If energy $E = G^p h^q c^r$,where $G$ is the universal gravitational constant,$h$ is Planck's constant,and $c$ is the speed of light,find the values of $p, q$,and $r$ respectively.

In an experiment,four quantities $a, b, c,$ and $d$ are measured with percentage errors of $1\%, 2\%, 3\%,$ and $4\%$ respectively. The quantity $P$ is calculated as $P = \frac{a^3 b^2}{cd}$. The percentage error in $P$ is ........ $\%$

In a system of units,if force $(F)$,acceleration $(A)$,and time $(T)$ are taken as fundamental units,then the dimensional formula of energy is:

Young-Laplace law states that the excess pressure inside a soap bubble of radius $R$ is given by $\Delta P = 4 \sigma / R$,where $\sigma$ is the coefficient of surface tension of the soap. The $EOTVOS$ number $E_0$ is a dimensionless number that is used to describe the shape of bubbles rising through a surrounding fluid. It is a combination of $g$ (acceleration due to gravity),$\rho$ (density of the surrounding fluid),$\sigma$ (surface tension),and a characteristic length scale $L$ (radius of the bubble). $A$ possible expression for $E_0$ is:

$E, m, l$ and $G$ denote energy,mass,angular momentum,and gravitational constant respectively. Then the dimensions of $\frac{El^2}{m^5G^2}$ are:

$A$ great physicist of this century ($P.A.M.$ Dirac) loved playing with numerical values of fundamental constants of nature. This led him to an interesting observation. Dirac found that from the basic constants of atomic physics ($c, e,$ mass of electron, mass of proton) and the gravitational constant $G$, he could arrive at a number with the dimension of time. Further, it was a very large number, its magnitude being close to the present estimate on the age of the universe ($\approx 15$ billion years). From the table of fundamental constants in this book, try to see if you too can construct this number. If its coincidence with the age of the universe were significant, what would this imply for the constancy of fundamental constants?