question_answer

If the speed of light \[c\left( =\text{3}\times \text{1}{{0}^{\text{8}}}\text{m}/\text{s} \right)\], Planck's constant \[h\left( =\text{6}\text{.6}\times \text{1}{{0}^{-34}}\text{J}-\text{s} \right)\]and gravitational constant \[G\left( =6.67\times {{10}^{-11}}mkx\,units \right)\]be chosen as fundamental units, find out the dimensions and value of unit of mass.

Answer:

                Let      \[m=k{{c}^{x}}{{h}^{y}}{{G}^{z}}\]                 \[\left[ {{M}^{1}}{{L}^{0}}{{T}^{0}} \right]={{\left[ L{{T}^{-1}} \right]}^{x}}{{\left[ M{{L}^{2}}{{T}^{-1}} \right]}^{y}}{{\left[ {{M}^{-1}}{{L}^{3}}{{T}^{-2}} \right]}^{z}}\]                                  ...        (i) \[={{M}^{y-z}}.{{L}^{x+2y+3z}}{{T}^{-x-y-2z}}\] Applying the principle of homogeneity of dimensions. \[y-z=1\]                                                                                                              ...       (ii) \[x+2y+3z=0\]                                                                                                   ...       (iii) \[-x-y-2z=0\]                                                                                                      ...      (iv) Adding (iii) and (iv) we get \[y+z=0\] Adding (ii) and (v) \[2y=1,\,y=1/2\] From (v),         \[\text{z }=-y=-\text{1}/\text{2}\] From (iii),         \[x=-2y-3z=-1+3/2=1/2\]                 \[\therefore \]from (i), \[m=k{{c}^{1/2}}{{h}^{1/2}}{{G}^{-1/2}}=k\sqrt{\frac{ch}{G}}\]                                 Taking \[k=1\]; \[m=\sqrt{\frac{3\times {{10}^{8}}\times 6.6\times {{10}^{-34}}}{6.67\times {{10}^{-11}}}}\approx {{10}^{-7}}kg\]