A particle of mass M moves under the action of an isotropic harmonic oscillator froce with potential energy U=kr^2 / 2 and angular momentum L.

1. find the possible orbits, the frequency of the motion for circular orbits and the frequency of small oscillations in the radial direction caused by perturbations that make the movement depart from circular motion.

2. The particle initially is moving in a circle of radius a. Find teh orbital speed v. It is then given a blow of impulse Mv in a direction making an angle alpha with its original velocity. Use teh conservation laws to determine the minimum and maximum distances from the origin during the subsequent motion. Explain your results physically for two limiting cases: alpha=0 and alpha=pi

1. find the possible orbits, the frequency of the motion for circular orbits and the frequency of small oscillations in the radial direction caused by perturbations that make the movement depart from circular motion.

2. The particle initially is moving in a circle of radius a. Find teh orbital speed v. It is then given a blow of impulse Mv in a direction making an angle alpha with its original velocity. Use teh conservation laws to determine the minimum and maximum distances from the origin during the subsequent motion. Explain your results physically for two limiting cases: alpha=0 and alpha=pi