If the time period $T \propto P^a d^b E^c$,where $P$ is pressure,$d$ is density,and $E$ is energy,then the value of $c$ is:

From the equation $\tan \theta = \frac{rg}{v^2}$,one can obtain the angle of banking $\theta$ for a cyclist taking a curve (the symbols have their usual meanings). Then say,it is

$A$ force defined by $F = \alpha t^2 + \beta t$ acts on a particle at a given time $t$. The factor which is dimensionless,if $\alpha$ and $\beta$ are constants,is:

$A$ massive black hole of mass $m$ and radius $R$ is spinning with angular velocity $\omega$. The power $P$ radiated by it as gravitational waves is given by $P=G c^{-5} m^{x} R^{y} \omega^{z}$,where $c$ and $G$ are the speed of light in free space and the universal gravitational constant,respectively. Then,

If energy $(E)$,velocity $(V)$,and time $(T)$ are chosen as the fundamental quantities,the dimensional formula of surface tension will be

Write and explain the principle of homogeneity. Check the dimensional consistency of the given equation: $x = x_{0} + v_{0}t + \frac{1}{2}at^{2}$.