac{Gm}{r}}$

The Attempt at a Solution

$v=\sqrt{\frac{Gm}{r}}$

$m=\frac{v^{2}r}{G}$

$m=\frac{(3.0*10^{8})^{2}(0.5*10^{-15})}{(6.67*10^{-11})}$

$m=6.75*10^{11} kg$

It says that this is wrong, but I can't find my mistake. Thanks!

 

[gneill]

The Schwarzschild radius is given by:
$$r_s = \frac{2 G M}{c^2}$$
I think you forgot the factor of 2.

EDIT: D'Oh. No square root! Fixed it.

 

[PirateFan308]

The Schwarzschild radius is given by:
$$r_s = \sqrt{\frac{2 G M}{c^2}}$$
I think you forgot the factor of 2.

Isn't the schwarzchild radius simply:

$R_s=\frac{2GM}{c^2}$

So this would rearrange to

$m=\frac{rc^2}{2G}$

and plugging in the values, I would get $m=3.37*10^{11}kg$

Is this now correct?

 

[gneill]

Isn't the schwarzchild radius simply:

$R_s=\frac{2GM}{c^2}$

So this would rearrange to

$m=\frac{rc^2}{2G}$

and plugging in the values, I would get $m=3.37*10^{11}kg$

Is this now correct?

Yes, and yes. Sorry about the square root distraction, I don't know where my head was at!

 

[PirateFan308]

Thanks for the help!