Question

Consider a new system of units in which c (speed of light in vacuum), h(Planck's constant) and G (gravitational constant) are taken as fundamental units. Which of the following would correctly represent mass in this new system?

• A. $\sqrt{\frac{hc}{G}}$
• B. $\sqrt{\frac{Gc}{h}}$
• C. $\sqrt{\frac{hG}{c}}$
• D. $\sqrt{hGc}$

Answer 1

The correct option is A $\sqrt{\frac{hc}{G}}$

Consideration anew system of units in which speed of light in vaccum$=e$Plank's constant$=h$

Gravitational constant$=G$

Mass in the new system

$F=\frac{1}{2}m{v}^2=h\gamma$

Now,

Using the method of dimension we say that,

Here

$e=L{T}^{-1}$

$\left[G\right]=M^{-1}{L}^3{T}^{-2}$

$\left[h\right]=M^1{L}^2{T}^{-2}$

Let, $ma{c}^x{G}^0{h}^z$

By substituting the dimension of each quantity in both the sides

$M=\left(L{T}^{-1}{)}^x\right(M^{-1}{L}^3{T}^{-2}{)}^y(M{L}^2{T}^{-1}{)}^{z}M=(M^{y+z}{L}^{x+3y+2z}{T}^{x-2y-x})$

By equating the power of $M,LandT$ in both the sides

$-y+z=1x+3y+2z=0-x-2y-z=0$

By solving the above equations

We get,

$x=\frac{1}{z},y=-\frac{1}{z},z=\frac{1}{z}$

So, $\left[m\right]=e^{1/2}{G}^{-1/2}{h}^{1/2}=\sqrt{\frac{hc}{G}}$