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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
