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Q.
Planck’s constant $(h)$, speed of light in vacuum $(c)$ and Newton’s gravitational constant $(G)$ are three fundamental constants. Which of the following combinations of these has the dimension of length?
NEETNEET 2016Physical World, Units and Measurements
Solution:
Step 1: Write the SI units of given physical quantities
The SI unit of $h =$ J.s or N.m.s
SI unit of $c = m / s$
S.I unit of $G =\frac{ N \cdot m ^2}{ Kg ^2}$
Step 2: Finding the units of all options
For A: $\sqrt{\frac{\left( N \cdot m ^2 / kg ^2\right)( m / s )}{( N \cdot m \cdot s )^{3 / 2}}}=\frac{ kg \cdot m ^3}{ N \cdot s ^5}=\frac{ kg \cdot m ^3}{\left( kg \cdot m / s ^2\right)\left( s ^5\right)}= m ^2 / s ^3$
For B: $\frac{\sqrt{( N \cdot m \cdot s )\left( N \cdot m ^2 / kg ^2\right)}}{( m / s )^{3 / 2}}=\frac{ Ns }{ kg }=\frac{\left( kg \cdot m / s ^2\right)\left( s ^2\right)}{ kg }= m$, it is the unit length.
For C: $\frac{\sqrt{( N \cdot m \cdot s )\left( N \cdot m ^2 / kg ^2\right)}}{( m / s )^{5 / 2}}=\frac{ Ns s ^3}{ m \cdot kg }=\frac{\left( kg \cdot m / s ^2\right)\left( s ^3\right)}{ m \cdot kg }= s$
For D: $\sqrt{\frac{( N \cdot m \cdot s )( m / s )}{ N \cdot m ^2 / kg ^2}}= kg$