General Physics Ques 14
- Planck’s constant $h$, speed of light $c$ and gravitational constant $G$ are used to form a unit of length $L$ and a unit of mass $M$. Then, the correct options is/are
(2015 Adv.)
(a) $M \propto \sqrt{c}$
(b) $M \propto \sqrt{G}$
(c) $L \propto \sqrt{h}$
(d) $L \propto \sqrt{G}$
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Answer:
Correct Answer: 14.( a, c, d )
Solution:
$ \begin{aligned} & M \propto h^a c^b G^c \\ & \mathrm{M}^1 \propto\left(\mathrm{ML}^2 \mathrm{~T}^{-1}\right)^a\left(\mathrm{LT}^{-1}\right)^b\left(\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right)^c \\ & \propto \mathrm{M}^{a-c} \mathrm{~L}^{2 a+b+3 c} \mathrm{~T}^{-a-b-2 c} \\ & a-c=1 …… (i) \\ & 2 a+b+3 c=0 …… (ii) \\ & a+b+2 c=0 …… (iii) \end{aligned} $
On solving (i), (ii), (iii),
$ a=\frac{1}{2}, b=+\frac{1}{2}, c=-\frac{1}{2} $
$\therefore M \propto \sqrt{c}$ only $\rightarrow$ (a) is correct.
In the same way we can find that, $L \propto h^{1 / 2} c^{-3 / 2} G^{1 / 2}$
$L \propto \sqrt{h}, L \propto \sqrt{G} \rightarrow$ (c), (d) are also correct.
BETA
