Question

if gravitational constant (G), planck's constant (h) and velocity of light (c) are fundamental quantities, the dimension of radius of gyration will be ​

Answer 1

Answer:

Dimension of the Gravitational constant

G

=

[

M

]

−

1

[

L

]

3

[

T

]

−

2

Dimension of Planck's constant

h

=

[

M

]

[

L

]

2

[

T

]

−

1

Dimension of the velocity of light

c

=

[

L

]

[

T

]

−

1

Radius of Gyration will have units of length. Let the dimension of length in new system be

[

L

]

=

[

G

]

x

[

h

]

y

[

c

]

z

Using first three dimensions we get

[

L

]

=

(

[

M

]

−

1

[

L

]

3

[

T

]

−

2

)

x

(

[

M

]

[

L

]

2

[

T

]

−

1

)

y

(

[

L

]

[

T

]

−

1

)

z

⇒

[

M

]

0

[

L

]

1

[

T

]

0

=

[

M

]

−

x

+

y

[

L

]

3

x

+

2

y

+

z

[

T

]

−

2

x

−

y

−

z

Comparing power of respective items on both sides we get

−

x

+

y

=

0

......(1)

3

x

+

2

y

+

z

=

1

......(2)

−

2

x

−

y

−

z

=

0

.....(3)

Adding (2) and (3) we get

x

+

y

=

1

.....(4)

Adding (1) and (4)

2

y

=

1

⇒

y

=

1

2

........(5)

Inserting this value in (4)

x

+

1

2

=

1

⇒

x

=

1

2

.......(6)

Inserting value of

x

and

y

in (2)

3

×

1

2

+

2

×

1

2

+

z

=

1

⇒

z

=

−

3

2

......(7)

Dimension becomes

[

L

]

=

[

G

]

1

2

[

h

]

1

2

[

c

]

−

3

2