NCERT Solutions for Chapter: Units and Measurement, Exercise 5: LA

A new system of units is proposed in which unit of mass is $\alpha \mathrm{kg}$ unit of length $\beta \mathrm{m}$ and unit of time $\gamma \mathrm{s}$. How much will $5 \mathrm{J}$ measure in this new system?

The volume of a liquid flowing out per second from a pipe of length $l$ and radius $r$ is written by a student as $V=\frac{\pi }{8}\frac{P r^4}{\eta l}$ where $P$ is the pressure difference between two ends of pipe and $\eta$ is coefficient of viscosity of the liquid having dimensional formula ${\mathrm{ML}}^{-1}{\mathrm{T}}^{-1}$. 

Check whether the equation is dimensionally correct or not.

A physical quantity X is related to four measurable quantities $a, b, c\text{ and }d$ as follows: $X=a^2 b^3 c^{5/2} d^{-2}$.
The percentage error in the measurement of $a, b, c\text{ and }d$ are $1\%, 2\%, 3\%$ and $4\%$ respectively. What is the percentage error in quantity $X$? If the value of $X$ calculated on the basis of the above relation is $2.763$, to what value should you round off the result?

 In the expression $P=E l^2 m^{-5} G^{-2}$ $E, m, l\text{ and }G$ denote energy, mass, angular momentum and gravitational constant, respectively. Show that $P$ is a dimensionless quantity.

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.

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.
 

Consider a sun like a star at a distance of $2 \mathrm{parsec}$. When it is seen through a telescope with magnification, what should be angular size of the star? Sun appears to be $\left(1/2\right)°$ from the earth due to atmospheric fluctuations, eye cannot resolve objects smaller than $1$ arc minute.
 

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.)