Question
A book with printing error contains four different formulae for displacement. Choose the correct formula/formulae (a) $y=a \sin \frac{2 \pi}{ T } t$
(b) $y=a \sin v t$
(c) $y=\frac{a}{ T } \sin \left(\frac{t}{a}\right)$
(d) $y=\frac{a}{T}\left(\sin \frac{2 \pi}{ T } t+\cos \frac{2 \pi}{ T } t\right)$

Answer

The arguments of sine and cosine function must be dimensionless so (a) is the probable correct formulae. Since
(a) $y=a \sin \left(\frac{2 \pi}{ T } t\right), \therefore\left[\frac{2 \pi t}{ T }\right]=\left[ T ^0\right]$ is dimensionless.
(b) $y=a \sin v t, \quad \because[v t]=[ L ]$ is dimensional so this equation is incorrect.
(c) $y=\frac{a}{t} \sin \left(\frac{t}{a}\right), \because\left[\frac{t}{a}\right]$ is dimensional so this is incorrect.
(d) $y=\frac{a}{t}\left(\sin \frac{2 \pi}{ T } t+\cos \frac{2 \pi t}{ T }\right):$ Though $\frac{2 \pi t}{ T }$ dimensionless $\frac{a}{ T }$ does not have dimensions of displacement so this is also incorrect.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free