question_answer 44)

                  On the basis of dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is not correct:                 (a) \[y=a\sin \,2\,\,\pi t/T\]                 (b) \[y=a\sin \,\upsilon t\]                 (c) \[y=a\sqrt{2}\left( \sin \frac{2\pi t}{T}-\cos \frac{2\pi t}{T} \right)\]                 (d) \[y=a\sqrt{2}\left( \sin \frac{2\pi t}{T}-\cos \frac{2\pi t}{T} \right)\]                

Answer:

                  (b, c) In \[\sin \,\upsilon t,\,\upsilon t\] should be dimensionless                 But \[\upsilon t=\frac{L}{T}\times T=L\](Dimension of length)                 In case of (c) \[\left( \frac{t}{a} \right)\] in \[\sin \left( \frac{t}{a} \right)\] should be dimensions \[[{{L}^{-1}}T]\]. More over \[\frac{a}{T}\ne \,y\]