If The Velocity Of Light C, Gravitational Constant G And Planks Const

1.	If The Velocity Of Light C, Gravitational Constant G And Planks Constant H Be Chosen As Fundamental Units, Find The Value Of A Gram, A Cm And A Sec In Term Of New Unit Of Mass, Length And Time Respectively. (take C = 3 X 1010 Cm/sec, G = 6.67 X 108 Dyn Cm2/gram2 And H = 6.6 X 10-27 Erg Sec)?
Answer» Given, c = 3 x 1010 cm/sec G = 6.67 x 108 dyn cm2/gm2 h = 6.6 x 10-27 erg sec Putting respective DIMENSIONS, Dimension formula for c = [M0L1T-1] = 3 x 1010 cm/sec …. (I) Dimensions of G = [M-1L3T-2] = 6.67 x 108dyn cm2/gm2 …(II) Dimensions of h = [M1L2T-1] = 6.6 x 10-27erg sec …(III) (Note: Applying newton’s law of gravitation, you can find dimensions of G i.e. G = Fr2/(mM) Similarly, Planck’s Constant (h) = Energy / frequency) To get M, multiply eqn-I and III and divide by eqn.-II, ⇒ [M0L1T-1].[M1L2T-1].[M1L-3T2] = ( 3 x 1010 cm/sec).( 6.6 x 10-27 erg sec)/ 6.67 x 108 dyn cm2/gm2 ⇒[M2] = 2.968 x 10-9 ⇒[M] = 0.5448 x 10-4 gm or 1gm = [M]/0.5448 x 10-4 = 1.835 x 10-4 unit of MASS To obtain length [L], eqn.-II x eqn-III / cube of eqn.-I i.e. [M-1L3T-2].[M1L2T-1].[M0L-3T3] = (6.67 x 108 dyn cm2/gm2 ).( 6.6 x 10-27erg sec)/(3 x 1010 cm/sec)3 ⇒ [L2] = 1.6304 x 10-65cm2 ⇒ [L] = 0.4038 x 10-32 cm or 1cm = [L]/ 0.4038 x 10-32 = 2.47 x 10-32unit of length In eqn-I, [M0L1T-1] = 3 x 1010cm/sec ⇒ [T] = [L] ÷ 3 x 1010cm/s ⇒ [T] = 0.4038 x 10-32 cm ÷ 3 x 1010cm/s = 0.1345 x 10-42 s or 1S = [T]/0.1345 x 10-42s = 7.42 x 1042unit of time Given, c = 3 x 1010 cm/sec G = 6.67 x 108 dyn cm2/gm2 h = 6.6 x 10-27 erg sec Putting respective dimensions, Dimension formula for c = [M0L1T-1] = 3 x 1010 cm/sec …. (I) Dimensions of G = [M-1L3T-2] = 6.67 x 108dyn cm2/gm2 …(II) Dimensions of h = [M1L2T-1] = 6.6 x 10-27erg sec …(III) (Note: Applying newton’s law of gravitation, you can find dimensions of G i.e. G = Fr2/(mM) Similarly, Planck’s Constant (h) = Energy / frequency) To get M, multiply eqn-I and III and divide by eqn.-II, ⇒ [M0L1T-1].[M1L2T-1].[M1L-3T2] = ( 3 x 1010 cm/sec).( 6.6 x 10-27 erg sec)/ 6.67 x 108 dyn cm2/gm2 ⇒[M2] = 2.968 x 10-9 ⇒[M] = 0.5448 x 10-4 gm or 1gm = [M]/0.5448 x 10-4 = 1.835 x 10-4 unit of mass To obtain length [L], eqn.-II x eqn-III / cube of eqn.-I i.e. [M-1L3T-2].[M1L2T-1].[M0L-3T3] = (6.67 x 108 dyn cm2/gm2 ).( 6.6 x 10-27erg sec)/(3 x 1010 cm/sec)3 ⇒ [L2] = 1.6304 x 10-65cm2 ⇒ [L] = 0.4038 x 10-32 cm or 1cm = [L]/ 0.4038 x 10-32 = 2.47 x 10-32unit of length In eqn-I, [M0L1T-1] = 3 x 1010cm/sec ⇒ [T] = [L] ÷ 3 x 1010cm/s ⇒ [T] = 0.4038 x 10-32 cm ÷ 3 x 1010cm/s = 0.1345 x 10-42 s or 1s = [T]/0.1345 x 10-42s = 7.42 x 1042unit of time

If The Velocity Of Light C, Gravitational Constant G And Planks Constant H Be Chosen As Fundamental Units, Find The Value Of A Gram, A Cm And A Sec In Term Of New Unit Of Mass, Length And Time Respectively. (take C = 3 X 1010 Cm/sec, G = 6.67 X 108 Dyn Cm2/gram2 And H = 6.6 X 10-27 Erg Sec)?

Answer»

Given,

c = 3 x 1010 cm/sec

G = 6.67 x 108 dyn cm2/gm2

h = 6.6 x 10-27 erg sec

Putting respective DIMENSIONS,

Dimension formula for c = [M0L1T-1] = 3 x 1010 cm/sec …. (I)

Dimensions of G = [M-1L3T-2] = 6.67 x 108dyn cm2/gm2 …(II)

Dimensions of h = [M1L2T-1] = 6.6 x 10-27erg sec …(III)

(Note: Applying newton’s law of gravitation, you can find dimensions of G i.e. G = Fr2/(mM)

Similarly, Planck’s Constant (h) = Energy / frequency)

To get M, multiply eqn-I and III and divide by eqn.-II,

⇒ [M0L1T-1].[M1L2T-1].[M1L-3T2]

= ( 3 x 1010 cm/sec).( 6.6 x 10-27 erg sec)/ 6.67 x 108 dyn cm2/gm2

⇒[M2] = 2.968 x 10-9

⇒[M] = 0.5448 x 10-4 gm

or 1gm = [M]/0.5448 x 10-4 = 1.835 x 10-4 unit of MASS

To obtain length [L], eqn.-II x eqn-III / cube of eqn.-I i.e.

[M-1L3T-2].[M1L2T-1].[M0L-3T3]

= (6.67 x 108 dyn cm2/gm2 ).( 6.6 x 10-27erg sec)/(3 x 1010 cm/sec)3

⇒ [L2] = 1.6304 x 10-65cm2

⇒ [L] = 0.4038 x 10-32 cm

or 1cm = [L]/ 0.4038 x 10-32 = 2.47 x 10-32unit of length

In eqn-I, [M0L1T-1] = 3 x 1010cm/sec

⇒ [T] = [L] ÷ 3 x 1010cm/s

⇒ [T] = 0.4038 x 10-32 cm ÷ 3 x 1010cm/s = 0.1345 x 10-42 s

or 1S = [T]/0.1345 x 10-42s = 7.42 x 1042unit of time