Density of wood is $0.5 \;gm / cc$ in the $CGS$ system of units. The corresponding value in $MKS$ units is.?

$A$ and $B$ possess unequal dimensional formula then following operation is not possible in any case:-

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 $\left[ T ^{-1}\right]$.

The force of interaction between two atoms is given by $F\, = \,\alpha \beta \,\exp \,\left( { - \frac{{{x^2}}}{{\alpha kt}}} \right);$ where $x$ is the distance, $k$ is the Boltzmann constant and $T$ is temperature and $\alpha $ and $\beta $ are two constants. The dimension of $\beta $ is

A force $F$ is given by $F = at + b{t^2}$, where $t$ is time. What are the dimensions of $a$ and $b$

Given that $v$ is the speed, $r$ is radius and $g$ is acceleration due to gravity. Which of the following is dimensionless?