Question

If the velocity of light , acceleration due to gravity and normal atmospheric pressure are taken as the fundamental units , what will be the units of mass , length, and time ,given velocity of light as3 *10 ⁸ ms ^-1 ,acceleration due to gravity 10 ms ^-2 and normal pressure = 10 ⁵ Nm^-2

Answer 1

We are calculating for the unit of mass.
Suppose,
$$\begin{gathered} \mathrm{m}={\mathrm{kc}}^{\mathrm{x}}{\mathrm{g}}^{\mathrm{y}}{\mathrm{p}}^{\mathrm{z}}---\left(\mathrm{i}\right) \\ \left[{\mathrm{M}}^1{\mathrm{L}}^0{\mathrm{T}}^0\right]=\left[{\mathrm{LT}}^{-1}{]}^{\mathrm{x}}\right[{\mathrm{M}}^0{\mathrm{L}}^1{\mathrm{T}}^{-2}{]}^{\mathrm{y}}\left[{\mathrm{M}}^1{\mathrm{L}}^{-1}{\mathrm{T}}^{-2}{]}^{\mathrm{z}} \\ \right[{\mathrm{M}}^1{\mathrm{L}}^0{\mathrm{T}}^0]={\mathrm{M}}^{\mathrm{z}}{\mathrm{L}}^{\mathrm{x}+\mathrm{y}-\mathrm{z}}{\mathrm{T}}^{-\mathrm{x}-2\mathrm{y}-2\mathrm{z}} \\ \mathrm{Applying}\mathrm{the}\mathrm{principle}\mathrm{of}\mathrm{homogeneity}\mathrm{of}\mathrm{dimensions}, \\ \mathrm{z}=1 \\ \mathrm{x}+\mathrm{y}-\mathrm{z}=0 \\ -\mathrm{x}-2\mathrm{y}-2\mathrm{z}=0 \\ \mathrm{So},\mathrm{x}+\mathrm{y}=1 \\ -\mathrm{x}-2\mathrm{y}=2 \\ \mathrm{On}\mathrm{solving}, \\ -\mathrm{y}=3\mathrm{or}\mathrm{y}=-3 \\ \mathrm{and}\mathrm{x}=1-(-3)=4 \\ \mathrm{Putting}\mathrm{these}\mathrm{values}\mathrm{in}(\mathrm{i}),\mathrm{we}\mathrm{get} \\ \mathrm{m}={\mathrm{kc}}^4{\mathrm{g}}^{-3}{\mathrm{p}}^1 \\ \mathrm{or}\mathrm{m}=\mathrm{k}\frac{{\mathrm{pc}}^4}{{\mathrm{g}}^3} \\ \mathrm{Taking}\mathrm{k}=1 \\ \mathrm{m}=\frac{{\mathrm{pc}}^4}{{\mathrm{g}}^3} \\ \mathrm{m}=\frac{{10}^5{\mathrm{Nm}}^{-2}\times (3\times {10}^8{\mathrm{ms}}^{-2}{)}^4}{[10{\mathrm{ms}}^{-2}{]}^3} \\ =\frac{{10}^5\times 81\times {10}^{32}}{{10}^3} \\ =8.1\times {10}^{31}\mathrm{kg} \end{gathered}$$
Similarly you can calculate for length and time.