If velocity of light $c$, Planck’s constant $h$ and gravitational constant $G$ are taken as fundamental quantities, then express time in terms of dimensions of these quantities.
If velocity of light $c$, Planck’s constant $h$ and gravitational constant $G$ are taken as fundamental quantities, then express time in terms of dimensions of these quantities.
$T = kc ^{ x } h ^{ y } G ^{ z }$
${\left[M^{0} L^{0} T\right]=\left[L^{-1}\right]^{x} \times\left[M L^{2} T^{-1}\right]^{y} \times\left[M^{-1} L^{3} T^{-2}\right]^{z}}$
$=\left[M^{y-z} L^{x+2 y+3 z} T^{-x-y-2 z}\right]$
Comparing powers
$y-z=0$
$x+2 y+3 z=1$
$-x-y-2z=1$
$y =\frac{1}{2}, z =\frac{1}{2}, x =-\frac{5}{2}$
$T = kc ^{-\frac{5}{2}} h ^{\frac{1}{2}} B ^{\frac{1}{2}}$
$T =k \sqrt{\frac{h G}{c^{5}}}$
Similar Questions
Match List $I$ with List $II$
List $I$
List $II$
$A$ Spring constant
$I$ $(T ^{-1})$
$B$ Angular speed
$II$ $(MT ^{-2})$
$C$ Angular momentum
$III$ $(ML ^2)$
$D$ Moment of Inertia
$IV$ $(ML ^2 T ^{-1})$
Choose the correct answer from the options given below
Match List $I$ with List $II$
| List $I$ | List $II$ |
| $A$ Spring constant | $I$ $(T ^{-1})$ |
| $B$ Angular speed | $II$ $(MT ^{-2})$ |
| $C$ Angular momentum | $III$ $(ML ^2)$ |
| $D$ Moment of Inertia | $IV$ $(ML ^2 T ^{-1})$ |
Choose the correct answer from the options given below
- [JEE MAIN 2023]
In the relation $y = a\cos (\omega t - kx)$, the dimensional formula for $k$ is
In the relation $y = a\cos (\omega t - kx)$, the dimensional formula for $k$ is
If velocity $v$, acceleration $A$ and force $F$ are chosen as fundamental quantities, then the dimensional formula of angular momentum in terms of $v,\,A$ and $F$ would be
If velocity $v$, acceleration $A$ and force $F$ are chosen as fundamental quantities, then the dimensional formula of angular momentum in terms of $v,\,A$ and $F$ would be
Velocity $(v)$ and acceleration $(a)$ in two systems of units $1$ and $2$ are related as $v _{2}=\frac{ n }{ m ^{2}} v _{1}$ and $a_{2}=\frac{a_{1}}{m n}$ respectively. Here $m$ and $n$ are constants. The relations for distance and time in two systems respectively are
- [JEE MAIN 2022]
Which of the following equations is dimensionally incorrect?
Where $t=$ time, $h=$ height, $s=$ surface tension, $\theta=$ angle, $\rho=$ density, $a, r=$ radius, $g=$ acceleration due to gravity, ${v}=$ volume, ${p}=$ pressure, ${W}=$ work done, $\Gamma=$ torque, $\varepsilon=$ permittivity, ${E}=$ electric field, ${J}=$ current density, ${L}=$ length.
Which of the following equations is dimensionally incorrect?
Where $t=$ time, $h=$ height, $s=$ surface tension, $\theta=$ angle, $\rho=$ density, $a, r=$ radius, $g=$ acceleration due to gravity, ${v}=$ volume, ${p}=$ pressure, ${W}=$ work done, $\Gamma=$ torque, $\varepsilon=$ permittivity, ${E}=$ electric field, ${J}=$ current density, ${L}=$ length.
- [JEE MAIN 2021]