The quantity with units of length only is [tex]\sqrt{\frac{hG}{c^3}}[/tex]
Explanation:
We have to combine the following constants:
- [tex]h[/tex], Planck constant, with units [tex][m^2][kg][s^{-1}][/tex]
- G, the Newton's gravitational constant, with units [tex][m^3][kg^{-1}][s^{-2}][/tex]
- [tex]c[/tex], the speed of light, with units [tex][m][s^{-1}][/tex]
The combination of these constant should have units of length only, so with meters (m).
First, we notice that [tex]h[/tex] has [kg] in its units, while G has [tex][kg^{-1}][/tex] in its units, so in order to make the [kg] disappear, we have to multiply them and they should have same power, so:
[tex]hG = [m^{2+3}][kg^{1-1}][s^{-1-2}]=[m^5][s^{-3}][/tex]
Now we have to make the seconds, [s], disappear. We do that by dividing the new quantity by [tex]c^3[/tex], so that the new units are:
[tex]\frac{hG}{c^3}=\frac{[m^5][s^{-3}]}{([m][s^{-1}])^3}=\frac{[m^5][s^{-3}]}{[m^3][s^{-3}]}=[m^2][/tex]
We are almost done: now the quantity has units of an area, squared meters. Therefore, in order to make it have it units of length, we just take its square root:
[tex]\sqrt{\frac{hG}{c^3}}=\sqrt{[m^2]}=[m][/tex]
Learn more about gravitational constant:
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