Dimensional Analysis and its Applications

The photograph of a house occupies an area of $1.75 {\mathrm{cm}}^2$ on a $35 \mathrm{mm}$ slide.  The slide is projected on to a screen, and the area of the house on the screen is $1.55 {\mathrm{m}}^2$.  What is the linear magnification of the projector-screen arrangement?

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$\left(m_{\mathit{e}}\right)$, mass of proton$\left(m_{\mathit{p}}\right)$) 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 ($~15 \mathrm{billion} \mathrm{years}$). From the table of fundamental constants in this book, try to see if you too can construct this number (or any other interesting number you can think of). If its coincidence with the age of the universe were significant, what would this imply for the constancy of fundamental constants?

A man walking briskly in rain with speed $v$ must slant his umbrella forward making an angle $\theta$ with the vertical. A student derives the following relation between $\theta$ and $v: \tan \theta =v$ and checks that the relation has a correct limit: as $v\mathit{\to }0$, $\theta \mathit{\to }0$, as expected. (We are assuming there is no strong wind and that the rain falls vertically for a stationary man). Do you think this relation can be correct? If not, guess the correct relation.

A famous relation in physics relates ‘moving mass’ $m$ to the ‘rest mass’ $m_0$ of a particle in terms of its speed $v$ and the speed of light, $c$. (This relation first arose as a consequence of special relativity due to Albert Einstein). A boy recalls the relation almost correctly but forgets where to put the constant $c$.

He writes : $m=\frac{m_0}{\mathit{(}1-v^2{\mathit{)}}^{1/2}}$

Guess where to put the missing $c$.

A book with many printing errors contains four different formulas for the displacement $y$ of a particle undergoing a certain periodic motion:

($a=\mathrm{maximum} \mathrm{displacement} \mathrm{of} \mathrm{the} \mathrm{particle}$, $v=\mathrm{speed} \mathrm{of} \mathrm{the} \mathrm{particle}$,  $T=\mathrm{time}-\mathrm{period} \mathrm{of} \mathrm{motion}$). Rule out the wrong formulas on dimensional grounds. 

A new unit of length is chosen such that the speed of light in vacuum is unity. What is the distance between the Sun and the Earth in terms of the new unit if light takes $8 \min$ and $20 \mathrm{s}$ to cover this distance?

A calorie is a unit of heat energy, and it equals about $4.2 \mathrm{J}$, where $1 \mathrm{J}=1 {\mathrm{kgm}}^2{\mathrm{s}}^{-2}$. Suppose we employ a system of units in which the unit of mass equals $\mathrm{\alpha } \mathrm{kg}$, the unit of length equals $\mathrm{\beta } \mathrm{m}$, the unit of time is $\mathrm{\gamma } \mathrm{s}$. Show that a calorie has a magnitude $4.2 {\mathrm{\alpha }}^{-1}{\mathrm{\beta }}^{-2}{\mathrm{\gamma }}^2$ in terms of the new units.