question_answer 78)

                  The displacement vector of a particle of mass m is given by \[r(t)=\hat{i}\,\,A\,\,\cos \,\omega t+\hat{j}B\,\sin \omega t\].                 (a) Show that the trajectory is an ellipse.                 (b)Show that \[F=-\,m{{\omega }^{2}}r\].

Answer:

                  (a) \[\vec{r}\,=A\,\cos \,wt\] \[\,\hat{i}+B\,\sin \,wt\,\,\hat{j}\]            ? (i)                 Compare it with \[\vec{r}\,=x\hat{i}\,+\,y\hat{j}\]                 \[\therefore \]\[x=A\,\cos wt\] or \[\frac{x}{A}=\,\cos wt\]      ?. (ii)                     and \[y=B\,\sin wt\]  or  \[\frac{y}{B}=\,\sin \,\,wt\,\]?. (iii)                 Square and adding eqns. (ii) and (iii), we get                 \[\frac{{{x}^{2}}}{{{A}^{2}}}+\,\frac{{{y}^{2}}}{{{B}^{2}}}=1\]                             ?(iv)                 Eqn. (iv) is the equation of an ellipse having major axis =A                 and semi-minor axis = B.                 (b) \[\vec{r}\,=A\,\cos \,\omega t\,\hat{i}\,+\,B\,\sin \,\omega t\,\hat{j}\]                 \[\therefore \]\[\vec{\upsilon }\,=\,\frac{d\vec{r}}{dt}\,=\,-A\omega \,\sin \,\omega \,\hat{i}\,+\,B\omega \,\cos \,\omega +\,\hat{j}\]                 and \[\vec{a}\,=\,\frac{d\vec{\upsilon }}{dt}\]                 \[=-A{{\omega }^{2}}\,\cos \omega t\,\hat{i}\,-B{{\omega }^{2}}\,\sin \,\omega t\,\hat{j}\]                 \[=-{{\omega }^{2}}\,(A\,\,\cos \,\omega \,t\,\hat{i}\,+\,B\,\,\sin \,\omega t\,\hat{j})\]                 \[=-\,{{\omega }^{2}}\vec{r}.\] Hence, \[F=\,ma\,=\,-m{{\omega }^{2}}\,\vec{r}\]