If the velocity of light c, universal gravitational constant G and planck's constant h are chosen as fundamental quantities. The dimensions of mass in the new system is: [2023]

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Q.
If the velocity of light c, universal gravitational constant G and planck's constant h are chosen as fundamental quantities. The dimensions of mass in the new system is: [2023]

1 $[h^{\frac{1}{2}} c^{- \frac{1}{2}} G^{1}]$

2 $[h^{1} c^{1} G^{- 1}]$

3 $[h^{- \frac{1}{2}} c^{\frac{1}{2}} G^{\frac{1}{2}}]$

4 $[h^{\frac{1}{2}} c^{\frac{1}{2}} G^{- \frac{1}{2}}]$

Ans.
(4)

Dimensional formale of mass is $H^{x} C^{y} G^{z}$

$M^{1} = {(M L^{2} T^{- 1})}^{x} (L T^{- 1}) {(M^{- 1} L^{3} T^{- 2})}^{z}$

$M^{1} L^{0} T^{0} = M^{x - z} L^{2 x + y + 3 z} T^{- x - y - 2 z}$

on comparing both side

$x - z = 1$

$2 x + y + 3 z = 0$

$- x - y - 2 z = 0$

On solving above equations we get

$x = \frac{1}{2}, y = \frac{1}{2}, z = \frac{- 1}{2}$

Solution

Q.
If the velocity of light c, universal gravitational constant G and planck's constant h are chosen as fundamental quantities. The dimensions of mass in the new system is: [2023]

1 $[h^{\frac{1}{2}} c^{- \frac{1}{2}} G^{1}]$

2 $[h^{1} c^{1} G^{- 1}]$

3 $[h^{- \frac{1}{2}} c^{\frac{1}{2}} G^{\frac{1}{2}}]$

4 $[h^{\frac{1}{2}} c^{\frac{1}{2}} G^{- \frac{1}{2}}]$

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Solution

Q. If the velocity of light c, universal gravitational constant G and planck's constant h are chosen as fundamental quantities. The dimensions of mass in the new system is: [2023]

1 [h12c–12G1]

2 [h1c1G–1]

3 [h–12c12G12]

4 [h12c12G–12]

Ans. (4) Dimensional formale of mass is HxCyGz M1=(ML2T–1)x(LT–1)(M–1L3T–2)z M1L0T0=Mx–zL2x+y+3zT–x–y–2z on comparing both side x–z=1 2x+y+3z=0 –x–y–2z=0 On solving above equations we get x=12, y=12, z=–12

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Q.
If the velocity of light c, universal gravitational constant G and planck's constant h are chosen as fundamental quantities. The dimensions of mass in the new system is: [2023]

1 $[h^{\frac{1}{2}} c^{- \frac{1}{2}} G^{1}]$

2 $[h^{1} c^{1} G^{- 1}]$

3 $[h^{- \frac{1}{2}} c^{\frac{1}{2}} G^{\frac{1}{2}}]$

4 $[h^{\frac{1}{2}} c^{\frac{1}{2}} G^{- \frac{1}{2}}]$