Question
A book with many printing errors contains four different formulae for the displacement $y$ of a particle undergoing a certain periodic function:
$i) \ y = a \sin \frac{2 \pi t}{T}$
$ii) \ y=a \sin v t$
$iii) \ y=\frac{a}{T} \sin \frac{t}{a}$
$iv) \ y =\frac{a}{\sqrt{2}}\left[\sin \frac{2 \pi t}{T}+\cos \frac{2 \pi t}{T}\right]$
Here, $a$ is maximum displacement of particle, $y$ is speed of particle, $T$ is time period of motion. Rule out the wrong formulae on dimensional grounds.
$i) \ y = a \sin \frac{2 \pi t}{T}$
$ii) \ y=a \sin v t$
$iii) \ y=\frac{a}{T} \sin \frac{t}{a}$
$iv) \ y =\frac{a}{\sqrt{2}}\left[\sin \frac{2 \pi t}{T}+\cos \frac{2 \pi t}{T}\right]$
Here, $a$ is maximum displacement of particle, $y$ is speed of particle, $T$ is time period of motion. Rule out the wrong formulae on dimensional grounds.