Find the ratio of time period of revolution of the planets $1$ and $2$.

A particle of mass $m$ moves along a straight line with constant velocity ${\overrightarrow{v}}_0$ in the $x$ direction at a distance $b$ from the $x$ axis (figure).

(a) Does the particle possess any angular momentum about the origin?
(b) Explain why the amount of its angular momentum should change or should stay constant.
(c) Show that Kepler's second law is satisfied by showing that the two shaded triangles in the figure have the same area when $t_{\left(D\right)}-t_{\left(\mathrm{C}\right)}=t_{\left(\mathrm{B}\right)}-t_{\left(\mathrm{A}\right)}$.

As seen in the figure shown below, two spheres of mass $m$ and a third sphere of mass $M$ form an equilateral triangle and the fourth sphere of mass $m_4$ is at the centre of the triangle. The net gravitational force on that central sphere from the three other spheres is zero.
(a) What is $M$ in terms of $m$?
(b) If we double the value of $m_4,$ then what is the magnitude of the net gravitational force on the central sphere?

Two planets $X$and $Y$ travel counterclockwise in circular orbits about a star as shown in figure. The radii of their orbits are in the ratio $3: 1$. At one moment, they are aligned as shown in Fig. (a), making a straight line with the star. During the next five years, the angular displacement of planet $X$ is $90.0°$as shown in Fig. (b). What is the angular displacement of planet $Y$ at this moment?

A thin rod with mass $M$ is bent in a scmicirclc of radius $R$ (figure).

(a) What is its gravitational force (both magnitude and direction on a particle with mass $m$ at $P$, the center of curvature?
(b) What would be the force on the particle if the rod were a complete circle?

A binary system consists of two stars that revolve in a circular orbit about a center of mass midway between them. This statement implies that the masses of the two stars are equal (figure). Assume the orbital speed of each star is $\left|\overrightarrow{v}\right|=200 \mathrm{km}/\mathrm{s}$ and the orbital period of each is $20$ days. Find the mass $M$ of each star.