Question

If volume is written as, V = Kgxcyhz

. Here, K is dimensionless constant and g, c, h are

gravitational constant, speed of light and Plank’s constant, respectively. Find the

value of x/z.​

Answer 1

Explanation:

Speed of light C=[LT

−1

]

gravitational constant G=[M

−1

L

3

T

−2

]

planck's constant h=ML

2

T

−1

Let M=C

x

G

y

h

z

[M]=[LT

−1

]

x

[M

−1

L

3

T

−2

]

y

[ML

2

T

−1

]

z

[M]=[M

−y+z

L

x+3y+2z

T

−x−2y−z

]

−y+z=1,x+3y+2z=0,−x−2y−z=0

On solving, we get

x=

2

1

;y=

2

−1

;z=

2

1

So, dimension of M=[C

1/2

G

−1/2

h

1/2

]

Answer 2

Answer:

The value of x/z is 1.

Explanation:

Given the volume, $V = KG^xc^yh^z$

where K is dimensionless constant, G, c, h are gravitational constant, speed of light and Plank’s constant, respectively.

Dimensional formula is the expression of a derived physical quantity written using in terms of the powers to which the fundamental units are to be raised to obtain one unit of a derived quantity.

The dimensional formula of the given quantities are given by

volume, $V=[L^{3}]$

speed of light, $c=[LT^{-1}]$

Gravitational constant, $G=[M^{-1}L^3T^{-2}]$

Planck's constant, $h=[ML^2T^{-1}]$

Using all the quantities in the formula,

$V = KG^xc^yh^z$

Since K is dimensionless constant, we get

$[L^{3}]= [M^{-1}L^3T^{-2}]^x[LT^{-1}]^y[ML^2T^{-1}]^z$

      $= [M^{-x}L^{3x}T^{-2x}][L^yT^{-y}][M^zL^{2z}T^{-z}]$

       $=[M^{-x+z}L^{3x+y+2z}T^{-2x-y-z}]$

Comparing the coefficients on both sides, we get

$-x+z=0$                                  ...(1)

$3x+y+2z=3$                           ...(2)

$-2x-y-z=0$                          ...(3)

From equation (1), we get

$-x+z=0\implies -x=-z$

$\implies x=z\implies \dfrac{x}{z} =1$

Therefore, the value of x/z is 1.

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