question_answer 31)

                  The displacement of a particle varies with time according to the relation                 \[y=a\sin \,\omega t+b\cos \,\omega t\] (a) The motion is oscillatory but not S.H.M. (b) The motion is S.H.M. with amplitude\[a+b\]. (c) The motion is S.H.M. with amplitude \[{{a}^{2}}+{{b}^{2}}\]. (d) The motion is S.H.M. with amplitude \[\sqrt{{{a}^{2}}+\,{{b}^{2}}}\].

Answer:

                  (d) \[y=a\sin \omega t+b\cos \omega t\]                 This wage is due to superposition of two waves which are represented by \[{{y}_{1}}=a\,\sin \,\omega t\] and \[{{y}_{2}}=b\cos \,\omega t=b\sin \,(\omega t+\frac{\pi }{2})\]                 \[\therefore \] Amplitude of resultant wave (represented by \[y\])                 \[A\,=\sqrt{{{a}^{2}}+\,{{b}^{2}}+\,2ab\,\,\cos \frac{\pi }{2}}=\,\sqrt{{{a}^{2}}+\,{{b}^{2}}}\]