Question

If the velocity of light c,the constant of gravitation G, and planks constant h be the chosen as fundamental units, Find the dimension of mass,length, time in terms of c, G, h. ​

Answer 1

Q. If the velocity of light c,the constant of gravitation G, and planks constant h be the chosen as fundamental units, Find the dimension of mass,length, time in terms of c, G, h.

Ans:-

$(m) = ( {h}^{ \frac{1}{2} } {c}^{ \frac{1}{2} } {g}^{ \frac{ - 1}{2} } )$

$(l) = ({h}^{ \frac{1}{2} } {c}^{ \frac{ - 3}{2} } {g}^{ \frac{ 1}{2} } )$

$(t) = ( {h}^{ \frac{1}{2} } {c}^{ \frac{ - 5}{2} } {g}^{ \frac{1}{2} } )$

$= [tex](t) = ( {h}^{ \frac{1}{2} } {c}^{ \frac{ - 5}{2} } {g}^{ \frac{1}{2} } )$[/tex]

$(t) = ( {h}^{ \frac{1}{2} } {c}^{ \frac{ - 5}{2} } {g}^{ \frac{1}{2} } )$

Explanation:-

We have,

$(c) = (l {t}^{ - 1} )$

$(g) = ( {m}^{ - 1} {l}^{3} {t}^{ - 2} )$

$(h) = (m {l}^{2} {t}^{ - 1} )$

$= > \frac{(h)(c)}{(g)} = \frac{m {l}^{2} {t}^{ - 1 } \times l {t}^{ - 1} }{ {m}^{ - 1} {l}^{3} {t}^{ - 2} }$

$= {m}^{2}$

$(m) = {h}^{ \frac{1}{2} } {c}^{ \frac{1}{2} } {g}^{ \frac{ - 1}{2} }$

Again,

$\frac{(h)}{(c)} = \frac{m {l}^{2} {t}^{ - 1} }{l {t}^{ - 1} }$

$= ml$

$2 = > (l) = \frac{h}{c(m)}$

$= \frac{h}{c {h}^{ \frac{1}{2} } {c}^{ \frac{1}{2} } {g}^{ \frac{ - 1}{2} } }$

$= ({h}^{ \frac{1}{2} } {c}^{ \frac{ - 3}{2} } {g}^{ \frac{ 1}{2} } )$

$(c) = l {t}^{ - 1}$

$(t) = \frac{(l)}{c}$

$= \frac{({h}^{ \frac{1}{2} } {c}^{ \frac{ - 3}{2} } {g}^{ \frac{ 1}{2} } )}{c}$

Answer 2

Answer:

Explanation:

Given :

If the velocity of light (c) ,the constant of gravitation (G) , and plancks constant (h) be the chosen as fundamental units,

Find the dimension of mass,length, time in terms of c, G, h.

Solution :

We know that,.

The unit of SPEED of light (c) = m/s = [LT⁻¹]

The unit of constant of GRAVITATION = Nm²/kg² = m³/kgs² = [M⁻¹L³T⁻²]

The unit for planck's constant = Js = kgm²/s² × s = kgm²/s = [ML²T⁻¹]

To find units for mass (m) , length (l)  , time (s),

We should use the 4 operators in such way that, we should get them,.

⇒ For mass(m),

$[ML^0T^0] = c^xG^yp^z$

⇒ $[ML^0T^0] = [M^{(z - y)}L^{(x+3y+2z)}T^{(-x-2y-z)}]$

⇒ z - y = 1 , x + 3y + 2z = 0 , -x - 2y - z = 0,

⇒ z - y = 1 .  y = - z,

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

⇒ MASS = $[c^xG^yp^z] = [c^{\frac{1}{2}}G^{ \frac{-1}{2}}h^{ \frac{1}{2}}]$

For length,

let $[M^0LT^0] = [c^xG^yh^z]$

⇒ $[M^0LT^0] = [M^{(z - y)}L^{(x+3y+2z)}T^{(-x-2y-z)}]$

⇒ z - y = 0 , x + 3y  + 2z = 1 , - x - 2y - z = 0

⇒ z = y , y + z = 1 , x + 5y = 1

⇒ $x = \frac{-3}{2} , y = \frac{1}{2} , z = \frac{1}{2}$

⇒ LENGTH = $c^xG^yh^z = c^{\frac{-3}{2}}G^{\frac{1}{2}}h^{\frac{1}{2}}$

For time,

let $[M^0L^0T] = [c^xG^yh^z]$

⇒ $[M^0L^0T] = [M^{(z - y)}L^{(x+3y+2z)}T^{(-x-2y-z)}]$

⇒ z - y = 0 , x + 3y + 2z = 0 , - x - 2y - z = 1

⇒ z = y , y + z = 1 , x + 5z = 0

⇒ $x = \frac{-5}{2} , y = \frac{1}{2} , z = \frac{1}{2}$

⇒ TIME = $c^xG^yh^z = c^{\frac{-5}{2}}G^{\frac{1}{2}}h^{\frac{1}{2}}$