BM Sharma Solutions for Chapter: Unit Dimension and Measurement, Exercise 4: DPP

A particle of mass $m$ is located in a region where its potential energy $\left[U\left(x\right)\right]$ depends on the position $x$ as, $U\left(x\right)=\frac{a}{x^{\mathit{2}}}-\frac{b}{x}$. Here, $a$ and $b$ are positive constants. $\left(\mathrm{i}\right)$ Write the dimensional formulae for $a$ and $b$.

$\left(\mathrm{ii}\right)$ If the time period of oscillation, which is calculated from the above formula, is stated by a student as $T=4\mathrm{\pi }a\sqrt{\frac{ma}{b^2}}$, check whether his answer is dimensionally correct.

A physical quantity $x$ depends on quantities $y$ and $z$ as follows, $x=Ay+B\tan \left(Cz\right)$ where $A$, $B$ and $C$ are constants. Which of the following do not have the same dimensions?

If the velocity of light $c$, gravitational constant $G$ and Planck’s constant $h$ are chosen as fundamental units, then the dimensions of mass, in the new system, is

If force $\left(F\right)$, length $\left(L\right)$ and time $\left(T\right)$ are assumed to be fundamental units, then the dimensional formula of the mass will be

Position of a body, with acceleration $a$, is given by $x=K{a}^m{t}^n$ where, $t$ is time and $K$ is a dimensionless constant. Find the values of $m$ and $n$.

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

The density of a liquid in the CGS system is $0.625 \mathrm{g} {\mathrm{cm}}^{-3}$. What is its magnitude in the SI system?

If the speed $v$ of a particle, of mass $m$, as a function of time $t$, is given by $v=\omega A\sin \left[\left(\sqrt{\frac{k}{m}}\right)t\right]$, where $A$ has dimension of length. Choose the correct option.