A particle moves in a central force field $$\overrightarrow {\bf{F}} = k{r^n}{\bf{\hat r}},$$   where k is constant, r is distance of the particle from the origin and $${{\bf{\hat r}}}$$ is the unit vector in the direction of $$\overrightarrow {\bf{r}} $$. Closed stable orbits are possible for

A mass m is constrained to move on a horizontal frictionless surface. It is set in circular motion with radius r₀ and angular speed ω₀ by an applied force $$\overrightarrow {\bf{F}} $$ communicated through an inextensible thread that passesthrough a hole on the surface as shown in figure given below. Then, this force is suddenly doubled.

The magnitude of the radial velocity of the mass

The speed of a particle whose kinetic energy is equal to its rest mass energy, is given by (c is the speed of light in vacuum)

The Hamiltonian corresponding to the Lagrangian $$L = a{{\dot x}^2} + b{{\dot y}^2} - kxy$$     is

A particle is moving in an inverse square field. If the total energy of the particle is positive, then trajectory of particle is