question_answer 55)

                  Show that the motion of a particle represented by \[y=\sin \,\omega t-\cos \,\omega t\]is simple harmonic with a period of \[2\pi \,l\,\omega \].

Answer:

                  \[y=\,\sin \,\omega t-\,\cos \,\omega t\]                 \[=\,\sin \omega \,t+\,\sin \,\left( \omega t-\,\frac{\pi }{2} \right)\]                 \[=2\,sin\,\left[ \frac{\omega t+\,\omega t-\,\frac{\pi }{2}}{2} \right]\,\,\cos \,\left[ \frac{\omega t\,-\omega t\,+\frac{\pi }{2}}{2} \right]\]                 \[=2\,\sin \,\left[ \omega t-\,\frac{\pi }{4} \right]\,\cos \left( \frac{\pi }{4} \right)\]                 \[=\,\frac{2}{\sqrt{2}}\,\sin \,\left( \omega t-\frac{\pi }{4} \right)\,=\,\sqrt{2}\,\sin \,\left( \omega t-\frac{\pi }{4} \right)\]                 \[\therefore \] Time period, \[T=\,\frac{2\pi }{\omega }\].