The speed of light ( c), gravitational c...

The speed of light ( c), gravitational constant (G) and plank's constant (h) are taken as fundamental units in a system. The dimensions of time in this new system should be.

`G^(1/2)h^(1/2)c^((-5)/2`

`G^((-1)/2)h^(1/2)c^(1/2)`

`G^(1/2)h^(1/2)c^((-3)/2`

`G^(1/2)h^(1/2)c^(1/2)`

Text Solution

Verified by Experts

The correct Answer is:

Let time `Tpropc^(X)G^(Y)h^(Z)`
`impliesT=kc^(X)G^(Y)h^(Z)`
Taking dimensions on both sides
`[M^(0)L^(0)T^(1)]`
`=[LT^(-1)]^(X)[M^(-1)L^(3)T^(-2)]^(Y)[ML^(2)T^(-1)]^(Z)`
i.e.
`[M^(0)L^(0)T^(1)]`
`=[M^(-Y+Z)L^(X+3Y+2Z)T^(-X-2Y-Z)]`
Equation powers of M,L and T on both sides we get
`-Y+Z=0`.........1
`X+3Y+2Z=0`.......2
`-X-2y-Z=1`..........3
From equation 1
`Y=Z`
Adding 2 and 3
`Y+Z=1`
or `2Y=1( :. Z=Y)`
`:.Z=Y=1/2`
Putting these values in (2) we get
`x+3/2+1=0` or `X=(-5)/2`
Hence `[T]=G^(1/2)h^(1/2)c^((-5)/2)]`

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If the velocity of light (c) , gravitational constant (G) and Planck's constant (h) are chosen as fundamental units, then the dimensions of mass in new system is

`c^(1//2)G^(1//2)h^(1//2)`

`c^(1//2)G^(1//2)h^(-1//2)`

`c^(1//2)G^(-1//2)h^(1//2)`

`c^(-1//2)G^(1//2)h^(1//2)`

If the velocity of light c, gravitational constant G and Planck's constant h be taken as fundamental units the dimension of mass in the new system will be-

`c^(1//2)h^(1//2)G^(1//2)`

`c^(1//2)h^(1//2)G^(-1//2)`

`c^(-3//2)h^(1//2)G^(1//2)`

`c^(-5//2)h^(1//2)G^(1//2)`

If velocity of light c, planck's constant h and gravitational constnat G are taken as fundamental quantities then the dimensions of the length will be

`sqrt((ch)/(G))`

`sqrt((hG)/(c^(5)))`

`sqrt((hG)/(c^(3)))`

`sqrt((hc^(3))/(G))`