Question

A particle of mass m is attracted towards the origin O by a central force of magnitude \frac{m\left(\alpha r+\beta^{2}\right)}{r^{3}} where (r, theta)polar coordinates and a,Bpositive constants. ) Show that the particle's path satisfies \frac{d^{2} u}{d \theta^{2}}+u=\frac{\alpha}{h^{2}}+\frac{\beta^{2}}{h^{2}} u where u = 1/r and h is a constant which you should define. [You may use expressions for r and ŕ in polar coordinates and the generic form of the ODE for u from the lecture notes without proof] ) If the particle starts at a distance B2/3a from the origin, moving with velocity \dot{\mathbf{r}}=-\frac{2 \alpha}{\beta} \mathbf{e}_{r}+\frac{2 \alpha \sqrt{3}}{\beta} \mathbf{e}_{\theta} \text { find expressions for } u \text { and } \frac{d u}{d \theta} \text { at } \theta=0 \text {, and the value of } h \text {. } =) Hence find the path of the particle. ) Show that the particle returns to its initial position after revolution, and then one flies off to infinity.  Fig: 1  Fig: 2  Fig: 3  Fig: 4  Fig: 5  Fig: 6  Fig: 7  Fig: 8  Fig: 9  Fig: 10  Fig: 11