If mass is written as then the value of will be (Constants have their usual meaning with a dimensionless constant)

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Solution

Q.
If mass is written as $m = k c^{P} G^{- 1 / 2} h^{1 / 2}$ then the value of $P$ will be (Constants have their usual meaning with $k$ a dimensionless constant) [2024]

1 1/2

2 1/3

3 2

4 - 1/3

Ans.
(1)

   $m = K C^{P} G^{- 1 / 2} h^{1 / 2}$ ...(1)

$G = \frac{F r^{2}}{m_{1} \cdot m_{2}} = \frac{M L T^{- 2} \cdot L^{2}}{M^{2}} = [M^{- 1} L^{3} T^{- 2}]$

Now using (1)

   $[M] = {[L T^{- 1}]}^{P} {[M^{- 1} L^{3} T^{- 2}]}^{- 1 / 2} {[M L^{2} T^{- 1}]}^{1 / 2}$

  Compare the dimension of T from L.H.S and R.H.S then

   $- P + \frac{1}{2} = 0 \Rightarrow P = \frac{1}{2}$

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