If velocity of light c , Planck’s constant h and gravitational constant G are taken as fundame

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1.Units, Dimensions and Measurement

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If velocity of light $c$, Planck’s constant $h$ and gravitational constant $G$ are taken as fundamental quantities, then express time in terms of dimensions of these quantities.

Option A

Option B

Option C

Option D

Solution

$T = kc ^{ x } h ^{ y } G ^{ z }$

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

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

Comparing powers

$y-z=0$

$x+2 y+3 z=1$

$-x-y-2z=1$

$y =\frac{1}{2}, z =\frac{1}{2}, x =-\frac{5}{2}$

$T = kc ^{-\frac{5}{2}} h ^{\frac{1}{2}} B ^{\frac{1}{2}}$

$T =k \sqrt{\frac{h G}{c^{5}}}$

Standard 11

Physics

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Similar Questions

A book with many printing errors contains four different formulas for the displacement $y$ of a particle undergoing a certain periodic motion:

$(a)\;y=a \sin \left(\frac{2 \pi t}{T}\right)$

$(b)\;y=a \sin v t$

$(c)\;y=\left(\frac{a}{T}\right) \sin \frac{t}{a}$

$(d)\;y=(a \sqrt{2})\left(\sin \frac{2 \pi t}{T}+\cos \frac{2 \pi t}{T}\right)$

$(a=$ maximum displacement of the particle, $v=$ speed of the particle. $T=$ time-period of motion). Rule out the wrong formulas on dimensional grounds.

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