Dimensional Analysis and its Applications

An artificial satellite is revolving around a planet of mass $M$ and radius $R$, in a circular orbit of radius $r$. From Kepler’s Third law about the period of a satellite around a common central body, square of the period of revolution $T$ is proportional to the cube of the radius of the orbit $r$. Show using dimensional analysis, that
  $T=$ $\frac{k}{R}$$\sqrt{\frac{r^3}{g}}$
where $k$ is a dimensionless constant and $g$ is acceleration due to gravity.
 

If velocity of light $c$ , Planck’s constant $h$ and gravitational constant $G$ are taken as fundamental quantities then express mass, length and time in terms of dimensions of these quantities.

Mars has approximately half of earth's diameter. When it is closest to earth, it is at about $\frac{1}{2}$ A.U. from the earth. Calculate what size it will appear when seen through the same telescope. (Comment: This is to illustrate why a telescope can magnify planets but not stars.)

Calculate the solid angle subtended by the periphery of an area $1{\mathrm{cm}}^2$ at a point situated symmetrically at a distance of $5\mathrm{cm}$ from the area.

Calculate the length of the arc of a circle of radius $31.0 \mathrm{cm}$ which subtends an angle of $\frac{\mathrm{\pi }}{6}$ at the center.

Why length, mass and time are chosen as base quantities in mechanics?

A function $f\left(\theta \right)$ is defined as:

$f\left(\theta \right)=1-\theta +\frac{{\theta }^2}{2!}-\frac{{\theta }^2}{3!}+\frac{{\theta }^4}{4!}-\dots \dots$

Why is it necessary for $f\left(\theta \right)$ to be a dimensionless quantity?

Express unified atomic mass unit in kg.

Name the device used for measuring the mass of atoms and molecules.

The radius of atom is of the order of 1 $1 Å$and radius of nucleus is of the order of fermi. How many magnitudes higher is the volume of atom as compared to the volume of nucleus?