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
$y-z=0$
$x+2 y+3 z=1$
$-x-y-2z=1$
$y =\frac{1}{2}, z =\frac{1}{2}, x =-\frac{5}{2}$
$T = kc ^{-\frac{5}{2}} h ^{\frac{1}{2}} B ^{\frac{1}{2}}$
$T =k \sqrt{\frac{h G}{c^{5}}}$
Write principle of Homogeneity of dimension.
The potential energy $u$ of a particle varies with distance $x$ from a fixed origin as $u=\frac{A \sqrt{x}}{x+B}$, where $A$ and $B$ are constants. The dimensions of $A$ and $B$ are respectively
The equation of a circle is given by $x^2+y^2=a^2$, where $a$ is the radius. If the equation is modified to change the origin other than $(0,0)$, then find out the correct dimensions of $A$ and $B$ in a new equation: $(x-A t)^2+\left(y-\frac{t}{B}\right)^2=a^2$.The dimensions of $t$ is given as $\left[ T ^{-1}\right]$.
Given that $\int {{e^{ax}}\left. {dx} \right|} = {a^m}{e^{ax}} + C$, then which statement is incorrect (Dimension of $x = L^1$) ?