Speed of light (c) , gravitational constant (G) and planck's constant (h) are taken as fundament

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1.Units, Dimensions and Measurement

hard

The speed of light $(c)$, gravitational constant $(G)$ and planck's constant $(h)$ are taken as fundamental units in a system. The dimensions of time in this new system should be

${G^{1/2}}{h^{1/2}}{c^{ - 5/2}}$

${G^{ - 1/2}}{h^{1/2}}{c^{1/2}}$

${G^{1/2}}{h^{1/2}}{c^{ - 3/2}}$

${G^{1/2}}{h^{1/2}}{c^{1/2}}$

(AIIMS-2008) (JEE MAIN-2019)

Solution

Let time, $T \propto c^{x} G^{y} h^{z}$

$\Rightarrow T=k c^{x} G^{y} h^{z}$

Taking dimensions on both sides $\left[M^{0} L^{0} T^{1}\right]=\left[L T^{-1}\right]^{x}\left[M^{-1} L^{3} T^{-2}\right]^{y}\left[M L^{2} T^{-1}\right]^{z}$

$i . e$

$\left[M^{0} L^{0} T^{1}\right]=\left[M^{-y+z} L^{x+3 y+2 z} T^{-x-2 y-z}\right]$

Equating power of $M, L,$ Ton both sides, we get

$-y+z=0 \quad \ldots(1)$

$x+3 y+2 z=0 \quad \ldots(2)$

$-x-2 y-z=1 \quad \ldots .(3)$

From $(1) \Rightarrow z=y$

Adding (2) and $(3) \Rightarrow y+z=1$

or $2 y=1 \quad[\text { From }]$

i.e, $y=\frac{1}{2}$

$\therefore z=y=\frac{1}{2}$

Putting these values in (2) we get $x+\frac{3}{2}+1=0$ or $x=\frac{-5}{2}$

Hence$,[T]=\left[G^{1 / 2} h^{1 / 2} c^{-5 / 2}\right]$

Standard 11

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The speed of light $(c)$, gravitational constant $(G)$ and planck's constant $(h)$ are taken as fundamental units in a system. The dimensions of time in this new system should be

Solution

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