The frequency of vibration of string is given by $\nu = \frac{p}{{2l}}{\left[ {\frac{F}{m}} \right]^{1/2}}$. Here $p$ is number of segments in the string and $l$ is the length. The dimensional formula for $m$ will be

An object is moving through the liquid. The viscous damping force acting on it is proportional to the velocity. Then dimension of constant of proportionality is

Planck's constant $h$, speed of light $c$ and gravitational constant $G$ are used to form a unit of length $L$ and a unit of mass $M$. Then the correct option$(s)$ is(are)

$(A)$ $M \propto \sqrt{ c }$ $(B)$ $M \propto \sqrt{ G }$ $(C)$ $L \propto \sqrt{ h }$ $(D)$ $L \propto \sqrt{G}$

A function $f(\theta )$ is defined as $f(\theta )\, = \,1\, - \theta  + \frac{{{\theta ^2}}}{{2!}} - \frac{{{\theta ^3}}}{{3!}} + \frac{{{\theta ^4}}}{{4!}} + ...$ Why is it necessary for  $f(\theta )$  to be a dimensionless quantity ?

Force $(F)$ and density $(d)$ are related as $F\, = \,\frac{\alpha }{{\beta \, + \,\sqrt d }}$ then dimension of $\alpha $ are

If the velocity of light $c$, universal gravitational constant $G$ and planck's constant $h$ are chosen as fundamental quantities. The dimensions of mass in the new system is