Planck's constant h, speed of light c and gravitational constant G are used to from a unit of length L and a unit of mass M. Then the correct option (s) is / (are)

Options

A. `M prop sqrt(c)`
B. `M prop sqrt(G)`
C. `L prop sqrt(h)`
D. `L prop sqrt(G)`

Answer

a, c, d

Explanation

For M
`[M]=[h]^(p)[C]^(q)[G]^(r)`
`[M]=[M^(1)L^(2)LT^(-1)]^(p)[L^(1)T^(-1)]^(p)[L^(1)T^(-1)]^(q)[M^(-1)L^(3)T^(-2)]^(r)`
`[M]=[M]^(p-r)[L]^(2p+q+3r)[T]^(p-q-2r)`
`:. P-r=1` …(i)
`2p+q+3r=0` …(ii)
`-p-q-2r=0` …(iii)
On solving Eqns. (i), (ii) and (iii), we get
`p=1/2, r=(-1)/2` and `q=1/2 rArr [M] prop sqrt(h)`
`[M] prop sqrt(C)`
`[M] prop 1/sqrt(G)`
Siilarly for `[L]`
`p-r=0` ...(iv)
`2p+q+3r=1` ...(v)
`-p-q-2r=0` ...(vi)
On solving Eqn. (iv), (v) and (vi)
`p=1/2, q=(-3)/2, r=1/2 rArr [L] prop sqrt(h)`
`[L] prop C^(3//2)`
`[L] prop sqrt(G)`