B M Sharma Solutions for Chapter: Dimensions and Measurement, Exercise 24: CONCEPT APPLICATION EXERCISE 1.2

You may not know integration, but using dimensional analysis you can check on some results. In the integral,

$\int \frac{dx}{{\left(2ax-x^2\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}=a^n{\sin }^{-1}\left(\frac{x}{a}-1\right)$, find the value of $n$.

Convert $1 \mathrm{MW}$ power on a new system having basic units of mass, length, and time as $10 \mathrm{kg}$, $1 \mathrm{dm}$, and $1\min$, respectively.

 Suppose we employ a system in which the unit of mass equals $100 \mathrm{kg}$, the unit of length equals $1 \mathrm{km}$ and the unit of time $100 \mathrm{s}$ and call the unit of energy eluoj ($\mathrm{joule}$ written in reverse order), then what is the relation between eluoj and joule?

If $1 \mathrm{g} \mathrm{cm} {\mathrm{s}}^{-1}=x \mathrm{N} \mathrm{s}$, then what is the value of $x$?

With the usual notations, check if the following equation $S_t=u+\frac{1}{2}a\left(2t-1\right)$ is dimensionally correct or not.

If the time period $\left(T\right)$ of vibration of a liquid drop depends on surface tension $\left(S\right)$, radius $\left(r\right)$ of the drop, and density $\left(\rho \right)$ of the liquid, then find the expression of $T$. 

 If $P$ represents radiation pressure, $C$ represents the speed of light, and $Q$ represents radiation energy striking a unit area per second, then non-zero integers $x, y$ and $z$ such that $P^{\mathit{x}}\mathit{ }{Q}^{\mathit{y}}\mathit{ }{C}^{\mathit{z}}$ is dimensionless, find the values of $x\mathit{,}\mathit{ }y\mathit{,}$ and $\mathrm{z}$.

If velocity $\left(V\right),$ force $\left(F\right),$ and energy $\left(E\right)$ are taken as fundamental units, then find the dimensional formula for mass.