In expression P = El^2m^{-5}G^{-2} , E , l , m and G denote energy, mass, angular momentum and

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1.Units, Dimensions and Measurement

medium

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

Option A

Option B

Option C

Option D

Solution

Given, expression is, $\quad P=E L^{2} m^{-5} G^{-2}$

where $\mathrm{E}$ is energy

$[\mathrm{E}]=\left[\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\right]$

$\mathrm{m}$ is mass

$[\mathrm{m}]=\left[\mathrm{M}^{1}\right]$

$\mathrm{L}$ is angular momentum $[\mathrm{L}]=\left[\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-1}\right]$

$\mathrm{G}$ is gravitational constant $[\mathrm{G}]=\left[\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}\right]$

Substituting dimensions of each term in the given expression,

$[\mathrm{P}] =\left[\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\right] \times\left[\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-1}\right]^{2} \times\left[\mathrm{M}^{1}\right]^{-5} \times\left[\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}\right]^{-2}$

$=\left[\mathrm{M}^{1+2-5+2} \mathrm{~L}^{2+4-6} \mathrm{~T}^{-2-2+4}\right]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]$

Hence, $P$ is a dimensionless quantity.

Standard 11

Physics

A length-scale $(l)$ depends on the permittivity $(\varepsilon)$ of a dielectric material. Boltzmann constant $\left(k_B\right)$, the absolute temperature $(T)$, the number per unit volune $(n)$ of certain charged particles, and the charge $(q)$ carried by each of the particless. Which of the following expression($s$) for $l$ is(are) dimensionally correct?

($A$) $l=\sqrt{\left(\frac{n q^2}{\varepsilon k_B T}\right)}$

($B$) $l=\sqrt{\left(\frac{\varepsilon k_B T}{n q^2}\right)}$

($C$)$l=\sqrt{\left(\frac{q^2}{\varepsilon n^{2 / 3} k_B T}\right)}$

($D$) $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{1 / 3} k_B T}\right)}$

normal

(IIT-2016)

The $SI$ unit of energy is $J=k g\, m^{2} \,s^{-2} ;$ that of speed $v$ is $m s^{-1}$ and of acceleration $a$ is $m s ^{-2} .$ Which of the formulae for kinetic energy $(K)$ given below can you rule out on the basis of dimensional arguments ( $m$ stands for the mass of the body ):

$(a)$ $K=m^{2} v^{3}$

$(b)$ $K=(1 / 2) m v^{2}$

$(c)$ $K=m a$

$(d)$ $K=(3 / 16) m v^{2}$

$(e)$ $K=(1 / 2) m v^{2}+m a$

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

Solution

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