Derive Planck Length in terms of c, G, and h_bar

[MCS5280]

Homework Statement

a. The first part of this problem was to derive the escape speed (Vesc) for a star of mass M and radius R.

b. The second part is where I am having trouble. It says to equate the Vesc calculated above and derive a formula for Planck length in terms of c, G, and h_bar.

Homework Equations

This is from the Heisenberg uncertainty section for position and momentum so:

$$\Delta$$X * $$\Delta$$P = h_bar/2

E = mc²

The Attempt at a Solution

The escape velocity I calculated from part a was Vesc = $$\sqrt{2GM/R}$$

Equating Vesc to C gives:

C = $$\sqrt{2GM/R}$$

Am I supposed to assume that $$\Delta$$p = Mc and solve for $$\Delta$$x using the Heisenberg uncertainty principle?

I have tried a couple of different methods and I can't get this to simplify down to the form I need. I am pretty sure this is an easy problem, I just am not seeing the trick.

 

[Delphi51]

According to Wikipedia the Planck length is defined as sqrt(hG/c^3)
I think it is just a way to get units of meters out of those fundamental constants.
I don't think it has a physical meaning, except in quantum mechanics which doesn't seem to apply to the problem you have.

 

[MCS5280]

Ok I think I am getting close to this one:

c=$$\sqrt{\frac{2GM}{R}}$$

M=$$\frac{c^2 R}{2G}$$

Using the Heisenberg Principal:
$$\Delta$$X $$\Delta$$P = $$\frac{h_bar}{2}$$

$$\Delta$$P = c M

Solving this for $$\Delta$$X
$$\Delta$$X = $$\frac{G h_bar}{c^3 R}$$

How do I get the square root and get rid of the R?

 

[pavi_elex]

Gravity level
c =under root of (2GM/R)
where R=L (Planck's Length) on gravity level
so c =under root of (2GM/L)

Uncertainty principle
uncertainty in momentum x uncertainty in position = reduced Planck constant/2
mv x L = reduced Planck constant/2
here on quantum level, Uncertainty in position is Planck length and velocity is c (velocity of light)
mcL= reduced h/2
m=reduced h/(2cL)

put the value of m in value of c in gravity level
c square = 2 G reduced h/(2cL*R)
c square = 2 G reduced h/(2cL*L)
where R=L (Planck length on gravity level)
Derive L (Planck length) from above equation.