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  • If the velocity of light c, universal gravitational constant G and Planck's constant h are chosen as fundamental quantities. The dimensions of mass in the new system is :<div class='qna-opt

If the velocity of light c, universal gravitational constant G and Planck's constant h are chosen as fundamental quantities. The dimensions of mass in the new system is :

Option: 1

\left[h^{1 / 2} c^{-1 / 2} G^1\right]


Option: 2

\left[h^{-1 / 2} c^{1 / 2} G^{1 / 2}\right]


Option: 3

\left[h^{1 / 2} c^{1 / 2} G^{-1 / 2}\right]


Option: 4

\left[h^1 c^1 \mathrm{G}^{-1}\right]


Answers (1)

best_answer

\begin{aligned} & {[\mathrm{M}]=[\mathrm{G}]^{\mathrm{x}}[\mathrm{h}]^{\mathrm{y}}[\mathrm{c}]^{\mathrm{z}}} \\ & {[\mathrm{M}]=\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]^{\mathrm{x}}\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]^4\left[\mathrm{LT}^{-1}\right]^{\mathrm{z}}} \\ & {\left[\mathrm{M}^2 \mathrm{~L}^0 \mathrm{~T}^0\right]=\left[\mathrm{M}^{-\mathrm{x}+\mathrm{y}}\right]\left[\mathrm{L}^{3 \mathrm{x}+2 \mathrm{y}+\mathrm{z}}\right]\left[\mathrm{T}^{-2 \mathrm{x}-\mathrm{y}-\mathrm{z}}\right]} \end{aligned}

\begin{aligned} & y-x=1 \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, ...(1)\\ & 3 x+2 y+z=0 \, \, \, \, \, \, .....(2)\\ \end{aligned}

-2 x-y-z=0\, \, \, \, \, ....(3)

On solving,  \mathrm{x}=-\frac{1}{2}, \mathrm{y}=\frac{1}{2}, \mathrm{z}=\frac{1}{2}

So   \mathrm{m}=\sqrt{\frac{\mathrm{hc}}{\mathrm{G}}}

Posted by

Rakesh

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