Which one of the following combinations of co | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The dimensions of the given constants are:$G = [M^{-1} L^{3} T^{-2}]$$h = [M L^{2} T^{-1}]$$c = [L T^{-1}]$Let the expression be $T = G^{a} h^{b}

Vedclass · Accepted Answer

$\left[\frac{G h}{c^{5}}\right]^{\frac{1}{2}}$

Vedclass · Answer

(A) The dimensions of the given constants are:$G = [M^{-1} L^{3} T^{-2}]$$h = [M L^{2} T^{-1}]$$c = [L T^{-1}]$Let the expression be $T = G^{a} h^{b} c^{d}$.Substituting the dimensions:$[T^{1}] = [M^{-1} L^{3} T^{-2}]^{a} [M L^{2} T^{-1}]^{b} [L T^{-1}]^{d}$$[M^{0} L^{0} T^{1}] = [M^{-a+b} L^{3a+2b+d} T^{-2a-b-d}]$Comparing the powers on both sides:For $M$: $-a + b = 0 \implies a = b$For $L$: $3a + 2b + d = 0 \implies 3a + 2a + d = 0 \implies d = -5a$For $T$: $-2a - b - d = 1 \implies -2a - a - (-5a) = 1 \implies 2a = 1 \implies a = 1/2$Thus,$a = 1/2, b = 1/2, d = -5/2$.The expression is $G^{1/2} h^{1/2} c^{-5/2} = \left[\frac{G h}{c^{5}}\right]^{\frac{1}{2}}$.

List-$I$	List-$II$
$A$. Meter $(L)$	$I$. $\sqrt{\frac{hc}{G}}$
$B$. Second $(S)$	$II$. $\sqrt{\frac{Gh}{c^5}}$
$C$. Kilogram $(M)$	$III$. $\sqrt{\frac{L^2c^3}{Gh}}$
$D$. Kelvin $(K)$	$IV$. $\sqrt{\frac{Gh}{c^3}}$

Which one of the following combinations of constants has the dimensions of time? $[G=$ gravitational constant,$h=$ Planck's constant,$c=$ velocity of light$]$

Similar Questions

Given a charge $q$,current $I$ and permeability of vacuum $\mu_0$. Which of the following quantity has the dimension of momentum?

In the equation $S = a + bt + ct^2$,where $S$ is measured in metres and $t$ is measured in seconds,the unit of $c$ is:

Density of a liquid in $CGS$ system is $0.625 \ g/cm^3$. What is its magnitude in $SI$ system?

Vedclass Test Series

Exam Paper Generator

Online Exam Module