Expression for time in terms of $\mathrm{G}$ (universal gravitational constant), $\mathrm{h}$ (Planck’s constant) and $\mathrm{c}$ (speed of light) is proportional to:

Question:

Expression for time in terms of $\mathrm{G}$ (universal gravitational constant), $\mathrm{h}$ (Planck’s constant) and $\mathrm{c}$ (speed of light) is proportional to:

  1. $\sqrt{\frac{\mathrm{hc}^{5}}{\mathrm{G}}}$

  2. $\sqrt{\frac{c^{3}}{G h}}$

  3. $\sqrt{\frac{\mathrm{Gh}}{\mathrm{c}^{5}}}$

  4. $\sqrt{\frac{\mathrm{Gh}}{\mathrm{c}^{3}}}$

JEE Main Previous Year Single Correct Question of JEE Main from Chemistry Laws of Motion chapter.

JEE Main Previous Year 2019


Correct Option: 3

Solution:

Let $t \propto G^{x} h^{y} C^{z}$

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

$\mathrm{h}=\left[\mathrm{ML}^{2} \mathrm{~T}^{-1}\right]$ and $\mathrm{C}=\left[\mathrm{LT}^{-1}\right]$

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

By comparing the powers of $\mathrm{M}, \mathrm{L}, \mathrm{T}$ both the sides

$$

\begin{aligned}

&-x+y=0 \Rightarrow x=y \\

&3 x+2 y+z=0 \Rightarrow 5 x+z=0 \\

&-2 x-y-z=1 \quad \Rightarrow 3 x+z=-1

\end{aligned}

$$

Solving eqns. (i) and (ii),

$\mathrm{x}=\mathrm{y}=\frac{1}{2}, \mathrm{z}=-\frac{5}{2} \therefore \mathrm{t} \propto \sqrt{\frac{\mathrm{Gh}}{\mathrm{C}^{5}}}$

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